Simple Harmonic Motion (SHM) Basics

Simple Harmonic Motion (SHM)

• Figure 1: A graph illustrating SHM (Simple Harmonic Motion) showing a sinusoidal pattern for displacement over time.

• Amplitude: The maximum displacement from equilibrium. Measured in centimeters (cm) or meters (m).

• Period (T): The time taken for one complete oscillation. Measured in seconds (s).

• Frequency (f): The number of oscillations per second. Calculated as f = 1/T. Measured in Hertz (Hz).

• Displacement Equation: y = y₀ cos(ωt + φ), where:

  • y₀ is the amplitude
  • ω is the angular frequency (ω = 2πf)
  • t is time
  • φ is the phase constant

Velocity and Acceleration in SHM

• Velocity Equation: The rate of change of displacement. In SHM, the velocity equation is derived from the displacement equation and is also sinusoidal.

• Acceleration Equation: The rate of change of velocity. In SHM, the acceleration is also sinusoidal and directly proportional to the displacement, but in the opposite direction of the displacement.

Specific SHM Example

• Equation of a particle: x = 10 sin (ωt), where x is in meters and t is in seconds. The period of oscillation is 30 s.

• Amplitude: 10 (m).

• Maximum Velocity: Calculated from the maximum value of the equation.

• Maximum Acceleration: Calculated from the maximum value of the equation.

• Displacement, Velocity, Acceleration at t = 15s: Given specific values in the example

Displacement as a Sine Function

• Displacement Equation (general): x = A sin(ωt + φ), or x= A cos (ωt +φ) where,

• A is the amplitude (max displacement),

• ω is angular frequency, and,

• φ is phase

Acceleration of SHM

• Acceleration as a Function of Displacement: a = -ω²x. This proves the acceleration is directly proportional to displacement but is in the opposite direction of displacement.

Force Graph in SHM

• Amplitude in a Force graph example: The amplitude is determined from maximum force.

• Angular Frequency, Period, and Max Velocity: These values are derived from the characteristics of the force graph in SHM.

SHM with Graph and Equation

• Displacement Equation: x = 15 sin(14πt + 2π) defines a particle undergoing SHM (time given in seconds). -Amplitude (A) and phase calculation.

• Displacement and Velocity at t=0.6 s: Displacement and velocity values calculated from the given equation.

• Acceleration when t=0.6 s: Calculate the acceleration.

Potential Energy and Force

• Relationship between Potential Energy and Displacement: In SHM a graph showing Potential Energy v/s Displacement is a parabola.

• Force when x= 0.2 m: The force at a specified displacement can be calculated from the Potential Energy curve.

Mass Vibrating in SHM (Numerical Example)

• Frequency (f): Derived from the given equation: x = 0.38 sin (6.50t). The angular frequency is 6.50 radians/sec.

• Total Energy: Calculating the total energy in SHM from the formula.

• Kinetic and Potential Energy: Evaluating kinetic and potential energies at x = 9.0 cm.

• Graph of x vs t: Sketching the graph of x vs t for the given equation.

SHM with Maximum Displacement

• Amplitude: Calculated based on maximum displacement.

• Frequency: Calculated from the period of repetition (4.0 s).

• Angular velocity (ω): Calculation of angular frequency.

• Equation for SHM: Deriving the equation for the particle's motion.

• Maximum Velocity: Calculated based on the derived equation.

• Acceleration Graph (a vs t): Sketching the graph of acceleration versus time.

• Position at Specific Time: Determining particle position at t = 50 ms.

This quiz covers the fundamental concepts of Simple Harmonic Motion, including amplitude, period, frequency, and the corresponding equations for displacement, velocity, and acceleration. Test your understanding of the key principles that govern SHM and its mathematical representations.