Problem 47 - Algebra and Trigonometry

Question

In Problems \(47-52,\) 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? $$ d(t)=-20 e^{-0.7 t / 40} \cos \left(\sqrt{\left(\frac{2 \pi}{5}\right)^{2}-\frac{0.49}{1600}} t\right) $$

Step-by-Step Solution

Answer

The motion is a damped harmonic. Initial displacement is -20 meters. As time increases, displacement approaches zero.

Step 1 - Identify given details

From the problem statement, the function describing the motion is given as: \[ d(t)=-20 e^{-0.7 t / 40} \, \cos \left(\sqrt{\left(\frac{2 \pi}{5}\right)^{2}-\frac{0.49}{1600}} \, t\right) \]. The function includes parameters related to the mass and damping factor, which should be identified next.

Step 2 - Describe the motion

The motion is described by the exponential decay function multiplied by a cosine function, representing a damped harmonic motion. - The mass of the bob, denoted by \(m\), is not explicitly given but can be inferred from context. - The damping factor is given in the exponential term as \(0.7 / 40 = 0.0175 \).

Step 3 - Initial displacement

Evaluate the displacement at \( t = 0 \): \[ d(0) = -20 e^{-0.7 \, \cdot \, 0 / 40} \, \cos \left(\sqrt{\left(\frac{2 \pi}{5}\right)^{2} - \frac{0.49}{1600}} \, \cdot \, 0 \right) \]. Simplifying, \[ d(0) = -20 \, \cdot \, 1 \, \cdot \, \cos(0) = -20 \, \cdot \, 1 = -20 \, \text{meters} \].

Step 4 - Graph the motion

Use a graphing utility to plot the function: \[ d(t) = -20 e^{-0.7t / 40} \, \cos \left(\sqrt{\left(\frac{2 \pi}{5}\right)^{2}-\frac{0.49}{1600}} \, t\right) \]. Observe the decreasing amplitude of the cosine oscillations over time.

Step 5 - Displacement at start of second oscillation

The second oscillation starts after one complete period of the cosine function. The angular frequency \( \omega \) is: \[ \omega = \sqrt{\left(\frac{2 \pi}{5}\right)^{2}-\frac{0.49}{1600}} \]. Calculate the period \( T \) as: \[ T = \frac{2 \pi}{\omega} \]. Then evaluate \( d(t) \) at \( t = T \).

Step 6 - Limit of displacement as time increases

Analyze the behavior of the displacement as \( t \rightarrow \infty \): \[ \lim_{t \to \infty} d(t) = -20 e^{-0.7 t / 40} \cos \left(\sqrt{\left(\frac{2 \pi}{5}\right)^{2} - \frac{0.49}{1600}} t \right) \rightarrow 0 \]. Due to the exponential term, the displacement approaches zero over time.

Key Concepts

pendulum motion

When discussing pendulum motion, it is essential to understand that a pendulum consists of a weight (called a bob) suspended from a pivot, which swings back and forth. This movement can be described by simple harmonic motion, especially when the disturbances are small. However, in real-life situations, additional factors such as air resistance and friction at the pivot point influence the motion, making it 'damped'. The given function for the distance of the bob, \[ d(t)=-20 e^{-0.7 t / 40} \, \cos \left(\sqrt{\left(\frac{2 \pi}{5}\right)^{2}-\frac{0.49}{1600}} \, t\right)\bigg)\bigg) \], combines these factors to describe the damped harmonic motion of the pendulum. Here, the cosine function represents the periodic nature of the motion, while the exponential term represents the damping effect, indicating a decrease in the amplitude over time.

damping factor

The damping factor in a damped harmonic motion quantifies how quickly the oscillations decrease due to resistive forces like friction or air resistance. Mathematically, it is identified with the term in the exponential function. In the given exercise, the damping factor is noted as \[ 0.7 / 40 = 0.0175 \]. This small value indicates that the pendulum's oscillations will lose energy slowly. Understanding the damping factor is crucial as it explains why the amplitude of pendulum motion reduces over time, leading eventually to the pendulum coming to a stop.

exponential decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. In the context of pendulum motion, exponential decay applies to the amplitude of the oscillations. The given function \[ -20 e^{-0.7 t / 40} \] shows this exponential decay. It is responsible for the amplitude gradually getting smaller, meaning that over time the pendulum swings get less and less pronounced. This is a reflection of the energy loss due to damping factors, which causes the motion to eventually cease.

cosine function

In harmonic motion, the cosine function is critical for representing the periodic nature of oscillations. For the given pendulum motion, \[ \cos\left(\sqrt{\left(\frac{2 \pi}{5}\right)^{2} – \frac{0.49}{1600}} \ t\right)\right)\right \], quantifies how the position of the pendulum bob varies with time. The cosine function oscillates between -1 and 1, showing that the pendulum moves back and forth symmetrically around the rest position. The period of the oscillation depends on the angular frequency, derived from the term inside the square root. This frequency influences how quickly the pendulum completes one full cycle of motion.