Module 5—Circular Motion

Understanding the Forces That Produce Circular Motion in a Vertical Plane

 

Previously, a simulation was used to investigate the direction and magnitude of the velocity, tension, and weight (gravitational force). Next, a free-body diagram (FBD) analysis will be used to determine the mathematical relationships among velocity, tension, and weight.

 

Three special cases will be illustrated:

In all cases, there are only two forces:

Ignore the magenta-coloured velocity vector. It is not part of a true FBD and is only there to indicate speed and direction of motion.

 

Case 1: At the Top of the Arc

 

If necessary, restart the Circular Motion: Vertical simulation. Press “Pause” when the ball reaches the top of the circle (as shown).

 

 

Turn on the vector free-body diagram display using the “Vectors” button (). At the infinitesimal instant that the ball is at the top of the circle, all of the forces are acting vertically.

 

Self-Check

 

SC 4. Define the positive direction as upward, and carefully label the tension force and weight force on the figure. What direction does each of these forces have in this case? Use + or − to label the direction on the diagram.

 

Check your work.
Self-Check Answers

 

SC 4.

 

 

Module 5: Lesson 2 Assignment

 

Remember to submit the answer to TR 2 to your teacher as part of your Module 5: Lesson 2 Assignment.

 

Try This

 

TR 2. The inward force is the force needed to deflect the ball in a circular path. At the top of the arc, Finward is the net force or sum of the weight and tension forces. Use the vector diagram of the ball at the top of the circle to answer the following questions.

 

  1. Using the FBD for the ball at the top of the circle, write an equation for Finward.

    Finward = _______ + _______

  2. What is the direction of the inward force at the top of the arc?

  3. Rewrite the equation of Finward, and include the direction of all of the forces. (Notice that at this infinitesimal instant, all of the forces are along the y-axis; so, you can use + or – to represent direction.)

    _______ = _______ + _______

  4. Manipulate this expression in terms of the tension force (T) in the rod.

    T = _______ + _______

  5. An inward force can also be defined by the expression . Remember that weight is defined as W = mg. Rewrite the equation from TR 2.d. by substituting these expressions.

Explore how speed varies during vertical circular motion by completing this Vertical Circular Motion – Minimum Speed tutorial.

 

In screen 4, after you have made the calculations, click on the rectangles at the bottom of the choices that you think are correct. 

 

Self-Check

 

SC 5. Re-start the Circular Motion: Vertical simulation, if necessary. Click the "Vectors" () box and the “Vertical mode” () button at the top of the screen. Set up the following conditions in the simulation. The mass and radius settings can be adjusted by moving the sliders. Double-clicking the slider or clicking () allows you to enter an exact value for the variable.

Press “Play.” Then press “Pause” when the ball is near the top of the arc. Click the ball, and position it at the exact top of the arc so the angle shows 90.0°.

 

  1. Record the speed at the top of the arc: _______ m/s

  2. Calculate the expected tension in the rod at the top of the arc by doing the following:

    1. State the equation you derived in TR 2.e.
    2. Calculate the tension acting at the top of the vertical loop using the mass, radius, and velocity at the top of the arc.
    3. Verify your answer by checking the tension measurement on the simulation when the ball is positioned at the top of the arc.

    tension measurement: ________ N
Check your work.
Self-Check Answers

 

SC 5.

  1. 8.094 m/s. 

  2.  




    3. tension measurement: 45.9 N

 

Module 5: Lesson 2 Assignment

 

Remember to submit the answers to TR 3 and TR 4 to your teacher as part of your Module 5: Lesson 2 Assignment.

 

Try This

 

TR 3. Set up the following conditions in the simulation. (The mass and radius settings can be adjusted by moving the sliders. Double-clicking the slider or clicking () allows you to enter an exact value for the variable.)

Press “Play.” Then press “Pause” when the ball is near the top of the arc. Click the ball, and position it at the exact top of the arc so the angle shows 90.0°.

 

  1. Record the speed at the top of the arc:  _______ m/s

  2. Calculate the expected tension in the rod at the top of the arc by doing the following:

    1. State the equation you derived in TR 2.e.
    2. Calculate the tension acting at the top of the vertical loop using the mass, radius, and velocity at the top of the arc.
    3. Verify your answer by checking the tension measurement on the simulation when the ball is positioned at the top of the arc.

      tension measurement: _______

TR 4. Set up the following conditions in the simulation. (The mass and radius settings can be adjusted by moving the sliders. Double-clicking the slider or clicking () allows you to enter an exact value for the variable.)

Press “Play.” Then press “Pause” when the ball is near the top of the arc. Click the ball and position it at the exact top of the arc so the angle shows 90.0°.

  1. Record the speed at the top of the arc:  _______ m/s.

  2. Calculate the expected tension in the rod at the top of the arc by doing the following:

    1. State the equation you derived in TR 2.e.
    2. Calculate the tension acting at the top of the vertical loop using the mass, radius, and velocity at the top of the arc.
    3. If the tension in the rod reaches zero at the top of the arc, the object is in a momentary state of free fall. Assuming the tension is zero, use the equation from TR 2.e. to derive an expression for the velocity at the top of the arc when an object undergoes free fall.
    4. Explain what this equation will be used to calculate. Refer to a bucket of water in your explanation.

      tension measurement: _______
Self-Check

 

SC 6. A 2.50-kg ball is attached to a 3.00-m bar and swung in a vertical circle.

 

  1. If the ball does not leave the circular loop, what minimum speed must it have at the top of the arc?

  2. By adjusting the initial velocity slider on the simulation when the radius is set to 3.0 m, find the initial velocity required to make the ball undergo free fall at the very top of the vertical loop.

    initial velocity: _________ m/s

  3. During your exploration in SC 6.b., you may have noticed that the bar can turn red when the ball moves very slowly at the top of the loop. What does this colour change symbolize? (Hint: Look at the tension measurement when the bar changes colour.)
Check your work.
Self-Check Answers

 

SC 6.



  2. initial velocity: 12.1 m/s

  3. According to the tension measurements, when the bar changes to a red colour, the tension has become negative (reversed directions). This symbolizes that the ball would be in free fall if it were attached to a rope instead of a bar.