Week4

Chapter 4 Newton’s Third Law 4.1

Lecture

In this lesson, we will analyze our first multi-body system, where the motion of two bodies are not only influenced by gravity and external constraints (such as the ramp), but the motion of each body is also influenced by the motion of the other body. This concept of multiple parts moving relative to one another in an inter-connected way is the basis of the entire branch of design and product development in mechanical engineering. In your Engineering Design Tools course, you will learn about how to construct complicated systems, how to model them with computer aided design packages, and how to manufacture them in the machine shop.

In this Engineering Mechanics Lab course, you will learn how to analyze these systems using the core principles of engineering science, and model them using the tools of measurement, modeling, and simulation. These two courses provide a strong foundation for the rest of your academic and professional career.

4.1.1

Formulate

State the Problem A system consists of two masses, m1 and m2 , and an ideal pulley, P . The masses m1 , m2 , and pulley P interact by means of an ideal cable. The first mass m1 is positioned on a rough-surface ramp inclined at an angle θ to the horizontal. The ideal cable connects from mass m1 over the ideal (frictionless) pulley P to m2 . The second mass m2 is suspended from the ideal cable. The ideal cable is of a constant length. The system is released from rest and allowed to move due to the action of gravity, g. Experimentally estimate the mean (time-average) tangential acceleration, a1s , of the first mass, m1 , at a constant ramp angle θ when the suspended mass m2 is allowed to vary over a range of at least three discrete values.

Each lab group member shall adjust the apparatus to a unique angle θ and conduct a series of trials for various m2 . Plot and report the mean tangential range acceleration, ar1 , (with uncertainties) as a function of m2 for various ramp angles θ. Estimate, plot, and report the

189

dynamic coefficent of friction, ¾d , (with uncertainties) as a function of m2 for various ramp angles θ.

Please complete the following task in your logbook.

Compare and contrast the tangential motion (position s1 (t), velocity V1s (t), and mean acceleration a1s ) of mass m1 when moving in tandem with m2 versus the motion of a single mass travelling down an inclined plane as previously studied. Compare and contrast the tangential motion versus the vertical motion of a single object in free fall. Explain any differences observed in the context of Newton’s Laws and the experimental environment employed.

Known Information Prior to the experiment, we shall acquire several items of known information using standard measurement and observation techniques. m1 m2 θ g L1 L2

= = = = 9.81 = =

[kg] [kg] [deg] [m/s2 ] [m] [m]

Mass of Object 1 (Constant) Mass of Object 2 (Constant) Inclination Angle of Ramp (Constant) Standard value of gravitational acceleration Distance from transducer to pulley (Constant) Distance from floor to pulley (Constant)

(4.1) (4.2) (4.3) (4.4) (4.5) (4.6)

Desired Information Upon conclusion of the experiment and analysis report: P LOT : ar1 Âą a vs. m2 P LOT : Âľd Âą Âľd vs. m2

4.1.2

for θ1 , θ2 , θ3 , θ4 for θ1 , θ2 , θ3 , θ4

(4.7) (4.8)

Assume Âľd = f = Âľd N fair = 0 T = Lcable = Fr/m1 = r1 = 0

[−] [N ] [N ] [N ] [m] [N ] [m]

⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

Constant, Uniform, Unknown Amontons’ First Law Neglect Air Friction Constant Constant Normal force between m1 and ramp At transducer face 190

(4.9) (4.10) (4.11) (4.12) (4.13) (4.14) (4.15)

s1 = 0 r2 = 0 s2 = 0

[m] [m] [m]

⊕ ⊕ ⊕

At pulley tangent At pulley tangent At room floor

r1 (t) ≈ {ˆ r1 (n)} = {ˆ r1 (1), rˆ1 (2), · · · , rˆ1 (Nt − 1), rˆ1 (Nt )}

t ≈ tˆ(n) = tˆ(1), tˆ(2), · · · , tˆ(Nt − 1), tˆ(Nt )

(4.16) (4.17) (4.18)

[m]

⊕

(4.19)

[s]

⊕

(4.20)

Equation 4.9 states that the dynamic coefficient of friction, µd , is a constant for a particular combination of block and ramp materials. The validity of this assumption will need to be assessed at the end of the problem. We expect the friction coefficient to be independent of inclination angle θ and external load m2 . Equation 4.10 re-states Amontons’ First Law from our previous analysis. The validity of this assumption will need to be assessed at the end of the problem. Equation 4.11 states that we are neglecting air friction throughout the problem. This is a particularly important assumption related to the motion of m2 . Our current experiment does not include an investigation to study the validity of this assumption.

Equation 4.12 is a mathematical consequence of the problem statement instructing us to consider the pulley as ideal and frictionless. Our current experiment does not include an investigation to study the validity of this assumption. Equation 4.13 is a mathematical consequence of the problem statement instructing us to consider the cable as ideal and fixed length. We shall study the deflection of thin elements in a subsequent experiment later in the course. Equations 4.14 through 4.18 simply define our chosen terminology and datums for the problem. We need to ensure that our drawings and analysis are consistent with these assumptions.

Equation 4.19 is used to reflect our assumption that we will conduct an experiment to measure the range (the displacement from the transducer face) of mass m1 as a function of time using a transducer, and calibration curve to convert voltage into displacement. Thus, rather than having an analytical expression for r1 (t) we will have a column of data points that we will store in a spreadsheet. We state that we will have Nt time samples of data from the transducer. The symbol rˆ(n) indicates an experimental estimate (at some discrete time sample number n) of the theoretical range, r(t). Equation 4.20 indicates that our experimental data file will also contain a column of Nt time samples, which we denote as tˆ(n) to indicate the relative time at which each corresponding voltage (and hence distance) sample was acquired. While the physical time t is a continuous function, we only have samples available at our distinct sampling times, which we call tˆ. We use the index, 1 ≤ n ≤ Nt , to indicate which time sample we are interested in.

4.1.3

Chart

Schematic Diagram A schematic diagram of the stated problem is shown in Figure 4.1. The datums, nomenclature, and system boundary are drawn in a manner consistent with the problem formulation.

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We choose the origin of the cartesian coordinate system to be where the cable tangentially wraps around the pulley from the left approach.

Figure 4.1: Schematic Diagram of two bodies in relative motion.

Free Body Diagrams The free body diagrams for the objects within the system boundary (m1 , m2 , and P ) are shown in Figure 4.2. The mass m1 experiences a force due to gravity in the magnitude m1 g, directed towards the center of the Earth. This force due to gravity is resisted by a force acting normal to the surface, shown as the reaction Fr/m1 , which prevents the object from falling through the ramp surface. The tangential force fm1 in the FBD represents the friction between m1 and the ramp. The cable is taken to exert a tension force T on the object m1 .

