Problem 45 - Algebra and Trigonometry

Question

An object of mass \(m\) (in grams) attached to a coiled spring with damping factor \(b\) (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is \(T\) (in seconds) under simple harmonic motion. (a) Find a function that relates the displacement d of the object from its rest position after \(t\) seconds. (b) Graph the function found in part (a) for 5 oscillations using a graphing utility. $$ m=10, \quad a=5, \quad b=0.8, \quad T=3 $$

Step-by-Step Solution

Answer

\(d(t) = 5e^{-0.04t} \times \text{cos}\bigg( \frac{2\text{π}t}{3} \bigg) \)

Step 1: Understanding Simple Harmonic Motion with Damping

The displacement function for an object in simple harmonic motion with damping is given by \[ d(t) = ae^{-bt/2m} \times \text{cos}\bigg( \frac{2\text{π}t}{T} \bigg) \]. Here, - \(a\) is the initial displacement - \(b\) is the damping factor - \(m\) is the mass - \(T\) is the period.

Step 2: Substitute Given Values into the Displacement Function

Substitute \(m = 10\) grams, \(a = 5\) cm, \(b = 0.8\) grams per second, and \(T = 3\) seconds into the formula. \[ d(t) = 5e^{-0.8t/2(10)} \times \text{cos}\bigg( \frac{2\text{π}t}{3} \bigg) \]

Step 3: Simplify the Displacement Function

Simplify the expression by performing the arithmetic operations: \[ d(t) = 5e^{-0.04t} \times \text{cos}\bigg( \frac{2\text{π}t}{3} \bigg) \]

Step 4: Graph the Displacement Function

Use a graphing utility to plot the function \(d(t) = 5e^{-0.04t} \times \text{cos}\bigg( \frac{2\text{π}t}{3} \bigg)\) for 5 oscillations. This can be done by setting the time range from 0 to around 15 seconds since one oscillation period is 3 seconds.

Key Concepts

displacement function

In the context of damped harmonic motion, the displacement function describes how far an object moves from its rest position over time. The formula for a damped harmonic oscillator is given by: \[ d(t) = ae^{-bt/2m} \times \text{cos}\bigg( \frac{2\pi t}{T} \bigg) \] where each term plays a role in describing the motion:

• \textbf{a} is the initial displacement
• \textbf{b} is the damping factor
• \textbf{m} is the mass
• \textbf{T} is the period

In a damped system, unlike a simple harmonic motion, the amplitude of oscillation decreases over time due to the damping effect. This is reflected in the term \[ e^{-bt/2m} \], which introduces an exponential decay, reducing the displacement as time progresses.

damping factor

The damping factor, denoted as \textbf{b}, represents the resistance that reduces the amplitude of oscillation over time. It is a vital parameter that influences how quickly the oscillations die out. In our given problem, \textbf{b} is 0.8 grams per second. The damping factor is part of the exponential term in the displacement function, \[ e^{-bt/2m} \]. As \textbf{b} increases, the system experiences more resistance, and the oscillations decay faster. Conversely, if \textbf{b} is small, the system will oscillate for a longer period before coming to rest. Understanding the damping factor helps in analyzing real-world systems where friction or other resistances play a significant role.

simple harmonic motion

Simple harmonic motion (SHM) refers to a type of periodic motion where an object moves back and forth around a central position. In ideal conditions without damping, the displacement function is described by: \[ d(t) = a \times \text{cos}\bigg( \frac{2\pi t}{T} \bigg) \] In SHM, the motion is purely sinusoidal, and the object oscillates with a constant amplitude and period. However, in real applications, damping effects like friction cause the amplitude to decrease over time, leading to damped harmonic motion. The period \textbf{T} depends on the physical properties of the system and remains constant even in damped systems. Understanding SHM is essential as it provides the foundational concepts that can be extended to more complex phenomena like damped and driven oscillations.