Part II-Basics of Torques



Activity 14 – Torque and Angular Momentum

Studio Physics I

Basics of Torques

Just as was the case for force, torque is a vector and so one must take its direction into account when applying Newton’s Second Law. Torques which act in opposite directions are assigned opposite signs (+ or –) and work against one another. Torques which act in the same direction are assigned the same sign (+ or –) and reinforce one another. Hence, we need to get good at determining the direction of a torque.

Part I – Determining the Direction of Torque Using [pic] (Analysis)

To answer questions regarding direction, DO NOT USE + OR – unless you have clearly defined what direction you mean when you say + or –. Instead, try to use words like up, down, right, left, into the page or out of the page.

1. What is the direction of the angular velocity of the disk shown at the left? (The axis points out of the page.)

2. What is the direction of the angular acceleration of this disk if it is speeding up?

3. What would the direction be of the torque necessary to make this disk spin in the direction shown with an increasing angular speed?

4. What is the direction of the angular acceleration of this disk if it is slowing down?

5. What would the direction be of the torque necessary to make this disk spin in the direction shown with an decreasing angular speed?

6. Consider the pulley shown at the left. (The axis is out of the page.) There is a mass attached to a rope which is wrapped around the pulley. If the mass is let free to fall, what is the direction of the angular velocity of the pulley? What is the direction of the angular acceleration? What is the direction of the torque that the mass and rope exert on the pulley at the point of attachment?

Part II – Determining the Direction of Torque Using [pic] (Analysis)

7. Use the right hand rule to state the direction of [pic] for each case shown below. (Hint: the answer to A is out of the page)

A) B) C) D)

8. Use the right hand rule to state the direction of [pic] for each case shown above. What is the relationship between [pic] and [pic]?

9. Consider the pulley shown in question #6. Draw a set of two arrows, one of which represents the direction of the tension in the string (where it attaches to the pulley) and one of which represents the radius of the pulley. Use these two arrows and [pic] to determine the direction of the torque produced by the tension. Note: The direction of the r vector is from the axis to the point where the force is applied. However, it will help you to determine the direction of the cross product if you draw the r and F vectors with their respective tails at the same point. You can always move a vector when you draw it as long as you preserve its direction. Compare your answer here to your answer for question #6 – they should be the same.

Part III – Angular Momentum of a Point Object (Analysis)

10. Suppose you have a stone of mass 200 grams tied to the end of a rope of length R=0.5 m. The stone is twirled in a horizontal circle of radius 0.5 m as shown in the figure below. The linear velocity of the stone at one instant is also shown. (But the object continues to move in circle the next instant.) What is the magnitude and direction of the stone’s angular momentum at this instant? Hint: [pic] for a point mass.

[pic]

11. Suppose everything about the problem statement above stays the same, except the rope breaks at the instant shown. Hence, the stone moves in a line rather than a circle in the moments that follow the instant shown. What is the magnitude and direction of the stone’s angular momentum at this instant?

12. Consider the stone tied to the end of the rope shown in question #10 above. The rope is not broken and the stone is moving in a circle. The stone has angular momentum — right? Does the stone also have linear momentum even though it is moving in a circle? If so, is it constant or changing with time? Explain (or justify) your answer.

13. Now suppose that the rope has broken and the stone is moving is straight line. (Ignore gravity.) The stone has linear momentum — right? Does the stone also have angular momentum? Explain (or justify) your answer.

14. When we want to talk about the linear position of an object, we have to choose an origin with respect to which we measure the position. (That is, we cannot just say “three meters away”, we have to say “three meters away” from where.) What do you have to choose to define an angular momentum for an object moving linearly? Why don’t you have to make this choice when the object moves in circle?

Part IV – Angular Momentum of a System of Two Objects (Exercise)

15. Consider the system of two point masses moving in the XY plane as shown below.

The mass of particle 1 is m1 = 0.001 kg and its velocity is 20 m/s in the +X direction. The mass of particle 2 is m2 = 0.002 kg and its velocity is 10 m/s in the –X direction. The origin of the coordinate system is at the center of mass of the two-particle system and the coordinates are in meters.

Find all three components (X,Y,Z) of the angular momentum of the system about the origin.

[pic]

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b

a

a

b

a

b

a

b

v

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