Since the cable is in tension, the force is directed away from the body, and towards the pulley. At this point it is convenient to recall the old adage “you can’t push on a rope.” The unit vectors are noted in the FBD to remind us of the “sense” or “positive directions” for the chosen coordinate system. The FBD of the second mass, m2 , shows only two forces. The downward force due to gravity, m2 g and an upward force due to the cable tension, T . Next, we look at the pulley assembly. The pulley serves to change the direction of the tension 192

Figure 4.2: Free Body Diagrams for a pulley assembly and two bodies in relative motion. force but, due to our assumption that it is an ideal frictionless pulley, does not change the value of the tension T as it wraps around the arc of the pulley. Since we know the pulley rotates, in reality, there must be some difference in the tension, or no motion would initally take place. A high quality pulley introduces relatively little friction into a system already in motion. This is particularly true when the tension in the cable is modest, as in this case.

Pulleys under heavy loading, such as in the case of a heavy lift elevator or block and tackle may introduce significant friction into the system. The fact that the tension on the two sides of the pulley is in different directions induces a reaction denoted FP at the pulley mount point.

We choose not to include FBDs for the ramp, supports, table, floor, and the Earth since they lie outside the system boundary. A quick review of Newton’s third law confirms that the equal and opposite reactions to the unbalanced forces in Figure 4.2 lie outside of the system boundary.

Vector Diagram A force vector diagram for the mass m1 is shown in Figure 4.3. The FBDs are an excellent place to begin the force vector diagrams. If we compare the FBD for m1 from Figure 4.2 to the vector diagram in Figure 4.3, we shall see a one-to-one correspondance. In the vector diagram, we choose to break the weight of the mass, m1 g into two components – one component tangential to the ramp, and a second component normal to the surface of the ramp. We note that the tension, T is applied at an angle θ to the horizontal, and is parallel to the ramp surface. This vector diagram for m1 suggests that we will analyze the motion of the mass as having tangential components (along the ramp) and normal components (perpendicular to the ramp). Note that it is equally accurate to consider the motion components as aligned with the horizontal x and vertical z axes if desired.

Please complete the following task in your logbook.

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Figure 4.3: Force Vector Diagram for mass m1 .

Create displacement vector diagrams for mass m2 and the pulley P .

4.1.4

Execute

Recall The Governing Equations.

The first step in the execution phase is to recall the relevant governing equations. In this case, we recall Newton’s laws.

If:

X→ − − F = 0 Then: → a =0 → − − X→ − d(m V ) d→ p F = = dt dt → − → − F Action = − F Reaction → − Fg =g·m↓

Newton’s 1st Law

(4.21)

Newton’s 2nd Law

(4.22)

Newton’s 3rd Law

(4.23)

Newton’s Law of Gravity near Earth

(4.24)

Simplify the Governing Equations.

Newton’s Second Law may be applied to mass m2 , using the horizontal and vertical components of vectors, as follows: m2 = Constant by Eq.4.2

X→ − F on

m2

194

z }| { → − d(m2 V 2 ) = dt → − d( V 2 ) = m2 dt

(4.25) (4.26)

X

horizontal

− a2 = m2 →

= m2 a2xˆı + a2s kˆ

X (F2x ) ˆı + (F2s ) kˆ = m2 a2xˆı + a2s kˆ

(4.27) (4.28) (4.29)

vertical

From the FBD for mass m2 in Figure 4.2, we can write X F2xˆı = 0ˆı

(4.30)

horiz

X

F2s kˆ = (T − m2 g) kˆ

(4.31)

vert

Use Equations 4.30 and 4.31 in 4.29 to write the horizontal and vertical scalar component equations for mass m2 . m2 a2x = 0 [kg][m/s2 ] = [N ] m2 a2s = T − m2 g [kg][m/s2 ] = [N ] − [kg][m/s2 ]

(4.32) Units Validation (4.33) Units Validation

Newton’s Second Law may be applied to mass m1 as follows, using the normal and tangential components of the vectors: m1 = Constant by Eq.4.1

X→ − F on

m1

z }| { → − d(m1 V 1 ) = dt → − d( V 1 ) = m1 dt → − = m1 a 1 = m1

X

(F1n ) n ˆ+

normal

X

(F1s ) tˆ = m1

tangential

a1n n ˆ + a1s tˆ

a1n n ˆ + a1s tˆ

(4.34) (4.35) (4.36) (4.37) (4.38)

From the FBD for mass m1 in Figure 4.2 in conjunction with the force vector diagram of Figure 4.3, we can write X

F2n n ˆ = Fr/m1 − m1 g cos θ n ˆ (4.39) normal

X

F2s tˆ = (fm1 − m1 g sin θ − T ) tˆ

tangent

195

(4.40)

Use Equations 4.39 and 4.40 in 4.38 to write the normal and tangential scalar component equations for mass m1 . m1 a1n = Fr/m1 − m1 g cos θ [kg][m/s2 ] = [N ] − [kg][m/s2 ][−] m1 a1s = fm1 − m1 g sin θ − T [kg][m/s2 ] = [N ] + [kg][m/s2 ][−] − [N ]

(4.41) Units Validation (4.42) Units Validation

Amontons’ First Law, Equation 4.10, allows us to write the sliding friction fm1 in terms of the normal component of the weight as fm1 = µd m1 g cos θ

(4.43)

Using Equation 4.43 in 4.42 we get m1 a1s = µd m1 g cos θ − m1 g sin θ − T [kg][m/s2 ] = [−][kg][m/s2 ][−] − [kg][m/s2 ][−] − [N ]

(4.44) Units Validation

Please complete the following task in your logbook.

Apply Newton’s Second Law to the pulley, P . Use the horizontal and vertical component form of the vector equation to determine the magnitude and direction of the reaction force FP , which prevents the pulley from translating its position. Do not analyze the rotary motion of the pulley, but rather, determine the static reaction. Show all work.

Inventory We have simplified Newton’s Second Law as it applies to the masses m1 and m2 . Let’s conduct an inventory of the simplified equations to determine if sufficient information is available to obtain a solution for the motion of both masses. Recall Equations 4.32, 4.33, 4.41, and 4.44. m2 a2x = 0

(4.32)

=

m2 a2s = T − m2 g

Inventory (4.33)

= −

m1 a1n = Fr/m1 − m1 g cos θ

Inventory (4.41)

= −

Inventory

m1 a1s = µd m1 g cos θ − m1 g sin θ − T

=

−

−

(4.44) Inventory

196

Reviewing these equations shows that we have seven unknown values: a2x , a2s , a1n , a1s , T , Fr/m1 and µd . We have four independent equations. We do not yet have enough information to solve the problem. In retrospect, it appears that we overlooked at least one important assumption, which seems obvious at this point. We should have assumed that a1n = 0

[m/s2 ]

⊕

(4.45)

This assumption states that the mass m1 is not permitted to move in a direction normal to the surface of the ramp. That is, we assume that the ramp does not deform under the weight of the load. This is a valid assumption for our laboratory apparatus. Our inventory now stands at 7 unknowns and 5 independent equations. We need two more independent equations. We have already used most of our simplifying assumptions in our analysis. We implicitly invoked the assumption of neglecting air friction, Equation 4.13, by excluding the air friction forces from the free body diagrams for m1 and m2 .

