Problem 48 - Algebra and Trigonometry

Question

The function \(d\) models the distance (in meters) of the bob of a pendulum of mass \(m\) (in kilograms) from its rest position at time \(t\) (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction. (a) Describe the motion of the object. Be sure to give the mass and damping factor. (b) What is the initial displacement of the bob? That is, what is the displacement at \(t=0 ?\) (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? $$ d(t)=-20 e^{-0.8 t / 40} \cos \left(\sqrt{\left(\frac{2 \pi}{5}\right)^{2}-\frac{0.64}{1600}} t\right) $$

Step-by-Step Solution

Answer

a) The mass and damping factor are given, b) Initial displacement is -20 meters, c) Use a graphing utility, d) Calculate numerically for t=5, e) Displacement approaches zero.

Step 1: Identify the function

Given the function: \[ d(t) = -20 e^{-0.8 t / 40} \, \cos \left(\sqrt{\left(\frac{2 \pi}{5}\right)^{2} - \frac{0.64}{1600}} \, t \right) \]This functions models the distance of the bob from its rest position in terms of time.

Step 2: Describe the motion and damping

Consider the general form of a damped oscillatory function: \[ d(t) = A e^{-bt} \, \cos(\omega t) \]Here: - The amplitude factor is \(A = -20\) meters.- The damping factor is \(b = \frac{0.8}{40} = 0.02\). Thus, the bob undergoes a damped oscillatory motion with amplitude 20 meters and damping factor 0.02.

Step 3: Determine initial displacement

Evaluate the function at \(t = 0\):\[ d(0) = -20 e^{-0.8 \cdot 0 / 40} \cos \left(0\right) = -20 \cdot 1 \cdot 1 = -20 \text{ meters} \]Therefore, the initial displacement of the bob is -20 meters.

Step 4: Graph the motion

To graph the function \(d(t)=-20 e^{-0.8 t / 40} \cos \left(\sqrt{\left(\frac{2 \pi}{5}\right)^{2}-\frac{0.64}{1600}} t \right)\), use a graphing utility. Enter the provided function to visualize the damped oscillatory behavior over time.

Step 5: Find displacement at the start of second oscillation

To determine the displacement at the start of the second oscillation, identify the period of the function. The angular frequency is \(\sqrt{\left(\frac{2 \pi}{5}\right)^{2} - \frac{0.64}{1600}}\). Solve for the period \(T\):\[ T = \frac{2 \pi}{\omega} \approx 5 \text{ seconds} \]Thus, at \(t = 5\) seconds (start of the second oscillation): \[ d(5) = -20 e^{-0.8 \cdot 5 / 40} \cos \left(\sqrt{\left(\frac{2 \pi}{5}\right)^{2} - \frac{0.64}{1600}} \cdot 5 \right) \]Calculate this numerically for the exact displacement.

Step 6: Analyze long-term displacement

As time increases without bound (\(t \rightarrow \infty\)), the exponent terms \(e^{-bt}\) approach zero. Thus:\[\lim_{t \to \infty} d(t) = \lim_{t \to \infty} -20 e^{-0.02 t} \cos(\omega t) = 0 \text{ meters} \]The displacement of the bob approaches zero as time increases indefinitely.

Key Concepts

distance function

The given distance function describes the position of a pendulum's bob over time. The function 's initial position.

damping factor

In damped oscillatory motion, the damping factor plays a critical role. It represents the rate at which the motion's amplitude decreases over time due to external forces like friction. The damping factor in our function is computed as follows: The given function is: Since the damping factor gives us information about how quickly the amplitude fades over time.

initial displacement

Initial displacement is the position of the bob at time zero. Plugging in t = 0 into our function: This tells us the initial displacement is -20 meters. It means the pendulum bob starts from a position 20 meters to the left of its central rest position.

graphing utility

To visualize the damped oscillatory motion, use graphing utilities like graphing calculators or software such as Desmos, GeoGebra, or any other relevant tools. Enter the function: Note how the amplitude decays and the motion oscillates about the equilibrium position. You'll see various peaks and troughs that gradually decrease over time, illustrating the damping effect.

long-term behavior

As time progresses, the damping factor causes the oscillations to damp out. Essentially, as t approaches infinity, the term e^{-0.02t} tends to zero, leading the function to zero, meaning: Here, we see that the bob comes to a rest position due to the damping effect.