In previous chapters we considered the effects of applying a force to an object that was free to move. In this chapter we study objects that are fixed to some axis so that the object cannot be displaced as a whole when a force is applied, but the object can rotate about that axis.

There are several ways to measure the amount of rotation of an object. Perhaps the simplest is to count the number of complete revolutions of the object about the pivot axis. Just as we were interested in the rate of change of linear position with time in the case of linear motion, we are interested in the rate of change of angular position when we investigate rotational motion. We can describe this in terms of the number of revolutions per second, but this method of measurement is not very satisfactory for subsequent analysis. There is another way of measuring rotational effects in which a circle is divided into 360 equal parts called "degrees." Thus there are 360 degrees (or 360^o) to a complete circle. We could measure the rate of rotation in terms of the number of degrees per second, but this, too, proves to not be a satisfactory unit for subsequent analysis.

We know that the circumference, C, or distance around a circle is given in terms of the radius, r, as C = 2 pi r where pi is a fundamental mathematical constant equal to 3.1416. If we define the measure of an angle as being the length along the arc divided by the radius of the arc, as shown in Figure 8.2 on page 136 in the text, we have a method of measuring angles that does work well in our subsequent analysis. For a complete circle, the length around the arc is called the circumference. This means the angle will be 2 pi r / r = 2 pi.

We refer to the unit of angular measurement defined in terms as the length along the arc divided by the radius as the radian. There are 2 pi (or 6.2832) radians to a complete circle. We stated above that a complete circle has 360^o. This gives us a conversion between the two methods of measuring angles, because 360^o = 2 pi radians. If we divide both sides of the equation by 2 we get 180^o = pi radians as a handy conversion formula from degrees to radians. Because the radian is defined in terms of the length along an arc divided by the radius of the arc, the length units cancel out, and the radian is, strictly speaking, a dimensionless quantity. Nevertheless we use the term radians to emphasize the presence of rotational motion. The rate of change of angular position is measured in radians per second.

There is a simple relationship between the velocity of an object moving in a circular path and the angular velocity, w, with which the object rotates about the pivot axis. The linear velocity tangent to the circle is given as the product of the radius of the circle and the angular velocity or v = r w . This equation is true only if the angular velocity, w, is measured in radians per second. It is not true if the angle is measured in degrees.

We define quantities to describe rotational motion that are analogous to the quantities that we used to describe linear motion. This means we do not have to memorize an entire new set of equations if we can identify what role the respective angular quantities play in the equations of the form used to describe linear motion. Instead of using x for linear position we use q for angular position. Instead of using v for linear velocity we use w for angular velocity. Instead of using a for linear acceleration we use a for angular acceleration. With these substitutions the equations for angular motion, as listed in Table 8.1 on page 145 in the text, now take on the exact same form as those we had for linear motion for position, velocity, and acceleration. The units will be in terms of radians, radians / second, and radians / second², respectively.

A torque about a pivot point is produced by a force acting on a lever arm. The torque, t , is defined as the product of the applied force, F, and the lever arm, l. Torque is a vector quantity for which the complete vector analysis is beyond the scope of the text. We can note the vector effect of a torque by identifying whether it produces a clockwise or a counterclockwise rotation. The conventional choice for the positive direction is to consider counterclockwise torques to be positive, although that choice is not required.

Torque has units of force times length or Newtons times meters. A torque tends to produce a change in the rotational motion of an object.

The concepts of torque and lever arms are used to identify the location of the center of gravity of an object. We know that the distribution of mass on a seesaw is important to the maintaining of balance. If the two persons on the seesaw are not of equal mass and if both persons are located the same distance from the pivot point the seesaw will not work. The product of the mass and the distance from the pivot point is the relevant quantity in this case. The single force at the pivot point is able to support the seesaw to achieve balance when the masses are properly located. So it is with other objects, even people. We can find the single point at which we can imagine all the force of gravity as acting, and we can balance the object by exerting an equal and opposite upward force through this point called the center of gravity, because the weight of the object itself exerts no net torque about the center of gravity. The distribution of the mass is important in this determination, because the mass that is at a greater distance from the pivot point will have a greater effect upon the rotation of the object. If we can locate the center of gravity, in many cases we can treat the object, regardless of its shape and mass distribution, as a point object with all of the mass considered to be concentrated at the center of gravity for purposes of calculating the effect of external forces.

The analogy between translational motion and rotational motion can be pressed a bit further if we consider Newton's Second Law. In the case of linear motion we wrote F_net= m a, where the mass was the proportionality constant relating the linear acceleration to the net force. For angular motion we can write a similar equation relating the net torque, t_net, to the angular acceleration by defining an appropriate proportionality constant. Thus we write t_net= I a where I is defined as the moment of inertia. The moment of inertia takes into account the amount of mass and how that mass is distributed with respect to the axis of rotation.

The moment of inertia depends upon the square of the distance of the mass from the axis of rotation. Fortunately the value of the moment of inertia for regular shaped objects such as disks, rings, and spheres has been calculated and is available in tables in textbooks and reference books. Some representative formulas are given in Figure 8.15 on page 151 in the text.

Because the constant of proportionality between the net torque and the angular acceleration in Newton's Second Law for Rotation is the moment of inertia, the distribution of the mass with respect to the pivot axis is important. Indeed, the moment of inertia depends upon the square of the distance from the axis to the mass, so this is a large effect; the greater the distance from the axis the greater is the moment of inertia. This is why it is more difficult to start a merry-go-round loaded with people than it is to start an empty merry-go-round. There is a large amount of mass at a great distance from the pivot point on a loaded merry-go-round.

In Chapter 7 we defined linear momentum as the product of the mass and the velocity or p = m v. By analogy we define the angular momentum as the product of the moment of inertia and the angular velocity or L = I w. Just as a force produced a change in the linear momentum, a torque produces a change in the angular momentum. When the net force was zero the linear momentum was conserved, so we should not be surprised to learn that when the net torque is zero, the angular momentum is conserved. We learned in a previous chapter that linear momentum is conserved when there is no net external force. Similarly for rotational motion, when there is no net external torque, angular momentum is conserved. Angular momentum depends upon the moment of inertia and the angular velocity, so if there is no net external torque and the moment of inertia is decreased the angular velocity must increase. We see this in the cases of spinning ballerinas or figure skaters when they begin their spins with their arms outstretched and then pull their arms closer to their torsos. When their arms are outstretched the moment of inertia is rather large, because the mass of the arms is located at a great distance from the pivot axis. When the arms are tucked in, the moment of inertia is reduced. The products of the moment of inertia and the angular velocity must be constant if there is no net external torque, so a smaller moment of inertia must lead to a greater angular velocity. Consequently the ballerina and figure skater begin to spin faster whenever they tuck in their arms.

The conservation of angular momentum is responsible for Kepler's Second Law, which was considered in Chapter 5. Equal areas are swept out in equal times by a radius vector between the Sun and a planet because of the conservation of angular momentum. The planets move faster when closer to the Sun in order to have the same angular momentum that they will have when they are farther from the Sun.