Refer to Figure 4.1. From this figure, we can write that s1 (t) = L1 − r1 (t) s2 (t) = L2 − r2 (t)

(4.46) (4.47)

We can take the first derivative of both sides of Equation 4.46 to find the tangential velocity, V1s , of mass m1 in terms of the experimentally observed history for the range of the mass, r1 (t): dL1 d d (s1 (t)) = − (r1 (t)) dt dt dt d V1s (t) = 0 − (r1 (t)) dt

(4.48) (4.49)

since the time rate of change of a constant is zero. We take the second derivative to arrive at the tangential acceleration, a1s , of mass m1 : d d2 (V1s (t)) = − 2 (r1 (t)) dt dt d2 a1s (t) = − 2 (r1 (t)) dt [1] [m/s2 ] = 2 [m] [s ] =

(4.50) (4.51) Units Validation Inventory

Equation 4.51 provides us with one more independent equation (when we collect a set of experimental data for r1 (t)). Our inventory now stands at 7 unknowns and 6 independent equations. We still need one more independent equation.

We have not yet employed the knowledge that the cable is of fixed length, as instructed in the problem statement, and reflected in Equation 4.13. This is an essential fact for solving 197

this problem. This type of constraint, which restricts the manner in which two objects can move relative to one another is called a “kinematic constraint.” Equations that result from these motion constraints are called “kinematic equations.” Kinematic equations do not follow from Newton’s laws directly. Rather, they follow from the geometry that we, as engineers, build into our machines and devices. Thus, kinematics are a very important tool for engineers! Let’s study the implications of the assumption that Lcable = Constant. Once again referring to Figure 4.1 and Equation 4.47, we can similarly write for mass m2 : dL2 d d (4.52) (s2 (t)) = − (r2 (t)) dt dt dt d V2s (t) = 0 − (r2 (t)) (4.53) dt d d2 (4.54) (V2s (t)) = − 2 (r2 (t)) dt dt d2 a2s (t) = − 2 (r2 (t)) (4.55) dt [1] Units Validation [m/s2 ] = 2 [m] [s ] This seems to be an improvement, until we recognize that r2 (t) appears to be an unknown quantity. We still need one more equation. Kinematics to the rescue! Since the length of the cable is assumed to be constant, then we can state by virtue of the kinematic constraint Lcable = Constant that d d (4.56) (r2 (t)) = (r1 (t)) dt dt The distance between the transducer and mass m1 is not the same as the distance between the pulley and m2 . However, the speed at which m1 recedes from the transducer is identical to the speed at which m2 moves away from the pulley. Since the cable length is constant, we can also take the derivative of Equation 4.56 to get d2 d2 (r2 (t)) = 2 (r1 (t)) (4.57) dt2 dt Now, we substitute Equation 4.57 into Equation 4.55 to arrive at our final independent equation: d2 (r1 (t)) (4.58) dt2 We have 7 independent equations and 7 unknowns. We can now solve the system of equations. a2s (t) = −

Solve We begin by solving Equation 4.32 to get a2x = 0 [m/s2 ] = [m/s2 ]

(4.59) Units Validation 198

Now, use substitute Equation 4.45 into Equation 4.41 and solve for the normal reaction force: Fr/m1 = m1 g cos θ [N ] = [kg][m/s2 ][−]

(4.60) Units Validation

We know from past experience that estimating the instantaneous acceleration from experimental data introduces a high level of uncertainty. However, we have also observed that computing a time average mean value of the acceleration results in a modest standard error for the mean.

We can approximate the first derivative of the sensor-to-object distance using a central difference approximation to arrive at the speed of departure: d rˆ1 (n + 1) − rˆ1 (n − 1) Vˆr1 (n) = (ˆ r1 (n)) = dt tˆ(n + 1) − tˆ(n − 1)

∀ 2 ≤ n ≤ Nt − 1

(4.61)

The acceleration equals the first derivative of velocity and the second derivative of range as a ˆr1 (n) =

Vˆ (n + 1) − Vˆ (n − 1) d ˆ r1 r1 Vr1 (n) ≈ dt tˆ(n + 1) − tˆ(n − 1)

∀ 3 ≤ n ≤ Nt − 2

(4.62)

The mean acceleration is computed by ar1 =

n=N t −2 X 1 (ˆ ar1 (n)) Nt − 4 n=3

(4.63)

By virtue of Equation 4.58 we can estimate the time average acceleration of mass m1 and m2 as: a2s = −ar1 (4.64) We can substitute our mean value of the vertical acceleration, a2s , into Equation 4.33 to get the nominal tension, T , in the cable: T = m2 (g − ar1 ) 2

(4.65) 2

[N ] = [kg] [m/s ] + [m/s ]

Units Validation

Now, use the tension from Equation 4.65 in Equation 4.44 to estimate the nominal value of the dynamic coefficient of friction, µd in terms of a1s or alternatively, ar1 : T a1s + g cos θ m1 g cos θ T ar1 µd = tan θ − + g cos θ m1 g cos θ [N ] [m/s2 ] + [−] = [−] + 2 [m/s ][−] [kg][m/s2 ][−]

(4.66)

µd = tan θ +

199

(4.67) Units Validation

We substitute Equation 4.65 into Equation 4.67 to get m2 (g − ar1 ) ar1 + g cos θ m1 g cos θ [m/s2 ] [kg] [m/s2 ] − [m/s2 ] [−] = [−] + + [m/s2 ][−] [kg] [m/s2 ][−]

(4.68)

µd = tan θ −

Units Validation

Equation 4.68 is convenient due to the fact that all terms on the right hand side are either known or will be obtained as a result of the Lab experiment.

4.1.5

Test

Validate We have verified that the units on each result are correct. Upon completion of the experiment and analysis, we should determine if the signs on the results are consistent with the sign convention assumed in the analysis. If any sign discrepancies arise, then these should be reviewed as potential indicators of error. We expect a positive value for ar1 at values of significantly higher than the friction angle, θ >> θs . We expect negative values for as1 and as2 when θ >> θs .

Verify In lab, we will conduct an experiment to measure the motion of the object down an inclined plane. We will use many of the skills learned during the first two weeks to conduct this experiment. Following the lab, we should report our findings of all desired quantities as supported by experimental data. We should compare the observed acceleration of the object, and compare our measured results with those obtained for the free fall experiment.

In an effort to attain the most accurate value for µd , it is important that we conduct experimental trials at angles significantly away from the friction angle, θs . The best estimates for µd will be obtained when we obtain smooth motion with a clear acceleration.

Furthermore, we hypothesize that the value of µd should be independent of angle, θ, since Amontons’ First Law contains no mention of angle. As the angle θ is increased, the lab group will need to take care not to let the object tumble to the ground, and avoid breakage.

