Question

You can use any angular measure—radians, degrees, or revolutions—in some of the equations in Chapter 9, but you can use only radian measure in others. Identify those for which using radians is necessary and those for which it is not, and in each case give your reasoning.

Short answer

The units of radians, degrees, or revolutions are mainly used in the measurement of angular kinematic quantities, and radian is only used for linear motion or work-energy.

Step-by-step solution

Concept/Significance of angular displacement

The angular displacement is the shortest angle or distance between the initial and the final position of an object in a circular motion with respect to a fixed point.

Explain the reasoning about the angular measure

The units of radians, degrees, or revolutions are mainly used in the measurement of angular kinematic quantities such as angular displacement, angular velocity, and angular acceleration because for these quantities only the relative distance around a circular path matters, and all three measures relate solely to angular motion.

However, any equation that relates angular motion to something besides angular motion, such as linear motion or work-energy, must use only radians, since angular displacement is technically unit less and radians serve as artificial "placeholder" units which have mathematical significance but no physical significance, and can be dropped without consequence when deriving other quantities.

Equations for which radians must be used instead of degrees or revolutions include tangential acceleration, radial acceleration, and kinetic energy.

Therefore, the units of radians, degrees, or revolutions are mainly used in the measurement of angular kinematic quantities, and radian is only used for linear motion or work-energy.