Galileo’s Paradox—Hinged Stick vs. Falling Ball

Introduction

In the absence of air resistance, all free-falling objects, regardless of their mass will accelerate toward the Earth at the same rate. This “law of motion” was proposed by Galileo Galilei (1564–1642) and later confirmed by Isaac Newton (1642–1727). However, in this discrepant event, a falling stick appears to break the laws of motion as it “falls” faster than a ball.

Concepts

• Acceleration due to gravity
• Torque
• Rotational acceleration
• Center of mass

Safety Precautions

Although this activity is considered non-hazardous, please follow normal laboratory safety guidelines.

Prelab Preparation

1. Attach the clamp and bracket to the 1.2 m stick as shown in Figure 1. The bracket needs to hold the stick firmly but do not fully tighten the nut and bolt. The stick will need to swing freely during the demonstration.
   {12591_Preparation_Figure_1}
2. Triple-loop the rubber band around the free end of the hinged stick (see Figure 2).
   {12591_Preparation_Figure_2}
3. Once the rubber band is twisted tightly around the stick, push the twisted rubber band down the stick until it is one inch from the end (see Figure 2c).
4. Connect the clamp and stick to a support stand as shown in Figure 3. Use the foam block as a base and connect the stick near the bottom of the support stand so the stick is parallel to the tabletop. The foam block acts as a cushion when the stick falls.
   {12591_Preparation_Figure_3}

Procedure

1. Raise the stick up to an angle of 30° from the horizontal. Use a protractor to measure the angle (see Figure 4).
   {12591_Procedure_Figure_4}
2. Use the support stick to hold the hinged stick at this 30° angle. Add a small piece of clay to the base of the support stick to prevent it from slipping on a smooth tabletop.
3. Place the Velcro-covered ball behind the rubber band on the stick so that the ball will remain balanced on the end of the hinged stick. The bottom of the ball and the upper lip of the cup should be at the same height, or the ball can be slightly lower than the lip of the cup (see Figure 4).
4. Ask the students to predict what will happen when the support stick is pulled from beneath the hinged stick. How will the ball fall? Will it stay on the stick? Will the ball free-fall without being affected by the hinged stick? Will the ball land in the cup? How will the stick fall? Is the stick really in free-fall?

   The most common response is that the ball and stick will fall at the same rate because of Newton’s law of gravity, especially if this topic was just covered in class. Since the ball’s height is at the same level as the lip of the cup, the ball cannot land in the cup because they should remain at the same relative height to one another as they fall. For students who predict either the stick or the ball will fall at a faster rate, ask them explain their reasoning.

5. After the initial discussion and student predictions, quickly pull the support stick out from beneath the hinged stick. For best results, pull the support stick out horizontally, away from the hinge so that the pulled stick will not "hang up" the falling stick. Avoid hitting the foam block when pulling out the stick (see Figure 5).
   {12591_Procedure_Figure_5}
6. Observe the ball land in the cup. What does this observation indicate? Did the stick fall faster than the ball? Does this violate Galileo’s observations and Newton’s laws of motion?

   Since the ball started out at the same height as the upper lip of the cup, the stick must have fallen faster than the ball. If the ball and stick had fallen at the same rate, the lip of the cup would have hit the ball as it swung down and would have knocked the ball away.

7. Discuss the results with your students.

Teacher Tips

• This is an excellent demonstration to perform before or after discussing Newton’s laws of motion, torque and/or angular momentum.
• If the Velcro^® disk detaches from the base of the cup, use glue or tape to secure it again.
• Occasionally, the connection between the clamp and the bracket may become loose and the stick will twist, making it difficult to balance. It is helpful to keep a Phillips screwdriver handy to tighten the screw, securing the bracket to the clamp.
• Do not discard the shipping tube. This large tube can be used as a storage container for meter sticks or posters. Or use it as a “thunder” tube when teaching about resonance. There are many uses for a long cardboard tube in a science classroom.
• For more advanced classes, students may experiment to determine the release angle when the ball begins to fall faster than the stick. A student should hold the ball next to the stick and release them at the same time, as a lab partner observes the separate motion of the ball and stick. Students should record their observations for each initial angle of the stick, and determine the transition angle when the ball begins to fall faster than the stick. Students may inquire into the effects of air drag on the stick compared to the ball. Lighter or heavier balls may be used to show that the air resistance has negligible effects on the falling rate of either the ball or the stick.
• The mathematics of the falling hinged stick is nonlinear and involves complex elliptical equations beyond the scope of this discussion. However, the Discussion section provides a basic foundation for the movement of the falling hinged stick.
• The Guinea and Feather Tube (Flinn Catalog No. AP4670), Ring and Discs (AP4634) and the Shoot the Monkey Demonstration Kit (AP6439), available from Flinn Scientific, are excellent supplemental demonstrations that show Newton’s laws and constant acceleration due to gravity.

