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The period and frequency of a mass on a spring

One interesting characteristic of the SHM of an object attached to a spring is that the angular frequency, and therefore the period and frequency of the motion, depend on only the mass and the force constant, and not on other factors such as the amplitude of the motion. We can use the equations of motion and Newton’s second law ( F net = m a ) to find equations for the angular frequency, frequency, and period.

Consider the block on a spring on a frictionless surface. There are three forces on the mass: the weight, the normal force, and the force due to the spring. The only two forces that act perpendicular to the surface are the weight and the normal force, which have equal magnitudes and opposite directions, and thus sum to zero. The only force that acts parallel to the surface is the force due to the spring, so the net force must be equal to the force of the spring:

F x = k x ; m a = k x ; m d 2 x d t 2 = k x ; d 2 x d t 2 = k m x .

Substituting the equations of motion for x and a gives us

A ω 2 cos ( ω t + ϕ ) = k m A cos ( ω t + ϕ ) .

Cancelling out like terms and solving for the angular frequency yields

ω = k m .

The angular frequency depends only on the force constant and the mass, and not the amplitude. The angular frequency is defined as ω = 2 π / T , which yields an equation for the period of the motion:

T = 2 π m k .

The period also depends only on the mass and the force constant. The greater the mass, the longer the period. The stiffer the spring, the shorter the period. The frequency is

f = 1 T = 1 2 π k m .

Vertical motion and a horizontal spring

When a spring is hung vertically and a block is attached and set in motion, the block oscillates in SHM. In this case, there is no normal force, and the net effect of the force of gravity is to change the equilibrium position. Consider [link] . Two forces act on the block: the weight and the force of the spring. The weight is constant and the force of the spring changes as the length of the spring changes.

An illustration of a vertical spring attached to the ceiling. The positive y direction is upward. In the figure on the left, figure a, the spring has no mass attached to it. The bottom of the spring is a distance y sub zero from the floor. In the middle figure, figure b, the spring has a mass m attached to it. The top of the spring is at the same level as in figure a, but the spring has stretched down a distance delta y, so that the bottom of the spring is now a distance y sub 1 equals y sub zero minus delta y from the floor. On the right, figure c, a free body diagram of the mass is shown with downward force m g and an upward force F sub s that equals k delta y which also equals k times the quantity y sub zero minus y sub 1.
A spring is hung from the ceiling. When a block is attached, the block is at the equilibrium position where the weight of the block is equal to the force of the spring. (a) The spring is hung from the ceiling and the equilibrium position is marked as y o . (b) A mass is attached to the spring and a new equilibrium position is reached ( y 1 = y o Δ y ) when the force provided by the spring equals the weight of the mass. (c) The free-body diagram of the mass shows the two forces acting on the mass: the weight and the force of the spring.

When the block reaches the equilibrium position, as seen in [link] , the force of the spring equals the weight of the block, F net = F s m g = 0 , where

k ( Δ y ) = m g .

From the figure, the change in the position is Δ y = y 0 y 1 and since k ( Δ y ) = m g , we have

k ( y 0 y 1 ) m g = 0 .

If the block is displaced and released, it will oscillate around the new equilibrium position. As shown in [link] , if the position of the block is recorded as a function of time, the recording is a periodic function.

If the block is displaced to a position y , the net force becomes F net = k ( y y 0 ) m g = 0 . But we found that at the equilibrium position, m g = k Δ y = k y 0 k y 1 . Substituting for the weight in the equation yields

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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