How to find the equation for simple harmonic motion

Problem statement:

A mass M = 3 kg attached to a spring of constant (stiffness) k = 1000 N/m is lying on a frictionless surface. The spring is initially stretched by A = 0.2 m. At time t = 0 the mass is released. Find:

  1. The amplitude, angular frequency, frequency and period of the simple harmonic motion undergone by M.
  2. The position of the mass as a function of time x(t).
  3. The position of the mass at t = 1 s.
  4. The instant t at which the mass first reaches x = 0.
  5. The velocity and acceleration of the mass as a function of time.

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Solution:

In the figure below is shown mass M at t = 0. We will take the coordinate x so that it is negative when the spring is compressed, zero at the natural length (when the spring is not deformed) and positive when the spring is stretched.

Since the force acting on the mass is restorative, when the mass is released it will begin to move to the left until it will reach x = -A. Then it will go back to x = A and the motion will repeat itself cyclically.

The amplitude of the simple harmonic motion undergone by the mass is equal to its initial displacement, because the restoring force is conservative and thus the total energy is conserved. Therefore:

The angular frequency ω of the simple harmonic motion related to the constant k of the spring is given by:

On the other hand, the relationship between the frequency ν, the period T and the angular frequency ω is:

And substituting their values into the formula:


The position as a function of time x(t) is given by:

Where δ is the phase constant of the simple harmonic motion, which can be determined using the initial conditions.

In this problem, at time t = 0 the position of the mass is x = A. Plugging this condition in x(t):

Finally, the equation x(t) is given by:


In order to find the position of the mass at any given time we substitute it into the equation x(t). Keep in mind that, since the unit for the angular frequency ω is rad/s, the argument in the cosine is an angle expressed in radians.

Substituting t = 1 into x(t) we get:

Note that, since the energy is conserved, the position x must be anywhere between x = -A and x = A.

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The time when the mass is passing through the point x = 0 can be found in two different ways. First, knowing that the period of a simple harmonic motion is the time it takes for the mass to complete one full cycle, it will reach x = 0 for the first time at t = T/4:

We can get the same result by substituting x = 0 into x(t):


The velocity of the mass is the first derivative and the acceleration the second derivative of the position function with respect to time:

And substituting the numerical values we get:

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