Discussion

This demonstration reveals an apparent contradiction to Galileo's original observations—all masses fall with the same acceleration. However, a careful examination of the apparatus shows that the stick and ball are not falling in the same manner. The falling stick is not truly in "free-fall" motion. The stick is attached at one end.

Gravity pulls down on all particles in an object equally. The net sum of all these forces can be assumed to be a single point in the object known as the center of mass. Therefore, gravity pulls down on the center of mass of the ball and the center of mass of the stick with a constant acceleration (see Figure 6).

However, since the stick is attached to a support stand by a hinge, the stick rotates about its attached end as it falls, instead of falling straight down as the ball does. Therefore, the stick falls with an angular acceleration (α_s) instead of linear acceleration. Angular acceleration is the acceleration of a rotating object. Angular acceleration (α_s) and tangential linear acceleration (a_t) are proportional according to Equation 1 (see Figure 7).

a_t = linear acceleration
r = distance the accelerating point is from pivot point
α_s = angular acceleration

The center of mass of the stick is still acted upon by the force of gravity and is pulled downward with a constant acceleration (see Figure 6). Therefore, the constant angular acceleration of the stick is given by Equation 2.

α_s = angular acceleration of the stick
g = constant acceleration due to gravity
r_cm = distance of center of mass from the pivot point (hinge)

Combining Equations 1 and 2 yields Equation 3 shows that any point on the board beyond the center of mass (r > r_cm) will have a linear downward acceleration that is faster than the center of mass (faster than the acceleration due to gravity, g). Points on the hinge-side of the center of mass (r < r_cm) have a slower downward acceleration than the center of mass. It is important to remember, however, that all the points on the stick still have the same, constant, angular acceleration, α_s. Therefore a cup placed beyond the center of mass of the stick falls with a faster acceleration than a free-falling ball. When both the cup and the ball are released from the same height, the cup will beat the ball to the benchtop.

There is a limit to the angle at which the stick can be released and still beat the ball. Unfortunately, this requires a thorough investigation of torque and angular acceleration which is explained in Further Mathematical Discussion. Simply stated, for initial angles between 0 and 35º, the end of the board will always beat the ball down when released from the same height. At angles greater than 35º, the ball will hit the ground first (as long as it does not hit the board as it falls).

Further Mathematical Discussion
To determine the “break-even” angle, the torque on the rotating stick produced by the pull of gravity must be analyzed. Torque, τ, is the “force-equivalent” for a rotating system and is equal to the moment of inertia of the system, I, multiplied by the angular acceleration, α. This is similar to the linear equation of force, F = ma. A torque produced by a force acting on a rotating lever arm is equal to the force multiplied by the distance between the pivot point and the position where the force acts, multiplied by the sine of the angle between the applied force and the lever arm (Equation 5). See Figure 8. For a rotating stick pivoting at one end, the moment of inertia equal to (mL²)/2 – 43, where L is the length of the stick. The torque about the center of mass of the stick (L/2) is, therefore, given by Equation 6. Solving for the angular acceleration gives Referring to Equation 1, the tangential linear acceleration at the end of the stick is equal to the distance from the pivot point (the length of the stick, L) multiplied by the angular acceleration. The vertical component of the tangential acceleration (a_vert) of the end of the board is equal to a_t multiplied by cos ϴ (see Figure 9). Note: sin δ is equal to sin (90 – ϴ), which also equals cos ϴ.
For the special case when the end of the stick has the same vertical acceleration as the falling ball, a_vert = g, the initial angle of the board is equal to: As the stick's release angle increases, it can be seen that the initial movement (rotation) of the board is slower compared to the quickness at which the ball falls. For small initial angles, the faster acceleration of the end of the stick is large enough to overcome its slow start. This is not the case for larger angles. The “break-even” initial angle at which the end of the stick falls with the same vertical acceleration as the ball is 35°. The end of the stick beats the ball when the initial angle is smaller than 35°. If the initial angle of the stick is larger than 35°, the end of the stick will not beat the ball to the benchtop. During the demonstration, it may be observed that the ball will not lose contact with the board when the initial angle is greater than 35°, so it may appear that the ball and board fall with the same acceleration. The ball is actually forcing the stick to fall with the same acceleration as the ball rides above it.