Oscillations 3

 

Here’s another example of SHM – a simple pendulum – “simple” because it consists of a string of zero mass (but length L) and a bob of zero size (but mass m). It is pulled back through some angle  and then released. We want to describe its subsequent motion. Once released the only forces acting on the bob are gravity pulling down and the tension in the string. The components of the gravitational force are  which is balanced by the tension in the string, and  which causes the bob to swing back towards equilibrium. This last force is shown in the diagram. Since the angle  decreases as the bob swings back, the force is negative. Now, the ball and string rotate and so produce a torque about the pivot point, and this torque provides an angular acceleration to the bob according to Newton’s law as applied to circular motion, i.e. . The torque , the moment of inertia , and the angular acceleration . Inserting these into Newton’s law provides
(1.1)    

After some arranging (1.1) may be written as

(1.2)

 

This is close to the equation for SHM – and if we confine the amplitude of swing to small angles we can use the (useful enough to remember relation for small angles measured in radians)  to arrive at

(1.3)

 

We’re done! Equation (1.3) is the same equation we solved for the mass on a spring (with  replacing x and g/L replacing k/m). The solution is then

(1.4)

 

this time with the natural frequency of the pendulum

 

One more generalization: If we rewrite Newton’s law without assuming a value for I we have

(1.5)

 

This is valid for any pendulum, e.g. a sack of potatoes swinging from a nail. (This type of pendulum is called a “physical pendulum”.)  The length L is measured from the nail to the center of mass of the sack, and the moment of inertia needs to be determined.

 

Moral: If a system moves under the influence of a linear restoring force, or reasonable approximation thereof, it satisfies an equation with the form of (1.5). It’s motion will be sinusoidal with period equal to the square root of the constant multiplying the dependent variable (  here, or x in the case of a mass on a spring).