## Simple Harmonic Motion Solver

Simple Harmonic Motion (SHM) is a periodic vibration or oscillation having the following characteristics:

• The force acting on the object and the magnitude of the object's acceleration are directly proportional to the displacement of the object from its equilibrium position.
• The force vector and the acceleration vector are directed opposite to the displacement vector making both directed toward the equilibrium position.
If an object is allowed to hang from a spring without moving, its weight is equal and opposite to the restoring force of the spring. When the object is pulled down below its equilibrium position, the restoring force of the spring is increased so that it is greater than the weight of the object. When you release the object from this lowered position, it is accelerated upward. After the object passes through the equilibrium position, the restoring force of the spring is less than the weight of the object. This results in a net downward force acting on the object and the object will decelerate until it stops. It then immediately begins to move downward. The net force acting on the object at the end of the spring always acts in a direction to return the object to the equilibrium position. In a similar manner, the object will be acted upon by the same two forces as it continues to move downward through the equilibrium position. A horizontal oscillating spring on a frictionless surface is analagous to a vertical oscillating spring.

The period (T) is the time required for one complete vibration and the frequency (f) is the number of complete vibrations which occur in one second. The amplitude (A) is the maximum displacement from the equilibrium position. A more detailed explanation of the remaining symbols can be found by clicking on the Directions button.

There is no need to enter units with the input values. The Units option box determines the necessary units and the value for g. When the amplitude (A) is assigned a value and the displacement (x) equals zero, the values v, a, Fr, KE, and PE are maximum values. If both the amplitude and displacement are assigned values, the values v, a, Fr, KE, and PE correspond to the displacement. The Pos option box indicates either a horizontal or vertical spring.

When entering a value, you may use an expression. For example, if the amplitude is 5 inches, you may enter 5/12 and the value will be converted to feet.

To provide the most information possible, all possible values are determined. For example, a stationary mass at the end of a spring does not have a period or frequency but they are calculated to include the example if such a mass was pulled down and released.

 T f m k F s h v a Fr KE PE x A ω t
 Units m cm ft Dec Places 0 1 2 3 Pos H V