Chapter 4.8, Problem 27E

Solution

a.

The frequency of the piston.

The frequency of the piston 8 cycles/sec.

Given:

Given, the wheel in a piston linkage like the one shown in the figure below has a radius of 6 in. It turns with an angular velocity of 16p 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:

  $\begin{array}{c}f=\frac{\omega }{2\pi }\\ f=\frac{16\pi }{2\pi }\\ f=8\end{array}$

The frequency of the piston is 8 cycles/sec.

Conclusion:

The frequency of the piston 8 cycles/sec.

b.

The equation that models the motion of the piston.

The equation that models the motion of the piston is $d=6\cos 16\pi t$

Given:

Given, the wheel in a piston linkage like the one shown in the figure below has a radius of 6 in. It turns with an angular velocity of 16p rad/sec. The initial position is the same as shown in the figure:

  

Calculation:

In the given model, $a=6,\omega =16\pi$

Hence,

The equation of the motion of the piston is $d=6\cos 16\pi t$ .

Conclusion:

The equation that models the motion of the piston is $d=6\cos 16\pi t$

c.

The distance from the initial position 2.85 sec after starting.

The distance from the initial position 2.85 sec after starting is approximately 4.15 in.

Given:

Given, the wheel in a piston linkage like the one shown in the figure below has a radius of 6 in. It turns with an angular velocity of 16p rad/sec. The initial position is the same as shown in the figure:

  

Calculation:

The distance between the positions at $t=0$ and $t=2.85$ is to be determined.

The initial position at $t=0$ is:

  $\begin{array}{c}d\left(0\right)=6\cos 16\pi \left(0\right)\\ d\left(0\right)=6\cos \left(0\right)\\ d\left(0\right)=6\left(1\right)\\ d\left(0\right)=6\end{array}$

The position at $t=2.85$ is:

  $\begin{array}{c}d\left(2.85\right)=6\cos 16\pi \left(2.85\right)\\ d\left(2.85\right)=6\cos \left(45.6\pi \right)\\ d\left(2.85\right)\approx 6\left(0.309\right)\\ d\left(2.85\right)\approx 1.85\end{array}$

The distance between the positions is approximately $6-1.85=4.15$

Conclusion:

The distance from the initial position 2.85 sec after starting is approximately 4.15 in.