1. Rotational Mechanics 1 We know that objects rotate due to the presence of torque acting on the object. The same principles that were true for what caused linear motion is true for rotational motion also. Newton’s 1 st Law stated that an object accelerates when there is an unbalanced force that acts on it. With rotating objects, the same principle applies. When there is a net (unbalanced) torque acting on an object, it will experience an angular acceleration.

2. 2 Rotational Inertia

3. 3 Moments of Inertia for Various Objects gray sphere red sphere rod

4. 4 A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of 1800 rev/s from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor. 53 mN

5. 5 0.385 kgm 2 7.07 kg

6. 6 Rotational Kinetic Energy In fact, some objects store both translational and rotational kinetic energy. Not only does the sphere rotate around the center of mass (pt. R), but the center of mass also translates down the ramp.

7. 7 This sphere is both moving with linear motion and with rotational motion. We have to account for both types of energy when doing our energy equation.

8. 8 8.22 m/s

9. 9 Angular Momentum In the same way that translational kinetic energy has a rotational analog, linear momentum has a rotational counterpart also.

10. 10

11. 11 Objects in a system have angular momentum relative to an axis of rotation even if the object is not actually in a state of rotation. A 34.0-kg girl runs with a speed of 2.80 m/s tangential to the rim of a stationary carousel. The carousel has a mass of 192 kg and a radius of 2.31 m. When the girl jumps onto the carousel, the system begins to rotate. What is the angular speed of the system? 0.317 rad/s girl running carousel girl

12. 12 A student sits at rest on a piano stool that can rotate without friction. The moment of inertia of the student/stool system is 4.1 kgm 2. A second student tosses a 1.5-kg mass with a speed of 2.7 m/s to the student on the stool, who catches it at a distance of 0.40 m from the axis of rotation. What is the resulting angular speed of the system? 0.37 rad/s

13. 13 In summary, these are some of the rotational analogs to linear motion equations. This list, however, is NOT exhaustive. Almost ALL equations that we have worked with in linear motion can have a rotational counterpart constructed.