Unit B4: Uniform Circular Motion and Gravity Vocabulary
Forces in Circular Motion
# centrifugal force – a fictitious force used to explain apparent equilibrium in an accelerating frame of reference. # centripetal acceleration – a radial acceleration directed towards the center of a circle. # centripetal force – a force that acts radially inwards on an object undergoing circular or elliptical motion # fictional force – an apparent force that acts on all masses in a non-inertial frame of reference, such as a rotating reference frame. # radial acceleration – acceleration directed along a radius of a circle # Uniform Circular Motion – motion in which an object travels in a circle at constant speed.
Uniform Circular Motion Circular motion is any type of motion where and object moves at a fixed distance away from the center of motion, that is, it moves in a cicle. An object that moves at constant speed (v) in a circle with radius of (r) is said to undergo Uniform Circular Motion (UCM). The magnitude of velocity is constant, but the direction of the motion constantly changes. Since there is a change in velocity, there must also be acceleration. This acceleration is a radial acceleration, and is directed towards the center of the circle. Another name for this type of acceleration is a centripetal acceleration. The magnitude of the centripetal acceleration is given by
a R=
v2 r
(B4-1)
where aR is the centripetal acceleration, v is the tangential speed and r is the radius Note that this equation holds true for elliptical motion, such as orbits.
Centripetal Force A force that acts towards the center of a circular path is called a centripetal force. The most commonly encountered centripetal force is gravity acting to hold planets in orbit around a star, or satellites in orbit around a planet.
Fictional Forces Sometimes, it is easier to explain the motion of a moving object by introducing fictional forces into the solution. These forces do not really exist, but are used to simplify the problem. For example, if you are in an elevator that is accelerating upwards, you feel as though you are being pushed down. The excess weight that you feel is only because the floor is pushing you upwards, but you perceive it as a real force. This non-existent force would be considered a fictional force.
Centrifugal Force A centrifugal force is a fictional force that is sometimes introduced in circular motion problems to simplify them from the standpoint of the moving object. A centrifugal force apparently acts in a direction opposite the centripetal force. This force can be used to explain why, for an observer on the Earth, the Sun does not pull the Earth towards it: the gravitational pull is offset by a fictitious centrifugal force. This force does not exist for an observer that is not in the same reference frame as the Earth, and is not the result of any of the fundamental forces of the Universe.
Kepler's Laws Johannes Kepler formulated three empirical laws relating to planetary orbits. He found that: * Planets orbit the Sun in elliptical orbits, with the Sun at one focus of the orbit; * Planets move faster when they are closer to the Sun, and slower when they are farther away; and
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2. What is Uniform Circular Motion? 3. What two quantities determine the value of the centripetal acceleration? 4. Why do we sometimes introduce fictitious forces into a problem? 5. What is the model that we currently use to describe the motion of planets in the Solar System? 6. What are the three key features of Kepler's Laws? 7. What are the key physical characteristics of gravity? 8. Why do satellites stay in orbit, rather than falling to Earth?
The period (T) of the orbit is related to the average orbital distance (R) by the relationship
T 2 ∝ R3
(B4-2)
Gravity Gravity is one of the three fundamental forces in the Universe. It is an interaction between particles that have mass. Newton’s Law of Universal Gravitation states that all objects with mass exert an attractive force on all other objects with mass. This force is 3 an attractive force; 3 directly proportional to mass (the more mass you have, the more force there is); 3 inversely proportional to the square of the distance (the greater the distance between the masses, the less force there will be). The formula that Newton developed was
F=
G m1 m2 r2
Problems
(B4-3)
where m1 is the mass of the first object, m2 is the mass of the second object, r is the distance between the centers of the objects, and G is known as the Universal Gravitational Constant which has a value of G = 6.67 x 10-11 N•m2/kg2.
The value of g We can look upon g in to different ways. First, g is the traditional acceleration due to gravity that we have been using for quite some time. The value of g varies with location. On the surface of the Earth, g is roughly equal to 10 m/s2. Another way to look at g is to recognize that it is the gravitation field strength, which is a measure of how strongly gravity acts The value of the acceleration due to gravity, g, on any planet can be determined from this equation
g=
GM 2 r
(B4-4)
where M and r are the mass and radius if the planet, respectively. Since G, M and r are all constants, we see that g must be a constant as well.
Questions 1. What is Circular Motion?
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1.
An F-22 Raptor traveling at 600 m/s pulls out of a dive by moving in an arc with a radius of 5000 m. What is the acceleration on the plane? 2. A car goes around a curve at 45 mph, which is equivalent to 20 m/s. What radius should the curve be so that the acceleration is 10 m/s2? 3. An F-86 Sabre traveling at 300 m/s pulls out of a dive by moving in an arc with a radius of 5000 m. If the mass of the plane is 6000 kg, what is the force on the wings of the plane? 4. The imaginary planet Mu Cambrensis III is 5×1010 m from the star Mu Cambrensis. The mass of Mu Cambrensis III is 5×1026 kg, and the mass of Mu Cambrensis is 5×1030 kg. What is the gravitational attraction between these two objects? 5. The mass of Mu Cambrensis III is 5×1026 kg. Its diameter is 36000 km. What is the acceleration due to gravity on the surface of Mu Cambrensis III?