8.4

Angular Variables and Tangential Variables

In the familiar ice-skating stunt known as “crack-the-whip”, a number of skaters attempt to maintain a straight line as they skate around the one person (the pivot) who remains in place. Figure 8-10 shows each skater moving on a circular arc and includes the corresponding velocity vector at the instant portrayed in the picture. For every individual skater, the vector is drawn tangent to the appropriate circle and, therefore, is called the tangential velocity . The magnitude of the tangential velocity is referred to as the tangential speed.

Figure 8-10 When doing the stunt known as “crack-the-whip”, each skater along the radial line moves on a circular arc. The tangential velocity of each skater is represented by an arrow that is tangent to each arc.

The physics of “crack-the-whip.”

Of all the skaters involved in the stunt, the one farthest from the pivot has the hardest job. Why? Because, in keeping the line straight, this skater covers more distance than anyone else. To accomplish this, he must skate faster than anyone else and, thus, must have the largest tangential speed. In fact, the line remains straight only if each person skates with the correct tangential speed. The skaters closer to the pivot must move with smaller tangential speeds than those farther out, as indicated by the magnitudes of the tangential velocity vectors in Figure 8-10.
With the aid of Figure 8-11, it is possible to show that the tangential speed of any skater is directly proportional to his distance r from the pivot, assuming a given angular speed for the rotating line. When the line rotates as a rigid unit for a time t, it sweeps out the angle shown in the drawing. The distance s through which a skater moves along a circular arc can be calculated from Equation 8.1, , provided is measured in radians. Dividing both sides of this equation by t gives . The term s/t is the tangential speed vT (e.g., in meters/second) of the skater, while is the angular speed (in radians/second) of the line:

| (8.9)

In this expression, the terms vT and refer to the magnitudes of the tangential and angular velocities, respectively, and are numbers without algebraic signs.

Figure 8-11 During a time t, the line of skaters sweeps through an angle . An individual skater, located at a distance r from the stationary skater, moves through a distance s on a circular arc.

It is important to emphasize that the angular speed in Equation 8.9 must be expressed in radian measure (e.g., in rad/s); no other units, such as revolutions per second, are acceptable. This restriction arises because the equation was derived by using the definition of radian measure, .
The real challenge for the “crack-the-whip” skaters is to keep the line straight while making it pick up angular speed—that is, while giving it an angular acceleration. To make the angular speed of the line increase, each skater must increase his tangential speed, since the two speeds are related according to . Of course, the fact that a skater must skate faster and faster means that he must accelerate, and his tangential acceleration aT can be related to the angular acceleration of the line. If the time is measured relative to t0 = 0 s, the definition of linear acceleration is given by Equation 2.4 as , where vT and vT0 are the final and initial tangential speeds, respectively. Substituting for the tangential speed shows that

Since according to Equation 8.4, it follows that

| (8.10)

This result shows that, for a given value of , the tangential acceleration aT is proportional to the radius r, so the skater farthest from the pivot must have the largest tangential acceleration. In this expression, the terms aT and refer to the magnitudes of the numbers involved, without reference to any algebraic sign. Moreover, as is the case for in , only radian measure can be used for in Equation 8.10.
There is an advantage to using the angular velocity and the angular acceleration to describe the rotational motion of a rigid object. The advantage is that these angular quantities describe the motion of the entire object. In contrast, the tangential quantities vT and aT describe only the motion of a single point on the object, and Equations 8.9 and 8.10 indicate that different points located at different distances r have different tangential velocities and accelerations. Example 6 stresses this advantage.

| Example 6 | | | A Helicopter Blade

A helicopter blade has an angular speed of and has an angular acceleration of . For points 1 and 2 on the blade in Figure 8-12, find the magnitudes of (a) the tangential speeds and (b) the tangential accelerations.

Figure 8-12 Points 1 and 2 on the rotating blade of the helicopter have the same angular speed and acceleration, but they have different tangential speeds and accelerations.

Reasoning Since the radius r for each point and the angular speed of the helicopter blade are known, we can find the tangential speed vT for each point by using the relation . However, since this equation can be used only with radian measure, the angular speed must be converted to rad/s from rev/s. In a similar manner, the tangential acceleration aT for points 1 and 2 can be found using , provided the angular acceleration is expressed in rad/s2 rather than in rev/s2.

Problem solving insight When using Equations 8.9 and 8.10 to relate tangential and angular quantities, remember that the angular quantity is always expressed in radian measure. These equations are not valid if angles are expressed in degrees or revolutions.

Solution

a. | Converting the angular speed to rad/s from rev/s, we obtain

The tangential speed for each point is

| (8.9)

| (8.9)

The rad unit, being dimensionless, does not appear in the final answers.
b. | Converting the angular acceleration to rad/s2 from rev/s2, we find

The tangential accelerations can now be determined:

| (8.10)

| (8.10)

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