Problem 42 - 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. $$ m=20, \quad a=15, \quad b=0.75, \quad T=6 $$

Step-by-Step Solution

Answer

The displacement function is given by \[ d(t) = 15e^{-0.01875t} \cos\left(\sqrt{\frac{\pi^2}{9} - 0.0003516} \cdot t \right) \].

Step 1: Identify the Differential Equation

The motion of an object in a damped harmonic oscillator is described by the differential equation \[ m \frac{d^2d}{dt^2} + b \frac{dd}{dt} + kd = 0 \] where \( d \) is the displacement, \( m \) is the mass, \( b \) is the damping coefficient, and \( k \) is the spring constant.

Step 2: Find the Spring Constant

Use the period \( T \) to find the spring constant \( k \). The period of a simple harmonic motion is given by \[ T = 2 \pi \sqrt{\frac{m}{k}} \] Rearrange to solve for \( k \): \[ k = \frac{4 \pi^2 m}{T^2} \] Given \( T = 6 \) seconds and \( m = 20 \) grams, \[ k = \frac{4 \pi^2 (20)}{6^2} = \frac{4 \pi^2 (20)}{36} = \frac{20 \pi^2}{9} \]

Step 3: Identify the Form of the Solution

The solution to the differential equation for a damped oscillator is of the form \[ d(t) = Ce^{-\gamma t} \cos(\omega t + \theta) \] where \( \gamma = \frac{b}{2m} \) and \( \omega = \sqrt{\frac{k}{m} - \gamma^2} \). Let's find \( \gamma \) and \( \omega \).

Step 4: Calculate Damping Factor \(\gamma\)

Substitute the given values for \( b \) and \( m \): \[ \gamma = \frac{b}{2m} = \frac{0.75}{2(20)} = \frac{0.75}{40} = 0.01875 \]

Step 5: Calculate \( \omega \)

Using the calculated value of \( k \) and \( \gamma \), solve for \( \omega \): \[ \omega = \sqrt{\frac{k}{m} - \gamma^2} = \sqrt{\frac{\frac{20 \pi^2}{9}}{20} - (0.01875)^2} = \sqrt{\frac{\pi^2}{9} - 0.0003516} \] Simplify further to get the numerical value for \( \omega \).

Step 6: Find the General Solution

With \( \gamma \) and \( \omega \) known, plug them into the general form of the solution: \[ d(t) = 15e^{-0.01875t} \cos(\omega t + \theta) \] The initial conditions (initial displacement \( a = 15 \) and initial velocity 0) can be used to solve for \( \theta \) and any constant if needed. For simplicity, we assume \( \theta = 0 \) given the problem’s information.

Step 7: Function Simplification

Finally, present the simplified function for displacement: \[ d(t) = 15e^{-0.01875t} \cos\left(\sqrt{\frac{\pi^2}{9} - 0.0003516} \cdot t \right) \]

Step 8: Plot the Graph

Using a graphing utility, plot the function \( d(t) \) over time for 5 oscillations. Set the time axis appropriately to cover approximately 30 seconds (since each oscillation takes around 6 seconds).

Key Concepts

differential equations

Differential equations are a kind of equation that involve functions and their derivatives. In the context of damped harmonic motion, we focus on a second-order linear differential equation to describe the motion. The general form for the damped oscillator is:
\[ m \frac{d^2d}{dt^2} + b \frac{dd}{dt} + kd = 0 \]

• \(m\) represents the mass of the object.
• \(b\) is the damping coefficient.
• \(k\) is the spring constant.
• \(d\) is the displacement from the rest position.

Understanding this differential equation is crucial because it frames how the object's motion will behave over time. The terms combine to reflect the impact of mass, damping, and the spring force on the motion.

spring constant

The spring constant (denoted as \( k \)) measures the stiffness of the spring. A higher spring constant means a stiffer spring that resists displacement more strongly. For our problem, we use the given period \( T \) (time for one complete cycle) to find \( k \). The relationship between the period and spring constant in simple harmonic motion is given by:
\[ T = 2 \pi \sqrt{\frac{m}{k}} \]
Rearranging to solve for \( k \):
\[ k = \frac{4 \pi^2 m}{T^2} \]
Substituting our values (\( m = 20 \) grams and \( T = 6 \) seconds):
\[ k = \frac{4 \pi^2 (20)}{6^2} = \frac{20 \pi^2}{9} \]
This result gives us the specific spring constant required to achieve the given period for the motion.

damping factor

The damping factor (\(b\)) quantifies the resistance force acting opposite to the velocity. It gradually reduces the amplitude of the oscillations over time, leading to a damped motion instead of perpetual motion. The formula for the damping factor is:
\[ \gamma = \frac{b}{2m} \]
For our problem, substituting the given values (\(b = 0.75\) grams/second and \(m = 20\) grams):
\[ \gamma = \frac{0.75}{2(20)} = 0.01875 \]
A small damping factor means light damping, where the system oscillates several times before stopping. Conversely, a large \(b\) would exhibit heavy damping, where oscillation quickly ceases.

harmonic oscillator

A harmonic oscillator refers to a system where the force acting to restore an object to its equilibrium position is directly proportional to the displacement. In our case, the equation governing a simple harmonic oscillator is modified by the damping factor:
\[ m \frac{d^2d}{dt^2} + b \frac{dd}{dt} + kd = 0 \]
In a non-damped scenario, the equation simplifies to:
\[ m \frac{d^2d}{dt^2} + kd = 0 \]
The introduction of damping results in complex solutions for displacement known as damped harmonic motion, characterized by both oscillatory and non-oscillatory exponential decay terms.

simple harmonic motion

Simple harmonic motion (SHM) describes the ideal situation where the object's restoring force is proportional to its displacement and acts in the opposite direction. It follows the relation:
\[ d(t) = A \cos(\omega t + \theta) \]
With:

• \(A\) as the amplitude (maximum displacement),
• \(\omega\) as the angular frequency,
• \(\theta\) as the phase constant.

However, when damping is present, the motion isn't purely sinusoidal. Mathematically, damped SHM is expressed as:
\[ d(t) = Ce^{-\gamma t} \cos(\omega t + \theta) \]
This equation includes an exponential decay term \(e^{-\gamma t}\), which represents the amplitude reduction over time due to damping.

displacement function

The displacement function describes how far from the rest position the object is at any given moment. For a damped harmonic motion, the function takes the form:
\[ d(t) = Ce^{-\gamma t} \cos(\omega t + \theta) \]
Here:

• 
  \(C\) is the initial amplitude, depending on initial conditions.


• \( \gamma\) is the damping factor.


• \( \omega\) is the modified angular frequency.


• \(\theta\) is the phase displacement (often taken as 0 in initial conditions).

For our solution, the displacement function was simplified to:
\[ d(t) = 15e^{-0.01875t} \cos \left( \sqrt{\frac{\pi^2}{9} - 0.0003516} \cdot t \right) \]
This reflects the combined oscillatory and exponential decay behavior of the damped motion.