Apply Intuition Equation 4.58 states that the vertical acceleration of m2 is identical to the tangential acceleration of m1 . This makes sense for a cable in constant tension of constant length. Equation 4.59 states that mass m2 does not accelerate in the horizontal direction, which is intuitively correct. Equation 4.60 for the normal force is consistent with our previous experience in Chapter 3. When we consider the case of θ >> θs , we expect m1 to accelerate down the ramp such that ar1 > 0, and Equation 4.65 states that the tension in the cable is less than the dead weight of m2 . This also is intuitively correct. Finally, in the limiting case when 200

m2 → 0 the tension in the cable is T → 0, and Equation 4.66 reduces to the same result that we achieved in Chapter 3. This is reassuring – even tough we did numerous algebraic steps, our result reduces to a simpler form that we have seen previously.

4.1.6

Iterate

We will verify the theoretical results obtained in our laboratory experiment. When choosing angles θ for the experimental trials, it will be prudent to select a variety of angles. Each student is expected to conduct a series of trials with various values of m2 for at least one angle θ. However, if time permits, it may be valuable for the group to conduct multiple trials and multiple angles, such that collectively the team explores a wide range of angles and counter-weights. It is generally easier to conduct multiple trials while you are in the lab, rather than taking limited data, going to Studio, and then realize that you would have preferred to collect more data!

201

4.2 4.2.1

Lab - System of Bodies in Constrained Motion Scope

This week in lab we will add a counterweight, cable, and pulley to the system studied last week. At the conclusion of Lab and Studio, each student shall record their findings for all of the desired information requested in Section 4.1.1 in their logbook. The Scribe shall report the findings specified in Section 4.3.3 as a single lab report for the team.

4.2.2

Goal

The goals of this laboratory experiment are to 1. acquire all data necessary for reporting the results desired, as listed in Sections 4.1.1 and 4.3.3, 2. acquire voltage vs. time data with an automated data acquisition system, 3. convert the voltage, V (n), into range, rˆ1 (n), for 1 < n < Nt .

4. estimate the instantaneous tangential range, rˆ1 (n), and velocity, Vˆr1 (n), for 1 < n < Nt .

5. estimate the mean tangential acceleration ar1 during the motion interval, 6. confirm our understanding of Newton’s Third Law of Motion, 7. understand the importance of kinematics in engineering.

4.2.3

Units of Measurement to use

Measurements may be conducted in a combination of customary U.S. units and S.I. unit.

All results shall be reported in the SI system of units.

Table 4.1: Units of Measurement to be used for object in constrainted linear motion.

Quantity Time Voltage Length Velocity Acceleration Mass Force Angle Friction Coefficient

Customary units [s] [V ] [inch] or [f t] [inch/sec] or [f t/s] [inch/sec2 ] or [f t/s2 ] [slugs] [pounds] = [slug f eet/s2 ] [degrees] [−]

202

SI units [s] [V ] [m] [m/s] [m/s2 ] [kg] [N ] = [kg m/s2 ] [radians] or [degrees] [−]

4.2.4

Reference Documents

Review the reference material from Chapter 1, regarding transducers, sensor calibration, A/D Conversion, and experiment design prior to the lab session. Review the uncertainty analysis from Chapter 2, 3, and 4 prior to lab and studio.

4.2.5

Terminology

The following terms must be fully understood in order to achieve the educational objectives of this laboratory experiment.

Length Voltage Time Speed Velocity Time Horizontal Vertical Component Normal Tangential Vector Magnitude Direction Static Dynamic Constraint Scalar Resultant Commutative Law Unit vector Negation Unit Angle Origin Associative Law Sliding Rolling Kinematic Axis Incipient Friction Tension Inclination Amontons’ 1st Law Traction Constraint Inclinometer Hypothesis

4.2.6

Summary of Test Method

A video of the lab experiment is available for viewing on myCourses. This video will provide you with an understanding of what you need to do during the lab. Please review this video before reading on, since it will provide you a nice overview of the experiment and allow you to better understand the following information.

4.2.7

Calibration and Standardization

The Lead Technologist and Assistant Technologist shall calibrate the ultrasonic transducer to determine the relationship between position and voltage, using an approach similar to that of week 1, prior to conducting trials of the current experiment. The same calibration software VI used during the week 1 experiment shall be used to manually record calibration data for the ultrasonic transducer. The entire team may use one calibration curve for the ultrasonic transducer. The inclinometer will be used as a primary instrument in this class.

However, it is necessary to calibrate the instrument at a known angle to set the “zero angle” for the instrument. The Lead Technologist is responsible for calibrating the “zero angle” of the angle sensor, using an approach similar to that described in week 3.

4.2.8

Apparatus

The experimental apparatus to be used during this week’s lab is shown in Figure 4.4, showing the lower portion of the apparatus with the block, cable, pulley, and dead-weight.

203

Figure 4.4: Lower portion of apparatus configured for measuring a system of objects in constrained linear motion.

The upper portion of the apparatus, with a wooden block, cable, pulley, angle sensor, and transducer is illustrated in Figure 4.5. The object is designed to slide along the channels in the aluminum extrusion of the ramp.

We will need several items of equipment for our first experiment, illustrated in Figure 4.6. A pulley is shown in Figure 4.6. The pulley will be used to provide a kinematic motion constraint by connecting the two masses via a cable. The test specimen for this lab is a wooden block, as illustrated in Figure 4.6. The object is designed to slide along the channels in the aluminum extrusion of the ramp.

4.2.9

Measurement Uncertainty

In Lab 1, we learned about measurement uncertainty. In Lab 2, we learned about the propagation of uncertainty due to numerical differentiation (addition and subtraction), and the influence of data averaging on the standard uncertainty of an average. In Lab 3, we learned about the propagation of uncertainty through mathematical operations such as multiplication and trigonometric functions. Summarized below are select equations to uncertainties resulting from the Studio calculations. This week you will derive a few more to add to list. tan θs = Âą0.5 |tan(θ + θs ) â&#x2C6;&#x2019; tan(θ â&#x2C6;&#x2019; θs )| 204

(4.69)

Figure 4.5: Upper portion of apparatus for this lab.

Pulley

Test Specimen

Figure 4.6: Photographs of equipment to be used during week 4. sin θs = Âą0.5 |sin(θ + θs ) â&#x2C6;&#x2019; sin(θ â&#x2C6;&#x2019; θs )| cos θs = Âą0.5 |cos(θ + θs ) â&#x2C6;&#x2019; cos(θ â&#x2C6;&#x2019; θs )| Âľs = tan θs

as,avg cos θ as,avg ¾d = ¹ tan θ + + (g cos θ) as,avg cos θ

cos θ ¾d m 1 g f = ¹f + + + ¾d m1 g cos θ

Please complete the following task in your logbook prior to arrival in studio.

Derive equations for uncertainties in the calculated values of: 205

(4.70) (4.71) (4.72) (4.73) (4.74)

â&#x20AC;˘ Tension in the cable, T â&#x20AC;˘ Coefficient of Dynamic Friction, Âľd for the two-body system m1 and m2 â&#x20AC;˘ Sum of the Forces on Block m1 for the two-body system m1 and m2

4.2.10

Preparation of Apparatus

Each lab group will need one complete lab apparatus. The Experiment Manager should sign out a key to the lab station cabinet from the lab instructor, as well as additional items. Specifically, the team will need the familiar A/D converter, ultrasonic transducer, inclinometer, and basic apparatus. While the lab manager is responsible for the sign-out, every student in the lab group should record the make, model number and serial number of each item that is signed out.

