Civil Engineering

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Kinetics of Particles-Newton's 2nd Law INTRODUCTION

 So far in the earlier chapter we did the motion analysis of moving particles without taking into account the forces responsible for the motion. From this chapter we begin our motion analysis involving the forces responsible for the motion. This analysis is known as kinetics. Here we will analysis motion of moving cars, elevators, blocks, airplanes, rockets etc. treating them as a particle, since rotation of these bodies about their own Centre of gravity, if any, is neglected.  In this chapter we will extensively use the Newton's Second law approach to kinetics. As stated further, this approach involves determination of acceleration of the moving particle by knowing the forces acting on the particle. Having determined the acceleration, the analysis is completedusing kinematic relations which we have studied in previous chapter. NEWTON'S SECOND LAW OF MOTION  Newton's second law of motion states "The rate of change of momentum of a body is directly proportional to the resultant force and takes place in the direction of the force”.  Consider a particle acted upon by several forces as shown in Fig. 10.1. Let F be the resultant force. Because of the resultant force, the particle would move in the direction of the resultant force. If u is the initial velocity, v is final velocity and this change takes place in t sec, we have from Newton’s second law:

 Rate of change of Momentum = Resultant Force Final momentum - Initial momentum i.e.  F Time

mv  mu  F t (v  u ) F  m t or F  m  a

…………10.1

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 Equation 10.1 is the mathematical expression of Newton's Second Law and this gives rise to another statement of Newton's Second Law, which is "If the resultant force acting on a particle is not zero, the particle will have an acceleration proportional to the magnitude of the resultant and in the direction o' the resultant force'.  Equation 10.1 is a vector relation since both the force and acceleration are vector quantities. The scalar relations from 10.1 can be developed as,

F x  m  a x

……

[10.2 (a)]

F y  m  a y

……

[10.2 (b)]

F z  m  a z

……

[10.2 (c)]

 Since we will normally restrict our analysis to coplanar forces, the equations 10.2 (a) and 10.2 (b) will be mainly used. D'ALEMBERT'S PRINCIPLE  Referring to equation 10.1 we have

F  m  a  This

is

Newton's

Second

Law

equation

and

relates

the

particle's

acceleration to the resultant of the external forces acting on the particle. The equation is vector a equation where the resultant force vector is equated to the ma vector.  Transposing the R.H.S. of the equation 10.1 we get,

F  ma  0

……

[10.3]

 The above equation is a dynamic equilibrium equation put forth by D'Alembert. The ma vector is treated as an inertia force and when added with a negative sign to all other forces, results in equilibrium state of particle.

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 Figure shows the dynamic equilibrium state of 1) A block sliding down a rough inclined plane 2) A sphere tied to a string, swinging as a pendulum to a lower position.

 Particle's acceleration can be found out by

D'Alembert's

principle, by

developing the figure representing a dynamic state and using the equilibrium equations used in static viz., Fx 0 and Fy 0

 Newton's Second Law approach to kinetics is more realistic than the D’Alembert's principle since it does not involve inertia forces and does not refer to an equilibrium state of a moving particle. We would therefore solve the problems in kinetics involving forces and acceleration using Newton's Second Law equation.

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N S L EQUATIONS APPLIED TO RECTILINEAR MOTION  Application of Newton's equations to determine the acceleration of rectilinear moving particles should be carried out by a systematic approach as explained below. step (1) Draw the FBD showing all the forces acting on the moving particle. I more than one particle is involved, the particles may be isolated and separate FBD may be drawn. step (2) By the side of FBD, draw the Kinetic Diagram (KD) which shows the particle with a ma vector acting on it. The magnitude of ma vector is the product of particle's mass and its acceleration. ma vector acts in the direction of particle's acceleration. step (3) Equations of Newton's second law viz 10.2 (a) and I0.2 (b) are now used, taking help of the FBD and KD drawn earlier. The particle's acceleration is thus obtained.  To understand the above outlined x steps, let us find out the acceleration of a block of mass m which is being pulled up an inclined plane by force P applied parallel to the plane. Let

k N

be the kinetic coefficient of friction between the block and plane. As the block moves up, the rough inclined surface develops a frictional force  k N down the plane.  The FBD of the block showing the forces and the corresponding Kinetic Diagram showing the ma vector is drawn as shown in figure. Taking the x and y axes as shown and using equation 10.2 (a) of the Newton's Second Law we have,

k

Substituting the value of N from equation (2) in (1)

P  mg sin   k  mg cos    ma a 

P  mg  sin   k cos   m

 Thus knowing the forces acting on the particle, its acceleration can be found out.

