Welcome back to Physics 211

1. Welcome back to Physics 211 Today’s agenda: Torque Rotational Motion

2. Extended objectsneed extended free-body diagrams • A point free-body diagram allows finding net force since points of application do not matter. • Extended free-body diagrams show point of application for each force and allow finding net torque.

3. Conditions for equilibrium of an extended object For an extended object that remains at rest and does not rotate: • The net force on the object has to be zero. • The net torque on the object has to be zero.

4. Restatement of equilibrium conditions m1r1+m2r2=0  W1r1+W2r2=0 i.e S force (W) x displacement (r) = 0 The quantity force x displacement is called torque (more shortly) Thus, equilibrium requires the net torque to be zero

5. EXAMPLE A 1-kg mass is fastened to a meter stick near one end. A person balances the system by placing a finger directly below point P which is just to the left of the mass. Is the center of mass of the system located 1. to the left of point P, 2. at point P, or 3. to the right of point P? 4. Unable to decide.

6. Tentative definition of torque: The torque on an object with respect to a given pivot point and due to a given force is defined as the product of the force exerted on the object and the moment arm. The moment arm is the perpendicular distance from the pivot point to the line of action of the force.

7. EXAMPLE: A meterstick is pivoted at its center of mass. It is initially balanced. A mass of 200 g is then hung 20 cm to the right of the pivot point. Is it possible to balance the meter-stick again by hanging a 100-g mass from it? 1. Yes, the 100-g mass should be 20 cm to the left of the pivot point. 2. Yes, but the lighter mass has to be farther from the pivot point (and to the left of it). 3. Yes, but the lighter mass has to be closer to the pivot point (and to the left of it). 4. No, because the mass has to be the same on both sides.

8. The Leaning Tower demo • Tower does not fall if the vertical line from its CM lies within base area

9. Reason There are two external forces on the tower: 1) Its weight W: This force effectively goes through the CM of the tower. If we choose the pivot point P at the base edge the weight force results in a torque about P. 2) Normal Force N: The normal force N of the ground on the tower. The force N is upward and on the opposite side of P relative to the weight force W. The torque created by N reinforces the torque created by W. If there is a net torque, equilibrium is not possible

10. Computing torque F |t|=|F|d =|F||r||sinq| =(|F sinq|)|r| Note that F sinθ is the component of the force at 900 to position vector times distance q r d O

11. Interpretation of torque • Measures tendency of any force to cause rotation • Torque is defined with respect to some origin – must talk about ‘torque of force about point X’ etc • Torques can cause clockwise (+) or anticlockwise rotation (-)

12. demo – fighting torque! • hold bar – add weights at different distances • effort increases with distance and magnitude of weight force

13. Conditions for equilibrium of an extended object For an extended object that remains at rest and does not rotate: • The net force on the object has to be zero. • The net torque on the object has to be zero.

14. What if t not zero ? • If the torque about some pivot point is not zero, the object will rotate about the pivot. • Rotation is consistent with direction of force

15. A T-shaped board is supported such that its center of mass is to the right of and below the pivot point. Which way will it rotate? 1. Clockwise. 2. Counter-clockwise. 3. Not at all. 4. Not sure what will happen. CM

19. Rotations about fixed axis • Every particle in a rigid body undergoes circular motion (not necessarily constant speed) with the same time period • v=(2pr)/T=w r. Quantity w is called angular velocity • Similarly can define angular acceleration a=Dw/Dt

20. Vector (or “cross”) product of vectors The vector product is a way to combine two vectors to obtain a thirdvector that has some similarities with multiplying numbers. It is indicated by a cross () between the two vectors. The magnitude of the vector cross product is given by: The direction of the vector AB is perpendicular to the plane of vectors A and Band given by the right-hand rule.

21. Right Hand Rule To get the direction of A x B do the following: Put the fingers of your right hand in the direction of A Then curl these fingers toward the direction of B Then outstretch your right thumb and its direction is that of A x B

22. REMINDER Scalar (or “dot”) product of vectors The scalar product is a way to combine two vectors to obtain a number (or scalar) that has some similarities with multiplying numbers (i.e., a product). It is indicated by a dot (•) between the two vectors.

23. Definition of torque: where r is the vector from the reference point (generally either the pivot point or the center of mass) to the point of application of the force F. where q is the angle between the vectors r and F.

24. The definition of torque T = r x F is our first application of the general concept of the cross product. Previously, we have utilized an application of the dot product in defining the concept of work W produced by a force F: W = F. r where r is the displacement There are many applications of both types of products