Circular Motion Paradox

Introduction

You accelerate around a sharp curve and you feel like you are being thrown toward the outside of the car. Are you being thrown into the car or is the car accelerating into you? What would happen to a helium-filled balloon floating inside the car when the car accelerates around the corner? Which way would it move?

Concepts

• Uniform circular motion
• Centripetal force
• Center of curvature

Materials

Safety Precautions

This demonstration is considered safe. Follow all normal laboratory safety rules.

Disposal

With proper care, the apparatus can be used indefinitely. Empty the water from the jars when the apparatus will not be used for long periods of time.

Prelab Preparation

The jars on the apparatus will need to be filled with water so that the buoys (fishing bobbers) are submerged beneath the water in each jar and floating freely above their tether line (see Figure 1).

Adjust the bobber along the length of the tether line to assure that the buoys are submerged. The entire apparatus will need to be inverted while the jar lids are being tightened in place on the rotating platform. Be sure to hold the lids and tighten the lids until there are no leaks.

Turn the axle into the threaded base and use the locknut to lock the axle in place. The tension on the axle at the rotation spot can be adjusted by raising and lowering the collar below the rotating platform. An Allen wrench is needed to loosen the set screw. Wax or silicon spray can be used to create less friction of the rotating platform on the shaft.

A preliminary run of the demonstration should be conducted prior to doing the demonstration with students. Be sure the apparatus is working smoothly before class use. The demonstration should be preceded by lessons on rotary motion and centripetal force.

Procedure

1. Explain the apparatus setup and allow time for discussion before actually performing the demonstration. Ask students to predict the movement of the buoys when the platform is spun and explain their predictions in detail and relate their predictions to real-life experiences.
2. To see the somewhat unpredicted results (buoys will actually move in toward the center of the circle—not outward), the apparatus needs to be spun. This can be done easily by hand and need not be with great gusto. As soon as the apparatus is spun and while it is slowing down, the inward movement of the buoys will be very obvious. The parabolic movement of the water outward will also be very obvious.
   {13878_Procedure_Figure_2_Position of water and buoys while spinning}
3. Repeat the spinning process several times until everyone sees the results clearly. Discuss the results.

Teacher Tips

• The apparatus is durable and with care can be used for many years. The axle and moving parts can be waxed from time to time for continual smooth operation.
• The axle can be removed from the base and placed in a variable speed rotator (if one is available) to show the result very dramatically with continuous and constant rotary motion.
• The effect is equally dramatic even if the jars are completely filled with water.

Discussion

Circular motion is more difficult to visualize than linear motion. Our bodies are pretty good accelerometers and we can often “feel” the forces involved in various types of motion (descending in an elevator, stopping in a jet plane, or going around a sharp curve in a car). Sometimes our intuitive guesses about the forces required for this acceleration can be incorrect based upon our tactile experiences. Circular motion in particular often defies the predictions of our senses.

Uniform circular motion results when the net force acting upon a mass moving at a constant speed changes the mass’ direction in such a way as to deflect it into a circular path. The force is always acting at a right angle to the direction that the mass is moving. The force causing this reaction is called the centripetal force.

Any motion in a curved path represents accelerated motion and requires a force directed toward the center of curvature of the path. Note that the centripetal force always acts at right angles to the instantaneous velocity of the mass. Therefore, the centripetal force cannot change the magnitude of the velocity (otherwise known as speed). Since velocity is a vector quantity having a magnitude and a direction, a change in direction is a change in velocity. A change in the mass’ velocity means that it is being accelerated. According to the second law of motion, acceleration is always in the same direction as the applied force. The force, F_c, acting on the mass is always directed towards the center of the circle. Thus, the acceleration is also directed toward the center of the circle. This acceleration is called centripetal acceleration.

With vector diagrams, uniform circular motion can be analyzed and equations for centripetal acceleration and centripetal force can be derived (most texts show the derivation). Note that the centripetal force is proportional to the square of the velocity, implying that a doubling of speed will require four times the centripetal force to keep the motion in a circle. The centripetal force is inversely proportional to the radius of curvature of the arc, r. (This explains why speeding around a sharp corner can be dangerous!)

The explanation for the observed behavior of the buoyed objects in the demonstration is as follows. The circular motion requires a net force directed toward the center of curvature. Since the tension exerted by the thread is directed outward, the force responsible for the circular motion can be exerted only by the fluid in which the buoy is immersed. The water contained inside the jar is not at rest in the inertial frame of reference, but rotates together with the jar. Since the water is fluid, not rigid, it is “thrown” to the outside of the turning jar because of its inertia (sometimes called the centrifugal force, which is a misnomer because there is no force that actually “throws” it outward). This rotation causes more water to flow to the outside and therefore the water pressure increases with distance from the rotation axis. Consequently, the force exerted on the liquid near the outer part of the jar is greater than the force acting on the fluid that is closer to the rotation axis. The net force is therefore directed toward the center of curvature which acts on the buoy causing it to move inward toward the center of curvature.