1. Rolling, Torque, and Angular Momentum Rolling: Translation and Rotation Friction and Rolling Yo-yo Torque: A Cross Product Angular Momentum Newton’s Second Law in Angular Form Angular Momentum Conservation of Angular Momentum Precession of a Gyroscope pps by C Gliniewicz

2. Although many object rotate in place about a fixed axis, other bodies rotate and translate simultaneously such as the wheel of an automobile. As a wheel rolls along the ground, the displacement along the ground is the same as the length of the circumference traveled. The velocity of the center of mass is the same as the tangential velocity of the wheel. Relative to the point on the ground where the wheel is touching, the bottom of the wheel is still. The center of the wheel appears to be moving at the velocity of the car. The top of the wheel appears to be moving at twice the velocity of the car relative to the point on the ground where the wheel is touching. A photograph from a stationary camera, of a wheel rolling past the camera will be blurry at the top of the wheel and nearly in focus at the bottom showing the relative speeds. The total kinetic energy of the rolling wheel is the sum of its translational and rotational kinetic energies. pps by C Gliniewicz

3. If the wheel were a disk, then one can write the formula as If the wheel were accelerating, one can use the fact that If the wheel were acceleration down an incline due to gravity and the force of friction, one realizes that the static friction force is causing the rotation of the wheel and the net force causing the rotation must be acting up the plane because a rolling wheel does not slide. But the direction of rotation of the wheel is opposite the direction of the acceleration of the center of mass, so when one substitutes a negative sign must be included. One now has the acceleration down the plane. pps by C Gliniewicz

4. Recall that previously the torque was defined with the sine of the angle and that it was a vector quantity. One can write the torque as This mean that the value of the torque can be calculated in terms of unit vectors in three dimensions with relative ease. The use of linear momentum and its conservation were important tools to solve problems. In rotation we use angular momentum,. One can calculate the vector in unit vector notation or find the magnitude using the angle between the r and v vectors when placed tail to tail, always using the value less than 180 º. Using the right hand rule one can determine the direction of the angular momentum vector from the plane containing the r and v vectors. Since one has the angular momentum in terms of linear momentum, the angular momentum of a particle passing a fixed point can also be determined. pps by C Gliniewicz

5. Since one has Newton’s second law, it can be rewritten in angular form. Since one knows that One has to be careful to keep the order of the cross products the same when performing these calculations. For a system of particles The angular momentum of a rigid body is In an isolated system, angular momentum is conserved. pps by C Gliniewicz

6. A simple gyroscope consists of a wheel connected to a shaft. The wheel can be made to spin freely about the shaft. If one end of the gyroscope’s shaft is placed on a support in a horizontal configuration while not rotating, it falls due to the torque created by the force of gravity and its unstable equilibrium. If the gyroscope is spinning, however, the torque changes the angular momentum of the gyroscope. A small change in L (dL) in a small time (dt) will produce a small change in the angle of motion (  ) which we call precession. If we solve for the rate of change of , we find the rate of precession. pps by C Gliniewicz