Chapter 4.8, Problem 28E

Solution

a.

The frequency of the piston.

The frequency of the piston 0.5 cycles/sec.

Given:

Given, the wheel in a piston linkage like the one shown in the figure below has a radius of 18 cm and turns with an angular velocity of p rad/sec. The initial position is the same as shown in the figure:

  

Calculation:

The equation modelling the motion is of the form $d=a\cos \omega t$

The frequency of the motion is given by:

  $\begin{array}{c}f=\frac{\omega }{2\pi }\\ f=\frac{\pi }{2\pi }\\ f=0.5\end{array}$

The frequency of the piston is 0.5 cycles/sec.

Conclusion:

The frequency of the piston is 0.5 cycles/sec.

b.

The equation that models the motion of the piston.

The equation that models the motion of the piston is $d=18\cos \pi t$

Given:

Given, the wheel in a piston linkage like the one shown in the figure below has a radius of 18 cm and turns with an angular velocity of p rad/sec. The initial position is the same as shown in the figure:

  

Calculation:

In the given model, $a=18,\omega =\pi$

Hence,

The equation of the motion of the piston is $d=18\cos \pi t$ .

Conclusion:

The equation of the motion of the piston is $d=18\cos \pi t$ .

c.

The number of cycles the piston makes in 1 min.

The number of cycles the piston makes in 1 minute is 30 cycles.

Given:

Given, the wheel in a piston linkage like the one shown in the figure below has a radius of 18 cm and turns with an angular velocity of p rad/sec. The initial position is the same as shown in the figure:

  

Calculation:

1 minute equals 60 seconds.

The piston makes 0.5 cycles/sec.

Hence, for 60 seconds,

The number of cycles made by the piston in 1 minute is $0.5\times 60=30$ cycles.

Conclusion:

The number of cycles the piston makes in 1 minute is 30 cycles.