Set 8 Problem number 10

• Important Note on Notation

• Problem

• Solution

• Generalized Solution

Problem

A turbine rotates through 19.49 radians while accelerating uniformly from 5 radians/second to 14.99 radians/second.

• How long does this take, and what is the angular acceleration of the turbine?

Solution

From the two angular velocities and the fact that the angular acceleration is constant we conclude that the average angular velocity is

• `omegaAve = ave angular velocity = ( 5 radians/second + 14.99 radians/second)/2 = 9.994 radians/second.

At this rate the time required to turn through 19.49 radians will be

• ( 19.49 radians)/( 9.994 radians/second) = 1.95 radians/second ( time interval = angular displacement / average angular velocity:   `dt = `d`theta / `omegaAve )

From the two velocities we can also determine that the change in velocity is 9.989 radians/second.

• To accomplish this change in 1.95 seconds requires an acceleration of ( 9.989 radians/second) / ( 1.95 seconds) = 19.47 radians/s^2.

Generalized Solution

If we know

`ds, v0 and vf we can determine vAve, then divide `ds by vAve to find `dt:

• vAve = (v0 + vf) / 2
• `dt = `ds / vAve
• We can calculate the result in two steps or combine the steps to obtain `dt  = `ds / [ (v0 + vf) / 2 ] = 2 `ds / (v0 + vf).

In the present example we know `d`theta, `omega0 and `omegaf.  So we can determine `omegaAve, then divide `d`theta by `omegaAve to find `dt:

• `omegaAve = (`omega0 + `omegaf) / 2
• `dt = `d`theta / `omegaAve
• We can calculate the result in two steps or combine the steps to obtain `dt = `d`theta / [ (`omega0 + `omegaf) / 2 ] = 2 `d`theta / (`omega0 + `omegaf).

Note that the reasoning is identical in the two situations.