Chapter 11: Periodic Motion

Introduction to Periodic Motion

Periodic motion refers to any motion that repeats itself at regular time intervals. A special case of periodic motion is simple harmonic motion (SHM), where the restoring force acting on an object is directly proportional to its displacement from equilibrium and acts in the opposite direction.

• Periodic Motion: Motion that repeats in a set time period.

• Simple Harmonic Motion (SHM): Motion where the restoring force is proportional to displacement: , where is the equilibrium position.

• Acceleration in SHM:

• The resulting motion is sinusoidal in time.

Simple Harmonic Motion: Key Quantities

Definitions and Relationships

Several quantities are used to describe SHM:

• Amplitude (A): The maximum displacement from equilibrium.

• Period (T): The time required to complete one cycle of motion (units: seconds).

• Frequency (f): The number of cycles per second (units: hertz, Hz).

• Relationship: ,

• Angular Frequency (\omega):

Mathematical Description of SHM

Equations of Motion

The position and velocity of an object in SHM as a function of time are given by:

• Position: (if at )

• Velocity:

Physical Systems Exhibiting SHM

• Spring-Mass System:

  • Angular Frequency:

  • Frequency:

  • Period:

• Pendulum (small angles):

  • Angular Frequency:

  • Frequency:

  • Period:

Geometric Interpretation of SHM

Simple harmonic motion can be visualized as the projection of uniform circular motion onto one axis. As a point moves in a circle at constant angular speed, its shadow on a diameter oscillates back and forth in SHM.

Energy in Simple Harmonic Motion

Conservation of Mechanical Energy

In the absence of friction, the total mechanical energy of an object in SHM remains constant and is exchanged between kinetic and potential forms:

• Total Energy:

• At maximum displacement (), all energy is potential:

• At equilibrium (), all energy is kinetic:

Velocity as a Function of Position

The velocity of the object at any position can be found using energy conservation:

• The maximum speed occurs at .

Worked Examples in SHM

Example 1: Spring-Mass System

A concrete block is hung from an ideal spring with force constant and stretches . Find:

• (a) Mass of the block: Use

• (b) Period of oscillation:

• (c) Period on the Moon: The period depends only on and , not gravity, so it remains the same.

Example 2: Finding the Spring Constant

A block of mass attached to a spring oscillates with amplitude and maximum acceleration . Find the spring constant .

• Maximum acceleration:

• Solve for :

• Given values:

Example 3: Two Springs in Parallel

Two identical springs ( each) are attached to a block of mass on a frictionless floor. Find the frequency of oscillation.

• Effective spring constant:

• Frequency:

• Given values:

Summary Table: SHM Quantities for Spring-Mass and Pendulum Systems

Additional info: For a simple pendulum, the small-angle approximation (in radians) is used to derive the SHM equations. This is valid for small oscillations where is much less than 1 radian.