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EXERCISE 1 1. A 100 kg crate is projected on a horizontal floor with an initial speed of 10 m/s. If tr1" = 0.4 and p1 = 0.3, calculate the distance traveled and the time taken by create to come to rest. 2. A block of mass 30 kg is placed on a plane. ď ­ k Between the block and plane is 0.3. If a force of 250 N is acting on the block, find its acceleration for the two cases shown.

3. Find force P required to accelerate the block shown in figure at 2.5 m/s2. Take p = 0.3.

4. Two blocks A and B are placed 15 m apart and are released simultaneously from rest. Calculate the time taken and the distance traveled by each block before they collide.

5. Two blocks A of weight 500 N and B of weight 300 N are 10 m apart on an inclined plane as shown in figure. ď ­ = O.2 for block A and 0.3 for block B. If the blocks begin to slide down simultaneously calculate the time and the distance travelled by each block when block A touches block B.

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6. Two blocks A and B rest on an inclined plane as shown. Find their acceleration on being released from rest. Also find the contact force between the blocks.

7. Three blocks m1 and m2 and m3 of masses 1.5 kg, 2 kg and 1 kg respectively are placed on a rough surface with ď ­ = O.2O as shown. If a force F is applied to accelerate the blocks at 3 m/s2 what will be the force that 1.5 kg block exerts on 2 kg block.

8. A package of total weight 4OO0 N is being pulled up a slope by a steel cable exerting a force F = (50 t2 + 25OO) N. Knowing that the velocity of the package was 5 m/s at t = 0, find the acceleration and velocity of the package at t=8 sec.

9. An airplane of mass 30,000 kg starts from rest and runs on the runway for 1500 m before it takes off. If the airplane engines developed a constant thrust of 50 kN and the air resistance force acting on the plane is F = 3 v2 (where F is in Newton and v is in m/s), find the takeoff velocity of the airplane.

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10. A particle of mass 3 kg initially at rest at the origin is set in rectilinear motion by a force varying with time as per the graph shown. Find a) The velocity and position of the particle at t =1O sec. b) The time and position at which the particle comes to rest c) The time at which the particle comes again to its original position.

11. A locomotive train weighing 5OOO kN exerts a constant pull of 120 kN. The train starts from rest on a level track and attains a speed of 9O kmph, after which steam is suddenly shut off, slowly bringing it to a halt. If the tractive resistance is 15 N/kN of the eight of the train, find the total distance covered by the train and the total time to cover this distance. What is the maximum power developed by the engine? 12. A vertical lift of total mass 750 kg acquires an upward velocity of 3 m/s over a distance of 4m moving with constant acceleration starting from rest. Calculate the tension in the cable. 13. A 400 kg lift-carrying a 60 kg person travels vertically up starting from rest and acquires a velocity of 4m/s in a distance of 3m of motion. Find the tension in the cable supporting the lift and the force transmitted by the person on the lift floor. Does the person feel normal or heavy or light. a) The lift is now brought to a halt from the speed of 4 mf s, in 2 sec time. What is the force exerted by the man on the floor of the lift and how does he feel now. 14. Two blocks A and B are connected by a rope and move on a rough horizontal plane under a pull of 60 N as shown. If p is 0.2 between the blocks and the surface, find the acceleration of the blocks and the tension in the rope.

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15. Two masses l.4kg and 7kg connected by a flexible inextensible cord rest on a plane inclined at 45 with the horizontal as shown in figure. When the masses are released what will be the tension T in the cord? Assume the coefficient of friction between the plane and the 14 kg mass as 0.25 and between the plane and the 7kg mass as 0.375. 16. Two particles of masses mA = 5 kg and m= 10 kg are supported as shown. acceleration of the blocks a) if the horizontal surface is smooth b) if the horizontal surface is rough coefficient of kinetic friction ď ­ k = Q.4 17. Calculate the vertical acceleration of the 10O kg block and the tension in the connecting string. Neglect friction.

18. Determine the tension in the string and the velocity of 1500N block shown in the figure 5 sec after starting from rest.

19. Determine the weight WA of block A required to bring the system to stop in 6 sec, if at the instant shown, the block C is moving down at 5 m/s. Given WB = 2OO N, Wc = 600 N. Take the pulley to be smooth. a) Find weight WA to bring the system to stop in 5 sec., If at the instant shown, vc = 3 m/s ď‚Ż , WB =150N and WC=500N.

