15.5: Frequency of Spring-Mass System

One interesting characteristic of the simple harmonic motion (SHM) of an object attached to a spring is that the angular frequency, and the period and frequency of the motion, depend only on the mass and the force constant of the spring, and not on other factors such as the amplitude of the motion or initial conditions. We can use the equations of motion and Newton's second law to find the angular frequency, frequency, and period.

Consider a 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; as a result, their sum is 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.

According to Hooke's law, as long as the forces and deformations are small enough, the magnitude of the spring force is proportional to the first power of displacement. Because of this, the spring-mass system is called a linear simple harmonic oscillator.

Substituting the expressions for acceleration and displacement in Newton's second law, the equation for angular frequency can be obtained.

The angular frequency depends only on the force constant and the mass, not the amplitude. It is also related with the period of oscillation using the given relation:

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

Consider a block of mass m connected to a horizontal spring, placed over a frictionless surface.

The net force on the block is the sum of the force due to its weight, the normal force, and the force due to the spring.

Since the weight and the normal force are of equal magnitude and opposite in direction, they cancel each other, and the net force becomes equal to the force due to the spring.

Here, the magnitude of force is proportional to the first power of displacement. Because of this, the spring-mass system is called a linear simple harmonic oscillator.

Using Newton's second law, the force can be expressed in terms of acceleration.

Substituting the expressions for acceleration and displacement, the equation for angular frequency is obtained.

The angular frequency is also defined as 2π over the period of oscillation.

Also, the inverse of the period is the frequency of oscillation.

A stiff spring produces rapid oscillations and a short period. In comparison, a heavy object tends to produce sluggish oscillations and a large period.