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How to Calculate the Angular Frequency of a Mass on a Spring

In physics, you can apply Hooke’s law, along with the concept of simple harmonic motion, to find the angular frequency of a mass on a spring. And because you can relate angular frequency and the mass on the spring, you can find the displacement, velocity, and acceleration of the mass.

Hooke’s law says that

F = –kx

where F is the force exerted by the spring, k is the spring constant, and x is displacement from equilibrium. Because of Isaac Newton, you know that force also equals mass times acceleration:

F = ma

These force equations are in terms of displacement and acceleration, which you see in simple harmonic motion in the following forms:

image0.png

Inserting these two equations into the force equations gives you the following:

image1.png

You can now find the angular frequency (angular velocity) of a mass on a spring, as it relates to the spring constant and the mass. You can also tie the angular frequency to the frequency and period of oscillation by using the following equation:

image2.png

With this equation and the angular-frequency formula, you can write the formulas for frequency and period in terms of k and m:

image3.png

Say that the spring in the figure has a spring constant, k, of 15 newtons per meter and that you attach a 45-gram ball to the spring.

The direction of force exerted by a spring.
The direction of force exerted by a spring.

What’s the period of oscillation? After you convert from grams to kilograms, all you have to do is plug in the numbers:

image5.png

The period of the oscillation is 0.34 seconds. How many bounces will you get per second? The number of bounces represents the frequency, which you find this way:

image6.png

You get nearly 3 oscillations per second.

Because you can relate the angular frequency,

image7.png

to the spring constant and the mass on the end of the spring, you can predict the displacement, velocity, and acceleration of the mass, using the following equations for simple harmonic motion:

image8.png

Using the example of the spring in the figure — with a spring constant of 15 newtons per meter and a 45-gram ball attached — you know that the angular frequency is the following:

image9.png

You may like to check how the units work out. Remember that

image10.png

so the units you get from the equation for the angular velocity work out to be

image11.png

Say, for example, that you pull the ball 10.0 centimeters before releasing it (making the amplitude 10.0 centimeters). In this case, you find that

image12.png
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