Problem 97 - University Calculus: Early Transcendentals

Question

Suppose that a piston is moving straight up and down and that its position at time \(t\) sec 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 some machinery breaks when you run it too fast.)

Step-by-Step Solution

Answer

Doubling the frequency doubles velocity, quadruples acceleration, and octuples jerk.

Step 1: Calculate the Velocity Equation

To find the velocity, differentiate the position function with respect to time: \[ v(t) = \frac{ds}{dt} = \frac{d}{dt} [A \cos(2 \pi b t)] = -A (2 \pi b) \sin(2 \pi b t) = -2\pi A b \sin(2 \pi b t) \] This expression represents the velocity of the piston.

Step 2: Effect of Doubling the Frequency on Velocity

If the frequency \( b \) is doubled to \( 2b \), substitute into the velocity equation: \[ v(t) = -2\pi A (2b) \sin(2\pi (2b)t) = -4\pi A b \sin(4\pi b t) \] Doubling the frequency doubles the coefficient of the velocity.

Step 3: Calculate the Acceleration Equation

Differentiate the velocity function to find acceleration: \[ a(t) = \frac{dv}{dt} = \frac{d}{dt} [-2\pi A b \sin(2\pi b t)] = -2\pi A b \cdot 2\pi b \cos(2\pi b t) = -4\pi^2 A b^2 \cos(2\pi b t) \] This expression represents the acceleration of the piston.

Step 4: Effect of Doubling the Frequency on Acceleration

Substitute \( 2b \) for \( b \) in the acceleration equation: \[ a(t) = -4\pi^2 A (2b)^2 \cos(4\pi b t) = -16\pi^2 A b^2 \cos(4\pi b t) \] Doubling the frequency quadruples the coefficient of the acceleration.

Step 5: Calculate the Jerk Equation

Differentiate the acceleration function to find jerk: \[ j(t) = \frac{da}{dt} = \frac{d}{dt} [-4\pi^2 A b^2 \cos(2\pi b t)] = 4\pi^2 A b^2 \cdot 2\pi b \sin(2\pi b t) = 8\pi^3 A b^3 \sin(2\pi b t) \] This expression represents the jerk of the piston.

Step 6: Effect of Doubling the Frequency on Jerk

Substitute \( 2b \) for \( b \) in the jerk equation: \[ j(t) = 8\pi^3 A (2b)^3 \sin(4\pi b t) = 64\pi^3 A b^3 \sin(4\pi b t) \] Doubling the frequency octuples the coefficient of the jerk.

Key Concepts

Velocity

Velocity is the rate at which an object's position changes with time. In our exercise, the position of the piston is given by the equation \( s = A \cos(2 \pi b t) \).

To find the velocity, differentiate this position function with respect to time. The velocity \( v(t) \) becomes:

• \( v(t) = -2\pi A b \sin(2 \pi b t) \)

This expression shows how fast and in what direction the piston moves at any given moment.

When we double the frequency \( b \) to \( 2b \), the velocity function changes:

• \( v(t) = -4\pi A b \sin(4 \pi b t) \)

Doubling the frequency directly doubles the amplitude of the velocity, making each oscillation's peak faster. This increase in velocity means the piston moves up and down more vigorously.

Acceleration

Acceleration is the change in velocity over time. It tells us how quickly the velocity of an object is changing. For the piston, the acceleration can be found by differentiating the velocity function. The derived formula is:

• \( a(t) = -4\pi^2 A b^2 \cos(2\pi b t) \)

Here, the acceleration's amplitude is proportional to the square of the frequency \( b \).

By increasing the frequency \( b \) to \( 2b \), the acceleration equation becomes:

• \( a(t) = -16\pi^2 A b^2 \cos(4\pi b t) \)

This shows that doubling the frequency results in an acceleration four times as great, as the amplitude quadruples. This substantial increase means the piston not only moves quicker but also accelerates faster, exerting more force on the mechanical structure.

Jerk

Jerk is the change in acceleration over time. It shows how quickly the acceleration itself is changing, giving us insight into the smoothness or abruptness of the motion. For the position function of the piston, the jerk can be calculated by differentiating the acceleration function:

• \( j(t) = 8\pi^3 A b^3 \sin(2\pi b t) \)

The term \( b^3 \) indicates that jerk's amplitude depends on the cube of the frequency.

Doubling the frequency \( b \) leads to a corresponding change in the jerk expression:

• \( j(t) = 64\pi^3 A b^3 \sin(4\pi b t) \)

When the frequency is doubled, the amplitude of jerk increases eightfold. This dramatic increase means that rapid changes in acceleration could make machinery experience intense forces that lead to wear or even damage, explaining why machines can break under high-speed operations.