Optional Unit V: Applications of Kinematics and Dynamics
D. Uniform Circular Motion

Key Concepts

If an object is travelling in a circular path at constant speed its velocity is changing because, even though the magnitude of the velocity remains constant, the direction of the velocity is changing continuously.

The velocity vector points along a tangent to the circle, in the direction that the object would tend to move if it were suddenly released.

The acceleration acts in the same direction as the change in velocity.

The acceleration is called centripetal acceleration. It always acts inward, toward the centre of the circle, in the same direction as the change in velocity, perpendicular to the velocity vector.

The instantaneous acceleration (_inst) at any point on the circular path is:

The magnitude of the centripetal acceleration is given by:

where R is the radius of the circle, T is the period of revolution, f is the frequency of revolution, and a_c is the magnitude of the centripetal acceleration.

The subscript c serves as a reminder of the vector nature of the acceleration. Its direction is constantly changing at every position along the circular path.

From Newton's Second Law:

The force acts in the same direction as the acceleration.

The force, directed towards the centre of the circle, is called centripetal force.

(Other relationships can be obtained by substituting the equations for centripetal acceleration into the equation for Newton's Second Law,

e.g.,F_c = (4¶²mR)/(T²), or F_c = 4¶²mRf²)

(It is important that students clear up any misconceptions they might have about centrifugal force, which is a fictitious force, that appears to act in an accelerated frame of reference.)

The minimum velocity needed at the top of the loop for an object to perform a loop-the-loop is:

To place a satellite into orbit, it must be travelling such that the force of gravity acting on it (i.e., its weight) provides a force equivalent to the centripetal force needed to maintain its motion.

The orbital velocity does not depend on the mass of the satellite.

Learning Outcomes

Students will increase their abilities to:

1. Define the following terms: centripetal acceleration, centripetal force.

2. Explain why an object travelling in a circular path at a constant speed undergoes a change in velocity.

3. Illustrate the direction of the velocity vector, the centripetal acceleration vector, and the centripetal force vector for a moving object at a specific position on a circular path.

4. Use a vector diagram to illustrate a change in velocity when the magnitude of the velocity vector remains constant but the direction changes.

5. Recognize that if an object were suddenly released from its circular path, it would tend to continue to move in the direction of the velocity vector, unless it was acted upon by some external force.

6. Explain that centripetal acceleration acts in the same direction as the change in velocity.

7. Explain that centripetal force acts in the same direction as centripetal acceleration.

8. Use mathematical relationships for centripetal acceleration and centripetal force to solve problems in- volving circular motion.

9. Recognize that to place a satellite into orbit, it must be travelling such that the force of gravity acting on it (i.e., its weight) provides a force equivalent to the centripetal force needed to maintain its motion.

10. Explain that the orbital velocity of a satellite does not depend on the mass of the satellite.

11. Describe some useful applications of satellites.

Teaching Suggestions, Activities and Demonstrations

1. Using data on Earth satellites (in which the period of revolution around the Earth can be obtained), or the Moon as an Earth satellite, determine the mass of the Earth. Assume that the satellites travel in circular orbits and that the Earth is spherical. From the apogee (farthest distance) and the perigee (closest approach) of the satellite, determine its average radius. Calculate the mass of the Earth using:

   

   Refer this to the next section on Universal Gravitaion.

2. Run a piece of thread through a 15 cm long piece of glass tubing. Attach a rubber stopper to one end of the thread, and washers to the other end. Spin the rubber stopper, keeping it moving at a constant speed. Record the radius and the number of washers at the other end. Change the number of washers, repeating a variety of tests. Attempt to vary the radius of revolution and determine the period. (Record the time for 10 complete revolutions, then divide by 10 to get the period.) Develop relationships relating centripetal force, mass, radius, and velocity for uniform circular motion.

   Actually manipulating the data to search for the relationships is to be preferred over simply verifying

   

3. Many useful applications of satellites have been developed. (Several different applications should be described. Stress STSE interrelationships. Some of the societal needs which have helped to promote the "space race" should be explained. The political, economic, and military exploitation of space, as well as cooperative international ventures, are important social and political phenomena which illustrate ways in which science and technology operate within other human influences.)

4. Use a Foucault pendulum to demonstrate the rotation of the Earth. A Foucault pendulum consists of a very heavy pendulum bob bolted securely to the ceiling with wire. (Some fancier types drop sand as they swing.) The inertial plane of the pendulum remains constant. As the pendulum begins to swing, mark the horizontal path it traces out on the floor. The pendulum should be relatively friction-free, so that it can swing for at least half an hour. After that time, note the apparent change in the direction in which the pendulum is swinging relative to the Earth. Careful observations should show that the amount of rotation of the plane of the pendulum corresponds to the same amount of rotation of the Earth in that time.

5. Show students how to calculate the magnitude of the velocity needed to maintain a satellite in orbit around the Earth at some given height. Relate this to universal gravitation.