12.7 Rotation About a Fixed Axis

In this section we’ll look at rigid bodies that rotate about a fixed axis. The problem-solving strategy for rotational dynamics is very similar to that for linear dynamics.

PROBLEM-SOLVING STRATEGY 12.1
Rotational dynamics problems

MODEL Model the object as a rigid body.

VISUALIZE Draw a pictorial representation to clarify the situation, define coordinates and symbols, and list known information.

Identify the axis about which the object rotates.
Identify forces and determine their distances from the axis. For most problems it will be useful to draw a free-body diagram.
Identify any torques caused by the forces and the signs of the torques.

SOLVE The mathematical representation is based on Newton’s second law for rotational motion:

τ_{net} = I α or α = \frac{τ_{net}}{I}

Find the moment of inertia in Table 12.2 or, if needed, calculate it as an integral or by using the parallel-axis theorem.
Use rotational kinematics to find angles and angular velocities.

REVIEW Check that your result has correct units and significant figures, is reasonable, and answers the question.

Workbook Exercise 28

EXAMPLE 12.11 Starting an airplane engine

The engine in a small airplane is specified to have a torque of $60 N m$ . This engine drives a 2.0-m-long, 40 kg propeller. On start-up, how long does it take the propeller to reach 200 rpm?

MODEL The propeller can be modeled as a rigid rod that rotates about its center. The engine exerts a torque on the propeller.

VISUALIZE FIGURE 12.28 shows the propeller and the rotation axis.

The figure illustrates an airline propeller in a vertical orientation.

FIGURE 12.28 Full Alternative Text

SOLVE The moment of inertia of a rod rotating about its center is found from Table 12.2:

I = \frac{1}{12} M L^{2} = \frac{1}{12} (40 kg) {(2.0 m)}^{2} = 13.33 {kg m}^{2}

The $60 N m$ torque of the engine causes an angular acceleration

α = \frac{τ}{I} = \frac{60 N m}{13.33 {kg m}^{2}} = 4.50 {rad/s}^{2}

The time needed to reach $ω_{f} = 200 rpm = 3.33 rev/s = 20.9 rad / s$ is

Δ t = \frac{Δ ω}{α} = \frac{ω_{f} - ω_{i}}{α} = \frac{20.9 rad / s - 0 rad / s}{4 {.5 rad/s}^{2}} = 4.6 s

REVIEW We’ve assumed a constant angular acceleration, which is reasonable for the first few seconds while the propeller is still turning slowly. Eventually, air resistance and friction will cause opposing torques and the angular acceleration will decrease. At full speed, the negative torque due to air resistance and friction cancels the torque of the engine. Then $τ_{net} = 0$ and the propeller turns at constant angular velocity with no angular acceleration.