11.1 Rolling, Torque, and Angular Momentum

Rigid Body

A rigid body is a collection of particles that act as a single-mass system; each component in a rigid body move, rotate, roll, etc all in sync with one another. Think of a series of LEGO blocks connected to one another: in the same way how LEGO blocks are connected and move/stick together, so too does a rigid body move in a similar capacity.

Translational Motion

Motion that moves purely in a linear fashion — either in the x, y, or z direction. There is no angular component attached to translational motion. For example, the center of mass of a wheel will ALWAYS move in a straight, linear path; therefore, the center of mass of a wheel follows translational motion.

Angular Motion

Motion that moves with respect to a change in the central angle of a circular object; for example, any point on the rim of a wheel will always move in a circular path as the central angle changes with respect to time.

Arc Length

The arc length made is the proportion of a total circle that has been swept out as a result of the central angle and the radius of the circle; for example, if the central angle has been swept out by 2pi, then the arc length is 2pi*r (the circumference of an entire circle).

Velocity of Center of Mass

Differentiating the formula s = (theta)(R) provides us with the velocity of the center of mass; recall that s represents the distance traversed, so its derivative must be velocity.

V_com = w * R

W = angular speed (the d(theta)/dt)

R = radial length

Translational AND Rotational Motion

Whenever an object rolls, its motion can be described as a combination of both translational and rotational motion.

We figure out the net effect (the combination) by identifying the instances in which we have a PURE rotational effect and when we have a PURE translational effect; combining these two produces a rolling motion.

Pure Rotation: The velocities of the particles on the edges of a circle will always be in the opposite direction of one another. The direction changing is what results in the opposing sign.

Pure translation: Assuming pure translation, then the entire wheel moves as a rigid body, so EVERY point on the circle will have the same velocity. If we know that the center of the circle has velocity V_Com, then so too do the points on the circumference have a translational velocity of V_com.

Effects of Rolling

The fact that the velocity at the top of a rolling wheel is twice the velocity in the center is why blurs are generated at the top of a wheel and the bottom is clearer.