Problem 95 - Thomas Calculus

Question

Running machinery too fast 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 machinery breaks when you run it too fast.)

Step-by-Step Solution

Answer

Doubling the frequency doubles the velocity, quadruples the acceleration, and increases jerk eightfold.

Step 1: Identify the position function

The position of the piston at time \( t \) is given by the function \( s(t) = A \cos(2\pi b t) \). \( A \) is the amplitude and \( b \) is the frequency of the piston.

Step 2: Differentiate to find velocity

The velocity \( v(t) \) of the piston is the first derivative of the position function with respect to time. Differentiate: \[ v(t) = \frac{ds}{dt} = -A \cdot (2\pi b) \cdot \sin(2\pi b t) \]. Thus, \( v(t) = -2\pi b A \sin(2\pi b t) \). Doubling \( b \) results in \( v'(t) = -4\pi b A \sin(4\pi b t) \).

Step 3: Differentiate to find acceleration

The acceleration \( a(t) \) is the derivative of velocity. Differentiate \( v(t) \): \[ a(t) = \frac{dv}{dt} = -A \cdot (2\pi b)^2 \cdot \cos(2\pi b t) \]. So, \( a(t) = -4\pi^2 b^2 A \cos(2\pi b t) \). Doubling \( b \) gives \( a'(t) = -16\pi^2 b^2 A \cos(4\pi b t) \).

Step 4: Differentiate to find jerk

The jerk \( j(t) \) is the derivative of acceleration. Differentiate \( a(t) \): \[ j(t) = \frac{da}{dt} = 8\pi^3 b^3 A \sin(2\pi b t) \]. Therefore, \( j(t) = 8\pi^3 b^3 A \sin(2\pi b t) \). Doubling \( b \) results in \( j'(t) = 64\pi^3 b^3 A \sin(4\pi b t) \).

Step 5: Conclusion: Effects of doubling frequency

Doubling the frequency \( b \) doubles the velocity, quadruples the acceleration, and increases the jerk by a factor of eight. This shows that the motion's impact on the machinery radically increases as the frequency rises, which can lead to mechanical failure when run too fast.

Key Concepts

Velocity

In physics, velocity refers to the rate at which an object changes its position. It's a vector quantity, which means it has both magnitude and direction. When dealing with the mechanics of a piston, the velocity is critical to understanding how fast the piston is moving at any given moment.
To calculate the velocity of the piston, we take the derivative of the position function with respect to time. In this exercise, this function is given by \( s(t) = A \cos(2\pi b t) \). Differentiating it, we find:

\[ v(t) = \frac{ds}{dt} = -2\pi b A \sin(2\pi b t) \]
The formula shows that velocity depends on the frequency \( b \). When frequency \( b \) is doubled, the velocity becomes:

\[ v'(t) = -4\pi b A \sin(4\pi b t) \]
This implies that doubling the frequency of the piston's movement doubles the maximum possible velocity, making the piston cycle faster. This increase in speed can cause stress on mechanical components, potentially leading to failure if not managed properly.

Acceleration

Acceleration measures how quickly velocity changes over time. Like velocity, it's a vector quantity. For the piston, acceleration indicates changes in speed and direction. In simpler terms, it's how quickly the piston speeds up or slows down.
To find acceleration, differentiate the velocity function with respect to time. From the given exercise, using the velocity expression:

\[ a(t) = \frac{dv}{dt} = -4\pi^2 b^2 A \cos(2\pi b t) \]
Acceleration is directly proportional to the square of the frequency \( b \). Therefore, when \( b \) is doubled, acceleration changes to:

\[ a'(t) = -16\pi^2 b^2 A \cos(4\pi b t) \]
Doubling the frequency leads to a fourfold increase in acceleration. This dramatic increase means forces acting on the piston increase significantly, which multiplies the stress experienced by the machinery. As such, high acceleration can increase wear and tear.

Jerk

Jerk is the rate of change of acceleration, simply the derivative of acceleration with respect to time. It is less common in daily conversations about motion, but crucial for understanding sudden shifts in acceleration. Jerk can cause jolting movements.
In the case of the piston, calculating jerk requires differentiating the acceleration function:

\[ j(t) = \frac{da}{dt} = 8\pi^3 b^3 A \sin(2\pi b t) \]
Jerk is cubic in relation to frequency \(b\). Thus, when frequency is doubled:

\[ j'(t) = 64\pi^3 b^3 A \sin(4\pi b t) \]
The jerk increases by a factor of eight. This rapid change can lead to significant impacts on the machine. Increased jerk means sudden shifts in motion, making the system prone to mechanical stress or failure at elevated frequencies.