Periodic Motion

A micro-course in Periodic Motion that is the section five of the course in classical mechanics available on Physics 12: Classical Mechanics. This course contains lectures about oscillation or periodic Motion, SHM (simple harmonic motion), and uniform circular motion.

Oscillation or Periodic Motion

Oscillation or Periodic motion is a motion that repeats in equal intervals of time.
 
Physical characteristics of a periodic motion are as follows:
-Period (T): the time to complete one cycle of motion.
-Frequency (f, or ν ): number of cycles per unit time.
-Amplitude (A): the magnitude of maximum displacement from the equilibrium point.

Equation of periodic motion is a periodic function of time, for example,

(or any linear combination of them) are equations of periodic motion, where ω is the angular velocity and it's equal to ω = 2πν = 2π/T .

Simple Harmonic Motion: Perfectly Elastic Spring

 Motion of an object connected to a perfectly elastic spring is a simple harmonic motion ( or periodic motion about a stable equilibrium point) if the restoring force (or spring force) is the only force acting on it.

Restoring force F_s  is the net force acting on the object connected to the spring with spring constant κ.

According to the Newton's second law, 

where 'a' is the acceleration of the object that is equal to the second derivative of 'x' with respect to time,

One of the solutions of the above equation is given by,

where 'A' is the maximum displacement (amplitude), ω angular velocity, and φ phase angle. By inserting equation (2) in equation (1), we can find ω as,

 and the period of motion as,

The mechanical energy of this system is given by,

that is the sum of kinetic energy and potential energy of the object. At x=±A , speed of the object is v=0, and by inserting these values in the equation (3), we get, 

So the mechanical energy of this system is conserved over time, and it depends on only spring constant and amplitude of motion. By inserting this value in equation (3), we get,

which gives speed of the object at any given point.

 

Simple Harmonic Motion: Simple Pendulum

Motion of a mass suspended from an unstretchable, massless string, about its equilibrium point on a path that can be approximated by a horizontal line, is a simple harmonic motion, if the gravitational force is the only external force acting on it (frictional and other possible forces are negligible). 

 Two forces are acting on mass 'm': gravitational force 'mg' and tension from string 'T'. The gravitational force has two components, 'mgcosθ' that is equal in magnitude and opposite to the direction of tension 'T',
                                  T=mgcosθ

and 'mgsinθ' that is opposite to the x-axis, and provides restoring force as,
                                  F=-mgsinθ

For very small displacement from equilibrium, θ is very small and,         
                                  sinθ~x/L

So the restoring force can be approximated as,

                                   F=-mg x/L

and according to the Newton's second law,

                                   ma=-mg x/L
                                      a=-gx/L

where 'a' is the acceleration of the mass 'm'.
The above equation is similar to the equation of motion of an object connected to the perfectly elastic spring (discussed earlier), so all of the equations there, are valid for this simple pendulum if we substitute 'κ'  by 'mg/L'.


Motion of simple pendulum is discussed in Grade-12 Physics: Mechanics.

Uniform Circular Motion

Motion of an object on a circle around the center of the circle with constant speed is called uniform circular motion. In uniform circular motion although the magnitude of the velocity (or speed) is constant, its direction changes uniformly due to the centripetal force that is constant in magnitude but its direction changes uniformly and it is always towards the center of the circle. 
Find more about uniform circular motion at Grade-12 Physics: Mechanics

Problems and Solutions

Problem 1

The equation of motion of a simple harmonic oscillator in SI system is given by, X=0.1 sin(πt+0.2π).
a) What are the period of motion, its amplitude and frequency, and phase angle?
b) If t=0 is origin of time, at which time is the oscillator at the equilibrium point for the first time?
c) Find the speed and acceleration of the oscillator in terms of time.
d) Find the distance from the equilibrium point when the kinetic energy of the oscillator is equal to its potential energy.

Problem solving lecture in periodic motion:1

Problem 2

A student is twirling a 155g ball on a 1.65m string in a horizontal circle. The string will break if the tension reaches 208N. What is the maximum speed at which the ball can move without breaking the string?

Problem solving lecture in periodic motion:2