1. Torque

2. Torque and Rotational Acceleration
What happens when you push on a tire toward the axle (axis of rotation)?
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3. Torque and Rotational Acceleration
What happens when you push continuously CCW and tangent to the axle?
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4. Torque and Rotational Acceleration
What happens when you stop applying the force?
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5. Torque and Rotational Acceleration
What happens if while the tire is still turning, you apply a force in the CW direction?
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6. Torque and Rotational Acceleration
In summary: An external force that produces zero torque does not change the tire’s rotation rate. When there is no force exerting torque on a tire, its rotational velocity remains constant. An external force that produces torque, causes the tire to accelerate in the direction of the torque (it can speed up or slow down).
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7. Net Force and Net Torque Comparison
Linear: A nonzero net force needs to be exerted on an object to cause its velocity to change. The greater the net force, the greater the linear acceleration. The net force and acceleration will always be in the same direction.
Rotational: A nonzero net torque must act on an object to cause rotational acceleration. The greater the net torque, the greater the rotational acceleration. The net torque and rotational acceleration will always be in the same direction. Negative torque will be in the clockwise direction and positive torque will be in the counterclockwise direction

8. Rotational inertia
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9. Rotational inertia
Rotational inertia is the physical quantity characterizing the location of the mass relative to the axis of rotation of the object.
The closer the mass of the object is to the axis of rotation, the easier it is to change its rotational motion and the smaller its rotational inertia.
The magnitude depends on both the total mass of the object and the distribution of that mass about its axis of rotation.
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10. Analogy between linear motion and rotational motion
Let’s look at some formulas…
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12. Expressions for the rotational inertia of standard-shape objects
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13. Expressions for the rotational inertia of standard-shape objects
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14. Newton’s Second Law

15. Newton's second law for rotational motion applied to rigid bodies
The rotational inertia of a rigid body about some axis of rotation is the sum of the rotational inertias of the individual point-like objects that make up the rigid body.
The rotational inertia of this two-block rigid body is twice the rotational inertia of the single block.
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16. Calculating rotational inertia
The rotational inertia of the whole leg is:
There are other ways to perform the summation process. Often it is done using integral calculus, and sometimes it is determined experimentally.
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17. Expressions for the rotational inertia of standard-shape objects
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18. Expressions for the rotational inertia of standard-shape objects
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19. Rotational form of Newton's 2nd Law

20. Tip

21. A 5m, 450N, beam on a hinge is suspended by a single cable
A 5m, 450N, beam on a hinge is suspended by a single cable. The beam’s center of mass is 3m from the hinge along the beam. What is the tension in the cable?
Ex:

22. Example 8.3
A 60-kg rollerblader holds a 4.0-m-long rope that is loosely tied around a metal pole. You push the rollerblader, exerting a 40-N force on her, which causes her to move increasingly more rapidly in a counterclockwise circle around the pole. The surface she skates on is smooth, and the wheels of her rollerblades are well oiled. Determine the tangential and rotational acceleration of the rollerblader.
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23. Example 8.3

24. Example 8.3

25. Example 8.6
A woman tosses a 0.80-kg soft-drink bottle vertically upward to a friend on a balcony above. At the beginning of the toss, her forearm rotates upward from the horizontal so that her hand exerts a 20-N upward force on the bottle. Determine the force that her biceps exerts on her forearm during this initial instant of the throw. The mass of her forearm is 1.5 kg and its rotational inertia about the elbow joint is kg•m2. The attachment point of the biceps muscle is 5.0 cm from the elbow joint, the hand is 35 cm away from the elbow, and the center of mass of the forearm/hand is 16 cm from the elbow.
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26. Example 8.6
ANSWER: 240N