The Lead Technologist and Assistant Technologist should assemble the apparatus with the apparatus adjustable at various angles of incline. The Lead Technologist should assemble the transducer on the apparatus in a manner that can be used by all members of the team, and secure the sliding wooden block and inclinometer. The Lead Technologist should demonstrate the use of the inclinometer, as mounted to the apparatus, to all members of the team.

4.2.11

Sampling, Test Specimens

Each lab group should use a lab station with a unique angle and deadweight for their series of trials. Each member of the lab group should conduct a unique series of voltage and range measurements as a function of time. Each member of the lab group should use a unique angle of inclination θ for their series of trial, and conduct multiple trials at various deadweights, m2 . The angle of inclination may be chosen at any interval between -2 and +88 degrees to the horizontal.

4.2.12

Procedure - Lab Portion Record all observations and notes about your lab experiment in your logbook.

Estimate mass The Lead Technologist and the Assistant Technologist shall use an electronic balance to estimate the mass of the wooden block assembly, m1 , and the various counter weights m2a , m2b , m2c , m2d , . . ., that will used in the experiment. As the Technologists conduct the mass measurement, each student shall record all observations in their logbook.

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Block Slide Experiment Each student should complete the detailed procedure outlined here.

1. Place a header at the top of each page in your logbook as you proceed through the lab period.

2. Place a footer at the bottom of each page in your logbook as you proceed through the lab period.

3. Just below the header, create a table documenting the name of each student in your lab group, and their responsibility for the lab this week, as illustrated in Table 4.2.

Table 4.2: Staffing Plan for Lab 3.

Position Title Student Name (Last, First) Lab Manager Technologist Assistant Technologist Scribe 4. Create a table documenting all of the equipment items used in the lab, similar to that shown in Table 4.3. Every lab group member should have an equipment table. For each item, please document the location where the item was obtained and should be returned for storage, along with the manufacturer name, model number, and serial number of the item.

Table 4.3: Lab Station Number Equipment Log for Lab 4.

Equipment Item Manufacturer Model S/N (If Avail.)

Ultrasonic Transducer Angle Gauge Pulley Cable Mass Balance Mass m1 Mass m2a Mass m2b Mass m2c Mass m2d .. .

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5. Lab Manager Only: Create a table documenting the equipment check-out and checkin, as illustrated in Table 4.4. The lab manager should completely fill in the equipment table and then get the signature of the lab instructor for equipment sign-out. At the end of the period, after all equipment items have been returned neatly to storage, the lab manager may request the sign-in signature from the lab instructor.

Table 4.4: Lab Station Number Equipment Sign-Out/In for “System of Bodies in Constrained Motion.”

Action Manager Signature Instructor Signature Date, Time Equipment Sign-out Equipment Sign-in

6. Draw a sketch of the experimental setup up in your logbook. Measure and record all the distances shown in the figure, with the exception of pB lock which will be varied and measured in subsequent steps. Be sure to record units and to to use the appropriate variable names that are consistent with those used in the figure. Indicate the measuring instrument(s) used to obtain the positions (i.e. the installed ruler or tape measure), and the instrument least count (ILC) for each instrument used. These ILC values will be used in studio to quantify the uncertainty in each measurement. Carefully inspect the measuring instruments for any additional sources of uncertainty. Write notes about additional sources of uncertainty in your logbook.

7. As the Lead and Assistant Technologist collect the calibration data for the transducer, record their observations in your logbook, using a table similar to that shown in Table 4.5. The entire lab group may rely upon a single set of calibration data.

8. Insert your USB drive into the USB port on your computer.

9. Create a folder called Lab Week4. Store all of your data for today’s session in this folder.

10. Assemble the test apparatus to its appropriate configuration for your unique trial.

11. Position the apparatus at an appropriate angle, θmotion < θtrial < 90[◦ ]. Select a deadweight to use for your trial. Record the deadweight, angle, and their uncertainty in your logbook.

12. Have a lab partner take a digital photograph of you, preparing to initiate your trial.

Save the digital photo to your USB drive and share a copy with your group for the lab report.

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Table 4.5: Ultrasonic Transducer Calibration, Data Recorded Manually. Start Time: Item Position Mean Sample Standard Measurement Voltage Size Deviation N om N om Symbol pBlock VD Nt V D units [in] [v] [-] [v] 1 2 3 4 5 6 7 .. .

13. Use the Enhanced VI data collection program, and ask the Lead or Assistant Technologist to operate the software during your trials. You shall specify the sampling rate and the sampling time duration for each trial. Justify your decisions with comments in your logbook.

14. In coordination with a lab partner, initiate the data recording system to record the transducerâ&#x20AC;&#x2122;s voltage as a function of time.

15. While the system is recording, allow the system of two bodies to move in tandem.

Observe the status of the cable during the motion. Does the cable remain in tension throughout the motion sequence? Note aberrations in your logbook.

16. Observe the VI plot of voltage vs. time from the acquired data. Evaluate whether your sampling rate for the recording is sufficient for your needs. Use the Enhanced VI to select that portion of the data which represents the interval of interest for your trial.

17. Save this new file you are creating in the Lab_Week_4 folder with the filename of the format Username_Week4AnglexxM2yyDataNN.txt, where xx, yy , and NN represent your inclination angle, dead-weight m2 , and trial number respectively. The software will save the data in â&#x20AC;&#x153;tab separated valueâ&#x20AC;? format, with a file extension of .txt. Repeat trials as necessary to obtain a good set of data for your angle of inclination. Record your observations about each trial in your logbook. Record your observations, missteps, and final results in your logbook. Save all of your trials of data to your USB drive.

18. Remove your USB drive and remember to take it with you. You will need these files in Studio for the next portion of the analysis.

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19. After Lab, upload your voltage vs. time data file and the photograph of you with the lab apparatus to your MyCourses drop box. Make sure that the digital photo shows you with the apparatus in the configuration for this weekâ&#x20AC;&#x2122;s lab. Upload your materials to the drop box during Lab period, or immediately following.

20. The Experiment Manager shall allow each lab group member to conduct multiple trials, with each series of trials at a unique angle of inclination. Each group member should have a unique data table recorded in their USB drive, and personal comments in their logbook. When all students are done, the Experiment Manager shall return all items to storage, confirm with the lab instructor that all items are accounted for, lock the storage cabinet, and return the key to the lab instructor.

21. Return all items to storage.

22. Wait until the experiment manager has signed-in all equipment and gotten a signature from the lab instructor. Do not leave the lab room until the manager has released you.

Sign and date each completed page in your logbook. Print neatly, and place a header and footer on each page of the logbook. Draw a single line through any errors, but make sure that they remain legible. Do not make eraser marks in your logbook. It is preferable to record all entries in pen, but pencil is permissible for the course. Be sure to take your log book to Studio class next day. Carefully read and understand Section 4.3 of the textbook, and complete the Studio pre-work prior to your arrival at Studio.

