## Pendulum Solver

Simple Harmonic Motion (SHM) is a periodic vibration or oscillation with 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 string without moving, its weight is equal and opposite to the tension in the string. When the object is pulled to the left or right of its equilibrium position and released, the weight of the object pulls straight down while the tension in the string pulls upward at an angle. These two concurrent forces produce a resultant or net force that accelerates the mass almost horizontally. When the mass returns to its equilibrium position, the weight of the object and the tension in the string are equal and opposite so they cancel out. Because there is no net force, the acceleration of the mass is zero but it has attained its maximum velocity. As soon as the weight passes through the equilibrium position, there again is an unbalanced force acting on the weight. However, this force is again acting toward the equilibrium position opposing the motion, so the weight slows down until it reaches zero velocity.

To summarize, this net force or restoring force depends on the displacement from the equilibrium position and is always directed toward the equilibrium position. The weight has its maximum acceleration at the amplitude (where it is changing direction) and a zero velocity. As the weight passes through the equilibrium position, it has a zero acceleration, a zero restoring force, and its maximum velocity.

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. If you are solving for g, you must indicate this by entering a "-1" for the value of g.

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.

 T f l m g v a Fr KE PE x A
 Units m cm ft Dec Places 0 1 2 3