Objective
If an object moves in a circular path there must be a Centripetal Force acting on it. This experiment determines this Centripetal Force and compares it with the balancing force of gravity on a hanging object.

Equipment
URL: http://www.phy.ntnu.edu.tw/java/circularMotion/circular3D_e.html

Definition
According to Mr. Isaac Newton, an object's "natural state of motion" is to stay at rest if it's already at rest or to continue in linear, uniform motion unless it's subjected to a net, external force. This means that if an object is moving at constant velocity (or speed) in a straight line, it will continue to move in a straight line, at that same velocity, unless some outside force changes its motion in some way.

So in order for an object to move in a circular path, some force is needed to pull it away from the straight-line trajectory it "wants" to follow (i.e., its natural state of motion). Some force needs to pull the rotating object in at every single point along its circular path in order for it continue moving in a circular fashion (instead of allowing it to follow its natural state of motion).

For example, imagine a ball attached to a string that's rotating on a table. The ball "wants" to continue in a straight line. But a force, transmitted via the string, pulls it in to the center at every point along its circular path. This force is called the centripetal force and is symbolized as F_c and is equal to the tension in the string. Mathematically, this force is equal to F_c = mv²/r, where m is the mass of the rotating ball, v is the ball's linear velocity (or speed), and r is the radius of its circular orbit; in words, this is:

                      mass x velocity2
Centripetal_Force = --------------------
                          radius

Now, imagine the string that's connected to the red ball goes through a hole in the table and connects directly to a black ball that simply hangs there. What forces act on the mass of the black ball? The acceleration due to gravity causes a downward force on the mass of the black ball; this downward force on the black ball's mass is commonly called its weight. Weight is different from mass. The weight of an object is equal to its mass times the acceleration due to gravity which can be written mathematically as: w = F_g = mg; in words, this is:

weight = Force_due_to_gravity = mass x acceleration_due_to_gravity

The tension in the string (which is equal to the centripetal force) is now produced by this force of gravity on the black ball's mass. This force is applied by the string and is supplied by the tension due to the weight of the black ball (which is simply the force due to gravity on the black ball mass). In this lab experiment, you will calculate the centripetal force (F_c) a rotating red ball feels; then, you'll compare this centripetal force to the weight of the hanging black ball (w = F_g = mg) to see if they truly are equal (in essence, you're determining if the centripetal force the red ball feels is produced by the force of gravity, or weight, of the black ball).

Procedure

1. Record the time, t, for 10 revolutions
   • Divide t by 10 to determine the period, T
2. Calculate the velocity v
   • Divide the distance traveled (2πr, where r is the radius) by T:
     • v = 2πr/T
3. Compute the centripetal force (F_c) of the rotating mass (the red ball):
   • F_c = m_{red_ball}a = m_{red_ball}v²/r, where m_{red_ball} = mass of rotating particle (the red ball)
4. Compute the Force exerted by the black ball (the hanging mass): weight = F_g = m_{black_ball}g
   • Given w_black, what is the mass_black?
5. Compare the two forces, F_g and F_c
6. Now increase the mass of the black ball (the hanging mass) and repeat steps 1-5 (above)
7. Finally, repeat the process for a 3^rd black ball mass (higher or lower, your choice).

Questions

1. How does the velocity of the rotating particle (the red ball) change as the hanging mass (the black ball) is increased? I.e., as you increase the mass of the black ball, what does the red ball have to do in order to maintain a constant radius of rotation?
   • Compare the velocities for at least two different black ball masses. How well do your values agree with the expected relation (that F_c = mv²/r)?

Notes

• Assume the mass of the rotating red ball to be m_red = 1 kg and the initial radius to be r = 10 m
• Check the "Trace" option to track the path, or trajectory, of the rotating red ball
• Use the "Clean" button to reset the clock and the traced path
• Do not change the L (angular momentum of the red ball) option and ignore the w (little w; should be omega, w), which is simply the angular velocity of the rotating red ball
• Right-click on the applet to pause it; right-click again to un-pause
• To change the mass of the hanging black ball:
  1. Right-click the applet to pause it
  2. Left-click and drag the red ball to increase the Weight of the hanging black ball (the weight of the hanging black ball is shown in the "Mg/m" box)
  3. Right-click again to unpause the applet
• Left-click and drag the hanging black ball to change the radius. Right-click and drag the hanging black ball to have it oscillate
• NOTE: Do not record any data, or fill any tables, directly on the handouts, templates, or scratch paper. Please record ALL data directly in your notebook (if you do write down anything on any scratch paper, staple those sheets directly into your lab notebook).