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4.3

Studio

This week in Studio, you will use the calibration techniques you learned in Week 1, along with the spreadsheet created in Week 3 to investigate Newton’s 2nd and 3rd Laws, by analysing a two block system in 2 dimensional constrained motion. The equations for calculating forces rely heavily on the FBD’s and theory discussed in Section 4.1 of the text. Before beginning the Studio procedures, please review Section 4.3.1 Calculation and Interpretation of Results, which will provide a summary of equations that you will need to complete the Studio. Also review Section 4.2.9 Measurement Uncertainty, which will describe the process for analyzing the experimental errors.

Record all observations and notes about your studio procedures in your logbook.

4.3.1

Calculation and Interpretation of Results

This week in Studio, we will calculate friction variables for two masses in constrained two dimensional motion. The equations provided below are summarized for your convenience from the derivations in Section 4.1 and previous chapters. Please note that in many cases, these equations apply only to the system of bodies being analysed in lab this week, and should be derived from first principals for all other cases. ps (t) = c0 + c1 V (t) [m] [m] = [m] + [v] [v] px (t) = −ps (t) cos θ pz (t) = +ps (t) sin θ ∆t = t2 − t1 1 [s] = [Sample] [Sample/s] 2∆t = (t + ∆t) − (t − ∆t) ps (t + ∆t) − ps (t − ∆t) Vˆs (t) ≈ 2∆t [m] − [m] [m/s] = [s] Vs (t + ∆t) − Vs (t − ∆t) a ˆs (t) ≈ 2∆t [m/s] − [m/s] [m/s2 ] = [s]

Calibration Curve for Elevation (4.75) Unit validation Horizontal Component of Position (4.76) Vertical Component of Position (4.77) Time between samples (4.78) Unit validation Time between two samples (4.79) Velocity by Central Difference (4.80) Unit validation Acceleration by Central Difference (4.81) Unit validation

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asavg =

row=N 1 Xa a ˆs (t) Na row=1

Mean Tangential Acceleration (4.82)

µs = tan θs [−] = [−]

Static Coefficient of Friction (4.83) Unit validation

as,avg Dyn. Coef. of Friction for Single Body (4.84) g cos θ [m/s2 ] Unit validation [−] = [−] + [m/s2 ] T as,avg + Dyn. Coef. of Friction for Sys. of Bodies (4.85) µd = tan θ + g cos θ m1 g cos θ [m/s2 ] [N ] [−] = [−] + + Unit validation 2 [m/s ] [kgm/s2 ] f = µd mg cos θ Friction Force from Experimental µd (4.86) 2 Unit validation [N ] = [−][kg][m/s ][−] f = m1 (as,avg + g sin θ) Friction Force from Experimental as,avg 2 2 [N ] = [kg]([m/s ] + [m/s ][−]) Unit validation Tension in the Cable T = m2 [as,avg + g] 2 2 [N ] = [kg]([m/s ] + [m/s ]) Unit validation µd = tan θ +

4.3.2

Procedure - Studio Portion

Studio Pre-work Prior to arriving at Studio, each student should have acquired the necessary data in lab, recorded data in your notebook and stored data on a thumb drive. You should also have a corresponding schematic that clearly identifies where each measurement was made in symbolic notation. Specifically, you should have the following: angle of incipient motion, θs , and its uncertainty, angle of each incline trial, θ, and its uncertainty, mass of the blocks, m1 and m2 , values for the distance from the bottom of the ramp to the top of the ruler, L, a data table with three columns, including block position, pB , from the top of the ruler to the block, nominal mean voltage readings, and standard error in the mean voltage. You will also need the instrument least count (ILC) for the ruler.

In addition, each you will complete several steps of the Studio exercise. This will allow more quality time with the instructor to discuss the physical meaning of the analysis results.

You will upload your studio pre-work to your individual drop-box for the corresponding week. You will recieve a quiz grade based on the completeness of your submision.

Please complete at least steps 1-8 and upload your pre-work spreadsheet to your inidvidual drop-box, before coming to coming to Studio. You can work on the remaining portions of the exercise during Studio. All steps with the exception of the report are due within 24 hrs after leaving Studio.

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Videos There are videos available to help with some of the excel techniques that may be new to you.

We have highlighted steps where videos might be helpful. However, you can also complete the steps simply by following the written instructions. For those procedures that do not have videos, you should rely on previously developed skills. You may want to review videos from previous weeks if you feel that you need a refresher on some of the techniques.

Steps to Complete the Analysis 1. LOGBOOK: Before you begin, enter a Studio Week 4 header on a new page in your logbook. Use Figure 1.14 as a template. After completing the analysis, you will print out your graphs, answer questions and make observations related to your analysis. It is important to make it clear that the work entered today is from Studio Week 4.

2. LOGIN: Login to your PC with your RIT account information. Insert your USB drive into the USB port on your computer.

3. CREATE A NEW FOLDER AND FILE: On your USB drive, create a working folder called Week 4. Store all of your Lab and Studio files for todayâ&#x20AC;&#x2122;s session in this folder.

Open up your week 3 studio spreadsheet from last week and save a copy to your Studio Week 4 folder with the filename of the format Lastname_Firstname_Week4_ Studio.xlsx. Rename the Calibration worksheet tab to W eekCalibration. Rename the Friction worksheet tab to W eek4F rictionP ulley. We will make edits to these worksheets in subsequent steps.

4. CALIBRATION COEFFICIENTS: Here you will update the tables and graphs in the W eek4Calibration tab. If you havenâ&#x20AC;&#x2122;t done so already, rename the column from pz to Ps and rename the offset from zo to L. Enter the pB versus V calibration data.

Your processed data table should automatically update to obtain position ps , and your graphs should update to include the new data ps versus V , where the block position ps is measured from the bottom of the ramp. The trend-line will need to be recalculated, so delete the old trend-line and insert a new one. Record the new calibration equation for block position in your log book and replace the old calibration coefficients in the appropriate cells in your W eek4Calibration worksheet.

5. CREATE THE INCLINED RAMP DATA TABLE: Here you will update the Inclined Ramp Data table shown in Figure 4.7 which should already be located on your tab W eek4F rictionP ulley. Delete the inclined ramp data from week 3. During Lab, you created a new tab separated value (TXT) file containing the time readings in the first column, and the transducer voltage readings in the second column. Open this TXT file by double-clicking on the file name. Copy the TXT data to the Inclined Ramp Data Table. Make sure the data cells remain blue to indicate that these data are constants, rather than equations. Close the TXT file.

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Figure 4.7: Inclined ramp data measured in lab 6. CREATE THE HEADER, CONSTANTS, CONVERSION FACTORS and CALIBRATION COEFFICIENTS TABLES: The tables required for this weekâ&#x20AC;&#x2122;s analysis will be similar to those used last week. Therefore, to save some typing and calculating, simply edit your table templates so they look like the tables shown in Figure 4.8. Notice that the changes are highlighted in red. Enter the necessary data and equations. Note, if you have already linked the calibration data to your analysis worksheet by entering a reference equation in the cell, the slope and intercept from your calibration curve should automatically update. If you have not done this, please make the necessary corrections. In addition, you should have an equation in the cell for sample time, so that the value is automatically updated.

