12.1. Rolling Motion
Figure 12.1. Rotational Motion of WheelA wheel rolling over a surface has both a linear and a rotational velocity. Suppose the angular velocity of the wheel is [omega]. The corresponding linear velocity of any point on the rim of the wheel is given by
Figure 12.2. Motion of wheel is sum of rotational and translational motion.An alternative way of looking at the motion of a wheel is by regarding it as a pure rotation (with the same angular velocity [omega]) about an instantaneous stationary axis through the bottom of the wheel (point P, Figure 12.3).
Figure 12.3. Motion of wheel around axis through P.
12.2. Kinetic EnergyThe kinetic energy of the wheel shown in Figure 12.3 can be calculated easily using the formulas derived in Chapter 11
Example Problem 12-1
Figure 12.4 shows a disk with mass M and rotational inertia I on an inclined plane. The mass is released from a height h. What is its final velocity at the bottom of the plane ?
The disk is released from rest. Its total mechanical energy at that point is equal to its potential energy
Figure 12.4. Mass on inclined plane.Conservation of mechanical energy implies that Ei = Ef, or
Figure 12.5. Problem 13P.Problem 15P
A small solid marble of mass m and radius r rolls without slipping along a loop-the-loop track shown in Figure 12.5, having been released from rest somewhere along the straight section of the track. From what minimum height above the bottom of the track must the marble be released in order not to leave the track at the top of the loop.
The marble will not leave the track at the top of the loop if the centripetal force exceeds the gravitational force at that point:
Figure 12.6. The yo-yo.Figure 12.6 shows a schematic drawing of a yo-yo. What is its linear acceleration ?
There are two forces acting on the yo-yo: an upward force equal to the tension in the cord, and the gravitational force. The acceleration of the system depends on these two forces:
Figure 12.7. Motion of particle P in the x-y plane.A particle with mass m moves in the x-y plane (see Figure 12.7). A single force F acts on the particle and the angle between the force and the position vector is [phi]. Per definition, the torque exerted by this force on the mass, with respect to the origin of our coordinate system, is given by
12.4. Angular MomentumThe angular momentum L of particle P in Figure 12.7, with respect to the origin, is defined as
A particle can have angular momentum even if it does not move in a circle. For example, Figure 12.8 shows the location and the direction of the momentum of particle P. The angular momentum of particle P, with respect to the origin, is given by
Figure 12.8. Angular momentum of particle P.
Example Problem 12-3
Figure 12.9 shows object P in free fall. The object starts from rest at the position indicated in Figure 12.9. What is its angular momentum, with respect to the origin, as function of time ?
The velocity of object P, as function of time, is given by
Figure 12.9. Free fall and angular momentum
Figure 12.10. Action - reaction pair.If we look at a system of particles, the total angular momentum L of the system is the vector sum of the angular momenta of each of the individual particles:
We conclude that
12.5. Angular Momentum of Rotating Rigid BodiesSuppose we are dealing with a rigid body rotating around the z-axis. The linear momentum of each mass element is parallel to the x-y plane, and perpendicular to the position vector. The magnitude of the angular momentum of this mass element is
12.6. Conservation of Angular MomentumIf no external forces act on a system of particles or if the external torque is equal to zero, the total angular momentum of the system is conserved. The angular momentum remains constant, no matter what changes take place within the system.
The rotational inertia of a collapsing spinning star changes to one-third of its initial value. What is the ratio of the new rotational kinetic energy to the initial rotational kinetic energy ?
The final rotational inertia If is related to the initial rotational inertia Ii as follows
Figure 12.11. Problem 61P.Problem 61P
A cockroach with mass m runs counterclockwise around the rim of a lazy Susan (a circular dish mounted on a vertical axle) of radius R and rotational inertia I with frictionless bearings. The cockroach's speed (with respect to the earth) is v, whereas the lazy Susan turns clockwise with angular speed [omega]0. The cockroach finds a bread crumb on the rim and, of course, stops. (a) What is the angular speed of the lazy Susan after the cockroach stops ? (b) Is mechanical energy conserved ?
Assume that the lazy Susan is located in the x-y plane (see Figure 12.11). The linear momentum of the cockroach is m . v. The angular momentum of the cockroach, with respect to the origin, is given by
12.7. The Precessing Top
Figure 12.12. The precessing top.A top, set spinning, will rotate slowly about the vertical axis. This motion is called precession. For any point on the rotation axis of the top, the position vector is parallel to the angular momentum vector.
The weight of the top exerts an external torque about the origin (the coordinate system is defined such that the origin coincides with the contact point of the top on the floor, see Figure 12.12). The magnitude of this torque is