Math Problem Statement

An object is attached to a coiled spring. It is pulled down a distance of 3 inches from its equilibrium position, then released.

If the time for one complete oscillation is 7 seconds, find the equation of the simple harmonic motion.

Solution

To find the equation of the simple harmonic motion, we need to determine the form of the motion and identify the parameters involved.

For simple harmonic motion, the displacement x(t)x(t) from the equilibrium position can be modeled as:

x(t)=Acos(ωt+ϕ)x(t) = A \cos(\omega t + \phi)

where:

  • AA is the amplitude (the maximum displacement from the equilibrium position).
  • ω\omega is the angular frequency.
  • tt is the time.
  • ϕ\phi is the phase constant.

Given dataTo find the equation of the simple harmonic motion, we need to determine the form of the motion and identify the parameters involved.

For simple harmonic motion, the displacement x(t)x(t) from the equilibrium position can be modeled as:

x(t)=Acos(ωt+ϕ)x(t) = A \cos(\omega t + \phi)

where:

  • AA is the amplitude (the maximum displacement from the equilibrium position).
  • ω\omega is the angular frequency.
  • tt is the time.
  • ϕ\phi is the phase constant.

Given

  1. The amplitude AA is 3 inches (since it is pulled down 3 inches from the equilibrium).
  2. The period TT is 7 seconds (time for one complete oscillation).

We start by calculating the angular frequency ω\omega. The relationship between the period TT and the angular frequency ω\omega is given by:

ω=2πT\omega = \frac{2\pi}{T}

Substituting the given period T=7T = 7 seconds:

ω=2π7\omega = \frac{2\pi}{7}

Next, we need to determine the phase constant ϕ\phi. Since the object is pulled down and then released, it starts at maximum displacement. This corresponds to a cosine function with no phase shift if we start measuring time from the moment of release. Hence, ϕ=0\phi = 0.

Thus, the equation of motion becomes:

x(t)=3cos(2π7t)x(t) = 3 \cos\left(\frac{2\pi}{7} t\right)

In summary, the equation of the simple harmonic motion is:

x(t)=3cos(2π7t)x(t) = 3 \cos\left(\frac{2\pi}{7} t\right)

Would you like any further details or have any questions?

Here are some questions you might ask next:

  1. How do we determine the velocity and acceleration of the object in simple harmonic motion?
  2. Can you explain the energy changes in simple harmonic motion?
  3. What would happen if the period of oscillation were to change?
  4. How does damping affect the equation of simple harmonic motion?
  5. Can we derive the equation of motion using differential equations?

Tip: Remember that in simple harmonic motion, the restoring force is always proportional to the displacement and acts in the opposite direction.

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Math Problem Analysis

Mathematical Concepts

Simple Harmonic Motion
Angular Frequency
Amplitude
Phase Constant

Formulas

Simple Harmonic Motion equation: x(t) = A cos(ωt + φ)
Angular Frequency: ω = 2π / T

Theorems

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Suitable Grade Level

High School