PYL 106: Pendulum


The Pendulum

A mass hanging from a string or attached to a rigid rod, i.e. a pendulum, is another example of a system which exhibits periodic motion. (In such a case, the mass is known as a bob.) The equilibrium position is when the string or rod hangs vertically. If the pendulum is not in this position, there is a restoring force. One of the quantities studied for a pendulum is the angle between the string and the vertical. Remember the angles we are talking about will be measured in radians.

Recall that the period T is the time taken before the motion starts repeating itself. We want to study the effect on the period of varying

(Note that we will use a mass on the end of a string to study the effect of mass and length variations, and we will use a rigid-rod pendulum to study the amplitude variations.) You should make sure that when varying one parameter, the others are held fixed to whatever extent possible.

The forces acting on the bob of a pendulum are its weight and the tension of the string. It is useful to analyze the pendulum in the radial/tangential coordinate system. The tension lies completely in the radial direction and the weight must be broken into components.

pendulum in radial and tangential components

The net radial force leads to radial acceleration, which is a centripetal acceleration.

Pendulum Equation 1

where the radius of the circular motion is l, the length of the pendulum. (Recall the formula for centripetal acceleration is v2/r.) The tension and the centripetal acceleration are both directed toward the center of the circle.

The net tangential force leads to a tangential acceleration.

Pendulum Equation 2

where the tangential acceleration is l α and α is the angular acceleration, d2θ/dt2. The negative sign indicates that the net force is a restoring force, i.e., that the tangential force is in the opposite direction of the displacement from equilibrium θ.

Another approach to the pendulum is conservation of energy. At an arbitrary angle, the energy is a combination of kinetic energy and gravitational potential energy. At the highest point of the swing, the energy is entirely potential; at the lowest point of the swing, the energy is entirely kinetic (if we take the lowest point to have zero gravitational potential energy).

Pendulum Equation 3

Substituting this expression for vmax which occurs at the equilibrium position into the expression for the tension at the equilibrium position θ=0 yields the following expression for the maximum tension: 

Pendulum Equation 4

which will be tested toward the end of the experiment.


Maximum tension