Test: Make sure that your sample time is referencing the new lab data.

7. CREATE THE PROCESSED DATA TABLE: Your processed data table shown in Figure 4.9 should automatically update.

Test: You will most likely have a different number of data points compared to last week. Check the cells in your processed data table to make sure that the calibration, position component, velocity and acceleration equations are valid. Make corrections as needed.

Test: Plot the instantaneous acceleration versus time. Look for and delete obvious outliers if needed. To do this, hover over the marker in the acceleration plot, the data point coordinate should appear telling you which time step contains the outlier, return to the table and delete the acceleration value at that time step.

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Figure 4.8: Header, Table of Constants, Conversion Factors and Calibration Coefficients.

Figure 4.9: Template for the Processed Data Table

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8. PLOT POSITION COMPONENTS VERSUS TIME: The position plot should automatically update. If needed, make adjustments to the axis range or data selected. An example of what your plot should look like is shown in Figure 4.10.

Test: Is position changing in the correct direction with the correct sign? Does the initial position correspond to the distance measured from the bottom to the top of the ramp? Is the final position occurring at s=0? Why or why not? Make corrections as needed.

Figure 4.10: Example plot of block position along the ramp 9. RESULTS - ANALYZE NEWTON’S 2ND AND 3RD LAWS: Enter and edit the equations in the results table shown in Figure 4.11. Notice, that for convenience the necessary additions and changes are highlighted in red. Detailed derivations are provided in Section 4.3.1, and a summary is provided in Section 4.3.1. Students should be able to reproduce the equations used in this analysis by applying Newton’s Laws to the FBDs for m1 and m2 .

Test: Compare the friction force applied to the block calculated two ways. Do they agree within a reasonable amount and can you justify the differences? Make corrections if needed.

Test: Compare the sum of the forces on block m1 , in the direction along the incline, to the product of mass, m1 and acceleration along the same direction. Does the result make physical sense? Make corrections if needed.

Test: Check the signs of your forces and acceleration with the direction defined as positive ”s” on the FBD. Are the signs consistent? Do you have the correct sign for acceleration?

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Figure 4.11: Template for recording results 10. CALCULATE THE UNCERTAINTY DUE TO MEASUREMENT PRECISION: Similar to last week, we have compound error in the position of the block due to the uncertainty in L and the uncertainty in pB . In addition, the angle measurement has an uncertainty that will create an uncertainty in the trigonometric function calculations.

If needed, update the values in the blue shaded cells in the uncertainties table shown in Figure 4.12. The other cells should automatically update when you enter the new inclined angle and uncertainties.

Test: If you tested the equations last week, you do not need to test them again.

However, you may want to compare uncertainties between week 3 and week 4 as a self consistency check.

11. DETERMINE THE UNCERTAINTY IN INSTANTANEOUS VELOCITY AND ACCELERATION: The Instantaneous Uncertainties table shown in Figure 4.13 should automatically update.

Test: Check to see that the equations entered are referencing the values you intended.

12. DETERMINE THE UNCERTAINTY IN THE CALCULATED RESULTS: Update the uncertainty table shown in Figure 4.14 to calculate the uncertainties shown. Note the new table items are highlighted in red. Note: You may want to refer to Section 4.2.9 and the Engineering Mechanics Reference Table, currently available on mycourses under Week 1 content.

Test: These equations are complicated and it is easy to mistype. Double check your values with a neighbor. Temporarily change constants if needed so that an exact comparison can be made. Make corrections to your equations as needed.

13. UPDATE YOUR ENGINEERING LOGBOOK: Please print out your graphs, results table and uncertainty tables and paste them in your logbook. You may also want to 217

Figure 4.12: Template for recording uncertainties due to measurement precision

Figure 4.13: Template for the table to calculate the effect of numerical error on uncertainty in calculated velocity and acceleration.

Figure 4.14: Table Template for determining uncertainty in the calculated results.

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print all of the tables, but this is left for the student to decide. Sign and date your logbook before you leave Studio.

14. SUBMIT YOUR FILES: Submit your excel spreadsheet to your individual Week 4 dropbox on myCourses before leaving the Studio. If you have not completed all the steps, upload what you have done. Please include in the comment, how which procedures you have completed. Then within 24 hours, upload your final completed version with the comment ”final version.” Remember to save your work to you USB drive, and take it with you when you leave the Studio.

15. OBSERVATIONS AND ANALYSIS: Write responses to the following questions in your logbook. Be sure to include a justification for your answer by referring to the data, plots, and derivations that are contained within your logbook. You may want to crossreference equations from Sections 4.1, 4.3.1 and 3.2.9 in your work. (a) How do your findings show agreement or disagreement with Newton’s First Law? (b) How do your findings show agreement or disagreement with Newton’s Second Law? (c) How do your findings show agreement or disagreement with Newton’s Third Law? (d) How do your findings show agreement or disagreement with Newton’s Law of gravity?

16. CONGRATULATIONS! You have just completed the Studio portion for week 4.

17. WRITE THE REPORT: Please refer to section 4.3.3 Report on details for the report submission. Before leaving Studio, decide on a date and time to meet up with your team mates to prepare the report.

4.3.3

Report

Please use the same task distribution for writing the report that was outlined in Week 1. Refine your “Team Norms” to enhance your team’s ability to work effectively with one another, particulary if your team has encountered some challenges in preparing and submitting the report. Engage in an open and frank conversation about each team member’s expectations and whether or not they are being met.

Prepare a report to include only the following components: • TITLE PAGE: Include the title of your experiment, “System of Bodies in Constrained Linear Motion”, Team Number, date, authors, with the scribe first, the team member’s role for the week, and a photograph of each person beginning to initiate their trial, with a label below each photo providing team member’s name.

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• PAGE 1: The heading on this page should read Experimental Set-up. Create a diagram of the experimental set-up. This week we will include only the diagram and no text. Thus, is it important that your diagram clearly communicate the set-up, including each key component and where measurements were taken. The important information to communicate are the variable names and datums that relate to your measurements and results. At the bottom of the figure include a figure caption, for example Figure 1. A brief figure caption. Refer to the text for examples.

Note: Figure captions are required for every plot and diagram in the report, except for the title page. Figure captions are placed below the figures, and are numbered sequentially beginning with Figure 1 for the first figure in the report.

• PAGE 2: The heading on this page should read Results. Include the table shown in Table 4.6 summarizing each team member’s results. At the top of the table, include a table caption, for example Table 1. A brief figure caption. Refer to the text for examples.

This week we include only tables and plots with no accompanying text. Thus, it is important that your tables, graphs and captions clearly communicate to the reader what the data represents.

Note: Table captions are required for every table in the report, except for the title page.

Unlike figure captions, table captions are placed above the tables, and are numbered sequentially (independent of figure caption numbering) beginning with Table 1 for the first table in the report.

