Oscillations (Chapter 3)

• Oscillation: To and fro motion of the object about mean position.
• Simple Harmonic Motion: To and fro motion in which the acceleration of the object is directly proportional to its displacement and the acceleration is always directed towards the mean position.

Difference between oscillation and simple harmonic motion

Consider the motion of an object on a ramp;

Here the driving force is mgsinθ (we assume no resistive forces are acting on the ball)

mgsinθ=ma

gsinθ=a

notice that both g and θ remain constant throughout the motion and hence acceleration is constant. This motion is thus known as oscillation.

Now consider the motion of a pendulum;

Here the driving force is again mgsinθ however the value of θ changes throughout the motion and hence the value of acceleration changes. This is a perfect example of simple harmonic motion.

Acceleration ∝ - displacement

a = -xω²

Motion Graphs of Simple Harmonic Motion

Displacement - Time Graph

Velocity-Time Graph

Acceleration-Time Graph

Velocity-Displacement Graph

Acceleration-Displacement Graph

Potential Energy - Displacement Graph

Kinetic Energy - Displacement Graph

Total Energy - Displacement Graph

Equations of Energy for Simple Harmonic Motion

Energy (total) = Energy (kinetic) + Energy (potential)

When Displacement = 0

Energy (total) = Energy (kinetic)

Energy = 1/2mv²

Now since v=ωx
Energy = 1/2mω²x²

Types of Oscillations 

1. Free Oscillation: Oscillation in which the net loss of energy is 0.
2. Forced Oscillation: Vibration of the particle at the frequency of the applied force.
3. Damped Oscillation: Decrease in the amplitude of vibrations due to external resistive forces.

Note: (To provide damping force to a system we can either change the medium of the oscillation of the system i.e if its in air put it in water or we can attach a light card perpendicular to the motion to provide resistive force)

Differences between Critical Damping and Over Damping

Critical Damping: In critical damping the system does not oscillate and the displacement of the system decreases to 0 in a short period of time

Over damping/Heavy damping: The system does not oscillate and the displacement decreases to 0 in a comparatively longer period of time.

• Resonance: Resonance occurs when the natural frequency of vibration of the object becomes equal to the driving frequency, giving maximum amplitude.

Note: Notice that as the damping increases, the curve shifts to the left and becomes flatter, in addition the amplitude of the vibration decreases

Applications of Resonance:

• Resonance is useful in nuclear magnetic resonance (we will learn about this later) and in RLC circuits.
• Resonance also has some undesirable effects including shattering of glass (at the right pitch of sound) and resonance in rigid structures.