Rotational N.2 nd Law

Transcription

1 Lecture 0 Chapter 1 Physics I Rotational N. nd Law Torque Course website:

2 IN THIS CHAPTER, you will continue discussing rotational dynamics Today we are going to discuss: Chapter 1: Parallel-Axis Theorem: Section 1.4 Torque: Section 1.5 Gravitational Torque: Section 1.5 Rotational Newton s nd Law: Section 1.6

3 Parallel Axis Theorem (w/out proof) Moment of inertia depends on the axis of rotation. How can we relate the moment of inertia measured relative to different axis of rotation? I m i r i The moment of inertia about any axis parallel to that axis through the center of mass is given by I I CM Md I : moment of inertia about any parallel axis I CM : moment of inertia about an axis through its center of mass M : total mass d : distance from a parallel axis to the center of mass. CM I CM d I Relative to which axis is the moment of inertia the smallest? BTW: The moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space.

4 Example Parallel Axis Theorem: Sphere For a uniform sphere of radius r 0 Moment of inertia for the sphere, rotating about an axis through its center of mass I CM Mr 5 0 Moment of inertia for the sphere about an axis going through the edge of the sphere? Apply Parallel Axis Theorem: I I CM Md Mr Mr Mr 5 0

5 Rotational Dynamics What causes rotation?

6 F (definition) When we apply the force, the door turns on its hinges (a turning effect is produced). r Axis of rotation It should depend on force, lever arm, angle: rf sin Torque is a turning force (the rotational equivalent of force). Sign of a torque convention: A positive torque tries to rotate an object in a CCW direction A negative torque tries to rotate an object in a CW direction In the 1st case, we are able to open the door with ease. In the nd case, we have to apply much more force to cause the same turning effect. Why? What causes rotation? There are three parameters (lever arm, force, and Arm positioning (angle)) seems to be very helpful in rotating doors. Let s construct a new quantity out of them

7 There are two ways of looking at Torque: F rf sin r F Let s arrange it like this: 1 r( F sin ) r( F ) Perpendicular component of force acting at a distance r from the axis r r F r sin Line of force (action) Or Let s arrange it like this: F(rsin) F( r ) Force times arm lever extending from the axis to the line of force and perpendicular to the line of force

8 ConcepTest The diagram shows the top view of a door, hinge to the left and door-knob to the right. The same force F is applied differently to the door. In which case is the turning ability provided by the applied force about the rotation axis greatest? Closing a Door A B C D E The torque is =Fd sin, and so the force that is at 90 to the lever arm is the one that will have the largest torque. (Clearly, to close the door, you want to push perpendicularly!!) So A or B? B has larger lever arm E A B C D

9 Example Torque A.0-m-long uniform beam (m=5 kg) shown in the figure lies on the frictionless table. Four forces shown in the figure are applied to the beam which can rotate about the axis going through its left end. The second force, F, is applied perpendicularly at the point of the CM. Calculate the net torque about the axis shown in the figure.

10 ConcepTest Loosening Nuts You are using a wrench to loosen a rusty nut. Which arrangement will be the most effective in loosening the nut? A B Because the forces are all the same, the only difference is the lever arm. The arrangement with the largest lever arm (case #) will provide the largest torque. C D E) all are equally effective Follow-up: What is the difference between arrangement A and D?

11 Torque due to Gravity We often encounter systems in which there is a torque exerted by gravity. CM The proof (Read only if you want) R CM W Mg The torque due to gravity is found by treating the object as if all its mass is concentrated at the center of mass. Example MgSin grav R CM An object will balance on a pivot only If the CM is directly above the pivot point. If the pivot point is not under the CM, the grav. torque will cause the object to rotate

12 Newton s nd law of rotation Force causes linear acceleration: (Translational N. nd law) Torque causes angular acceleration: (Rotational N. nd law) F ma I Torque (rotational equivalent of force) Angular acceleration I is the Moment of Inertia (rotational equivalent of mass)

13 Example Falling rod What is the angular acceleration of the rod, if it is released from rest, at the moment it is released? What is the linear acceleration of the tip?

14 Happy Thanksgiving