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20. At a given instant 5O N block A is moving downwards with a speed of 1.8 m/sec. determine its speed 2 sec. later. Block 'B' has a weight 2O N, and the coefficient of kinetic friction between it and the horizontal plane is  k = 0.2. Neglect the mass of pulleys and chord. Use D’Alembert’s principle.

21. Two blocks A and B are connected by an inextensible string as shown. Neglecting the effect of friction, determine the acceleration of each block and tension in the cable.

22. Masses A (5 kg), B (10 kg), (2O kg) are connected as shown in the figure by inextensible cord passing over massless and frictionless pulleys. The coefficient of friction for masses A and B with ground is 0.2. If the system is released from rest, find acceleration of the blocks and tension in the cords.

23. The two blocks starts from rest. The horizontal plane and the pulley are frictionless. Determine the acceleration of each blocks and tension in the cord.

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24. Block A (7 kg), B (12 kg), - (30 kg) are connected by an inextensible string as shown. If the system is released from rest find the accelerations of each block and tension in the cord. Assume smooth surfaces.

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N S L EQUATIONS APPLIED TO CURVILINEAR MOTION  For curvilinear moving particles, the kinetic diagram shows the components of ma vector viz, man, and mat. The man component acts towards the centre of path while the mat component acts tangent to the path. Both the components lie in the plane of curvilinear motion.

EXERCISE 2 1. A steel bob of mass 5 kg tied to a string of 3 m length is whirled with a constant speed, such that the bob moves in a circle in the horizontal plane. If the string makes an angle of 30" with the vertical, find the speed of the bob and tension in the string. 2. A car weighing 12 kN goes round a flat curve of 90 m radius. Determine the uniform limiting speed of the car in order to avoid outward skidding. Take µ = 0.35 3. Figure shows two vehicles moving at 90 kmph on a road. Knowing that µ= 0.5 between road and the tyres. Determine total acceleration of each vehicle when the brakes are suddenly applied and the wheels skid.

4. A vehicle of weight 8OO0 N travels with a constant speed of 72 kmph over a vertical parabolic curve as shown. Find the pressure exerted by the tyres on the road at the peak B.

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5. In a roller coaster ride, the coaster traveling around the loop ABC needs to have a certain minimum velocity at the peak B. If the loop diameter is 10 m, find this velocity.

6. A pendulum of mass 600 gms, has a speed of 3.6 m/s at the position shown. Find the tension in the string and the total acceleration at this instant.

EXERCISE 3 Theory Questions Q.1 State Newton's Second Law of Motion. Q.2 Derive the mathematical expression of Newton's Second Law. Explain D'Alembert's Principle. UNIVERSITY QUESTIONS 1. A ball thrown down the incline strike the incline at a distance of 75 m along. If the ball rises 20 m above the point of projection. Complete the initial velocity and angle of projection with horizontal.

(10 Marks)

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2. The

system

shown

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figure

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is

released from rest. What is the height lost by bodies A, B and C in 2 seconds?

Take coefficient of kinetic

friction at rubbing surfaces as o.4. Find also the tension in wires. (10 Marks) 3. A 50 kg block kept on a 150 inclined plane is pushed down the plane with an initial velocity of 20 m/s. If μk =0.4 , determine the distance travelled by the block and the time it will take as it comes to rest.

(4 Marks)

4. Two block mA = 10 kg and mB = 5 kg connected with cord and pulley system as shown in fig. Determine the velocity of each block when system is started from rest and block B gets displacement by 2 m. Take μk =0.2 between block A and Horizontal surface.

(6 Marks)

5. A body of mass 25 kg resting on a horizontal table is connected by string passing over a smooth pulley at the edge of the table to another body of mass 3.75 kg and hanging vertically as shown. Initially, the friction between the mass A and the table is just sufficient to prevent the motion. If an additional 1.25 kg is added to the 3.75 kg mass, find the acceleration of the masse (4 Marks) 6. State D’ Alembert’s principle with two examples.

(4 Marks)

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7. Find out min. value of P to start the motion.

(8 Marks)

8. Sphere A is supported by two writes AB, AC. Find out tensions in wire AC:(i) before AB is cut (ii) just after AB is cut.

(4 Marks)

9. A block of mass 5kg is released from rest along a 40 degree inclined plane. Determine the acceleration of the block when it travels a distance of 3m using D’ Alemberts principle. Take coefficient of friction as 0.2. (4 Marks)

10. The mass of A is 23kg and mass of B is 36kg. The coefficient are 0.4 between A and B, and 0.2 between ground and block B.

Assume smooth

drum. Determine the maximum mass of M at impending motion. (8 Marks)

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