The Simple Pendulum

Simple Pendulum: there is a point mass that is attached to a string of negligible mass that does not stretch. The string is attached to a support system that is frictionless. The motion of the simple pendulum is periodic, as the pendulum swings back and forth in one direction due to the force of gravity. This is only when it is released from a certain vertical height.

Above is a depiction of a simple pendulum where the green ball is the mass that is attached to the string of length L.

The displacement (s) is represented and related to the angular displacement as follows:

s=LθNote:

Displacement to the right of the equilibrium position is positive Displacement to the left of the equilibrium position is negative

Forces: Acting on the mass mg Tension from the string T

Axis Systems: Radial axis along the length of the string Tangential axis tangent to the circular motion of the mass Both axis are perpendicular to each other and their directions change as the

mass oscillates

The Radial Axis:

The mass (green ball) makes an angle θ with the radial axis The mass along the radial axis is mgcosθ and it points away from the

suspension pointT−mgcosθ=0T=mgcosθ

The Tangential Axis The weight is mgsinθ Using Newton’s 2nd law of motion for the tangential component

Fnet , t=−mgsinθ¿ma

Then where a denotes the acceleration and θ=sL

a=−gsinθ

¿−gsin( sL )Note:

Acceleration is proportional to sin(s/L) not to displacement (s)

Function for small-angle approximation is as followssinx=x

Thus:

sin( sL )= sLLeading to:

a=−( gL ) sThe mass is proportional to the displacement and opposite in sign thus the angular frequency is obtained by comparing the above equation with the standard equation

for simple harmonic motion: ω=√ gL

The period and frequency of the oscillation is then given by

T=2πω

¿2π √ Lgand

f= ω2π

¿ 12π √ gL

Period:

Period of a simple pendulum depends on the length of the pendulum Acceleration is due to gravity

Question to check you knowledge:1. What is the tangential acceleration of the mass (green ball) when θ=π?

a. –gb. –πc. 2πd. 0

Answer: D: 0

Equation used: a=−gsinθsin (θ )=0a=−g (0 )a=0