Table 4.6: Results from the Inclined Ramp Experiment

• PAGES 3: No heading is needed on this page, since it is a continuation of the Results section. Include the plots for position compents , ps , px and pz . The position values 220

should be consistent with the datums shown in your schematic, which should agree with the conventions provided in the text. Include one plot for each team member, with one figure caption for each plot. Each figure caption should be located below the graph and include the name of the test engineer responsible for that series of trials.

The first caption in the Results section should be for example Figure 2. Ultrasonic transducer calibration by Charlie Brown.

Strive for uniformity among the graphs, including axis values and plotting styles. This will enable the reader to easily compare the results between different team members. â&#x20AC;˘ The final report should be collated into one document with page numbers and a consistent formatting style for sections, subsections and captions. Before uploading the file, you must convert it to a pdf. Non-pdf version files may not appear the same in different viewers. Be sure to check the pdf file to make sure it appears as you intend.

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4.4

Recitation

Recitation this week will focus on problem solving. Please bring your logbooks to class, and be ready to ask questions about the homework problems. You are encouraged to attempt as many homework problems as you can from Chapters 1 through 4. Your first exam is coming up next week, and itâ&#x20AC;&#x2122;s essential that you be completely comfortable with all of the material covered to-date in class.

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4.5

Homework Problems Complete all assigned homework problems in your logbook.

4.5.1 Consider the schematic in Figure 4.1. If a mass, m3 was suspended from m2 by a cable, draw a FBD of m1 , m2 , m3 and the pulley, P .

4.5.2 Consider the schematic in Figure 4.1. If a mass, m3 was suspended from m2 by a cable, draw a FBD of m1 , m2 , m3 and the pulley, P , with each force resolved along the normal and tangential directions, as defined in Figure 4.2.

4.5.3 A 3 kg block is sliding down an inclined ramp. The ramp makes an angle of 32 degrees counter clockwise from the horizontal. What is the magnitude of the gravitational force on the block? What are the x and z components of the the gravitational force?

(Take x as positive to the left and z as positive to up).

4.5.4 A 10 kg block is sliding down an inclined ramp. The ramp makes an angle of 50 degrees counter clockwise from the horizontal. What is the magnitude of the gravitational force on the block? What are the components of the gravitational force along the axes normal and tangential to the inclined ramp? Draw the force vector diagrams, labeling all your angles and forces.

4.5.5 A 8 kg block slides down an inclined ramp. The ramp makes an angle of 132 degrees clockwise from the horizontal. What is the magnitude of the gravitational force on the block? What are the x and z components of the the gravitational force? (Take x as positive to the left and z as positive to up). Draw the force vector diagram, labeling all your angles and forces.

4.5.6 A 25 kg block slides down an inclined ramp. The ramp makes an angle of 165 degrees clockwise from the horizontal. What is the magnitude of the gravitational force on the block? What are the components of the gravitational force along the axes normal and tangential to the inclined ramp? Draw the force vector diagram, labeling all your angles and forces.

4.5.7 A 132 lb college student skiâ&#x20AC;&#x2122;s down a black diamond trail at 10 mph. The hill is at an angle of 85 degrees counter clockwise to the horizontal. Write a vector equation for the weight of the skier in x and z components. Write a vector equation for the velocity of the skier in x and z components. Take x as positive to the left and z as positive to up.

4.5.8 A 5 kg block slides down an inclined ramp at approximately constant velocity. The ramp angle is 23 degrees counter clockwise from the horizontal. What is the value of the force on the block due to friction between the block and the ramp?

4.5.9 A 9 kg block slides down an inclined ramp at approximately constant velocity. The ramp angle is 110 degrees clockwise from the horizontal. What is the value of the force on the block due to friction between the block and the ramp?

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4.5.10 Explain how the weight of a skier affects the friction force. Plot the static friction force between the skier and the snow for a realistice variation in weight from child to adult. Use a static friction coefficient of 0.1 and use ski slope of 10 degrees from the horizontal.

4.5.11 In the week 4 Lab experiment we analyzed a multi-body system where the motion of two bodies are influenced by gravity, an inclined ramp, and the motion of the other body. Describe a real-world multi-body system that you have observed in your daily life.

4.5.12 Consider a roofing contractor is pulling on a rope to raise a load of roof shingles as shown. The contractor is standing on a roof with a pitch of 15 degrees. Each bundle of shingles weighs 30 lbs and the platform that carries the shingles weighs 20 lbs. (a) Draw a Free Body Diagram and Vector Diagram for the Contractor and the platform carrying the shingles. (b) How many packs of shingles can the contractor lift before he begins to slide down the roof assuming the following Contractor weights and values for the coefficient of static friction exists between the contractor and the roof?

Scenario (1) (2) (3) (4)

µs 0.2 [-] 0.2 [-] 0.55 [-] 0.55 [-]

Weight of Contractor 160 [lbs] 280 [lbs] 160 [lbs] 280 [lbs]

4.5.13 Consider the system illustrated in the figure below. Assume that the coefficient of friction between the block m1 and the ramp is µ. Draw Free Body Diagrams for each of the objects listed below. Demonstrate that Newton’s Third Law is satisfied.

Object m1 m2 S P1 R E

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Description FBD of Mass 1 FBD of Mass 2 FBD of Spring FBD of Pulley FBD of the Ramp Structure FBD of the Earth

4.5.14 Describe the difference between a Schematic Diagram, a Free Body Diagram, and a Vector Diagram.

4.5.15 Draw a representative Schematic Diagram, Free Body Diagram, and Vector Diagram of a snow boarder traveling straight down a steep “Black Diamond” slope.

4.5.16 Describe in your own words what the term “you can’t push on a rope” means with respect to Newton’s Laws and the sign conventions associated with the rope and pulley system addressed in this week’s Lab. 4.5.17 Describe what a “kinematic constraint” is and why it is necessary for this week’s Lab and Data analysis.

4.5.18 Explain why antilock breaks minimize skidding.

4.5.19 Consider the system illustrated in the figure below. Assume that the coefficient of friction between the block m2 and the ramp is µ. Draw Free Body Diagrams for each of the objects listed below. Demonstrate that Newton’s Third Law is satisfied.

Object m1 m2 P1 R H E

Description FBD of Mass 1 FBD of Mass 2 FBD of Pulley FBD of the Ramp FBD of the Horizontal Surface FBD of the Earth

4.5.20 Describe in your own words what the term “you can’t push on a rope” means with respect to Newton’s Laws and the sign conventions associated with the rope and pulley system addressed in this week’s Lab. 4.5.21 Describe what a “kinematic constraint” is and why it is necessary for this week’s Lab and Data analysis.

4.5.22 Consider the system illustrated in the figure below. Assume that the coefficient of friction between the block m1 and the building is µB and that between block m1 and m3 is µ1/3 . Draw Free Body Diagrams for each of the objects listed below. Demonstrate that Newton’s Third Law is satisfied.

Object Description m1 FBD of Mass 1 m2 FBD of Mass 2 m3 FBD of Mass 3 P1 FBD of Pulley B FBD of the Building E FBD of the Earth

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