Question

Running Machinery Too Fast Suppose that a piston is moving straight up and down and that its position at time \(t\) seconds is $$s=A \cos (2 \pi b t)$$ with \(A\) and \(b\) positive. The value of \(A\) is the amplitude of the motion, and \(b\) is the frequency (number of times the piston moves up and down each second). What effect does doubling the frequency have on the piston's velocity, acceleration, and jerk? (Once you find out, you will know why machinery breaks when you run it too fast.)

Short answer

Doubling the frequency will quadruple the velocity, increase the acceleration by a factor of 8, and jerk by a factor of 16.

Step-by-step solution

Find the Velocity

The velocity of the piston is given by the first derivative of the position function with respect to time (t). Hence, the derivative of \(s=A \cos (2 \pi b t)\) with respect to time yields the velocity function. Using chain rule, the derivative of the cosine function is -sin function multiplied by the derivative of its inside function which gives, \(v=-2\pi bA \sin(2\pi b t)\)

Find the Acceleration

Acceleration can be calculated by taking the derivative of the velocity function. This involves applying the chain rule twice. The acceleration is given by the derivative of \(v=-2\pi bA \sin(2\pi b t)\) which yields \(a=-4\pi^2 b^2 A \cos(2\pi b t)\). This is because the derivative of the sine is the cosine, multiplied by the derivative of its inside function.

Find the Jerk

Jerk, the rate of change of acceleration, can be calculated by taking the derivative of the acceleration function. This involves applying chain rule thrice. So the jerk is the derivative of \(a=-4\pi^2 b^2 A \cos(2\pi b t)\), which gives \(j=8\pi^3 b^3 A \sin(2\pi b t)\).

Analyze changes when doubling frequency

From the above steps, it can be seen that the frequency, \(b\), gets increased power when moving from Position (b's power = 1) to Velocity (b's power = 2), to Acceleration (b's power = 3) and to Jerk (b's power = 4). Therefore, when the frequency, \(b\), is doubled, the velocity is quadrupled, acceleration increases by a factor of 8, and jerk increases by a factor of 16.