Simple Harmonic Motion | General Awareness for SSC CGL PDF Download

An oscillatory motion of constant amplitude and of single frequency under a restoring force whose magnitude is proportional to the displacement and always acts towards mean position, is called Simple Harmonic Motion.
For example – Equation of SHM is given by x = Asin (ωt + δ), where (ωt + δ) is known as phase.

Characteristics of SHM

• The particle's movement should be a back-and-forth motion along a straight path centered around the mean position. 
• The force restoring the particle to the mean position must always be directly proportional to the distance of the particle from that mean position.

Some Definitions Related to SHM

Time Period and Frequency 

• Time taken by the particle to complete one oscillation, is known as time period (T). 
• The number of oscillations completed by the particle in one second, is called frequency ( ν ).
  Its unit is second⁻¹ or hertz.
• The product of frequency with a factor 2π, is called angular frequency (ω).
• Angular frequency
  Its unit is second⁻¹ or hertz.

Displacement and Amplitude

• A physical quantity which changes uniformly with time and also mean position in a periodic motion, is called displacement (y ). 
• The maximum displacement in any direction from the mean position, is called amplitude (a).
  Displacement in SHM at any instant is given by y = a sin ωt or y = a cos ωt
  
• Velocity of a particle executing SHM at any instant is given by v = dy / dt = aω cos ωt
  At mean position (y = 0), velocity is maximum.
  v_max = aω
  At extreme position ( y = a ), velocity is zero.
  
• Acceleration of a particle executing SHM at any instant is given by α = d²y / dt²A
  or α = − ω²y
• At mean position (y = 0), the acceleration is zero and at extreme position ( y = a ), the acceleration is maximum. α_max = − aω² 
  Time period in SHM is given by
  

Simple Pendulum

• A heavy point mass suspended from a rigid support by means of an elastic inextensible string, is called a simple pendulum. 
• Time period of a simple pendulum is given by
  where, l = effective length of the pendulum
  g = acceleration due to gravity.
• The time period of a simple pendulum of infinite length is 84.6 min. The time period of a second’s pendulum is 2 s. Its length on the earth is nearly 100 cm.
• Acceleration due to gravity decreases with altitude (height) and therefore time period of a pendulum clock will increase and clock becomes slow. 
• If the bob of a simple pendulum is suspended from a metallic wire, then the length of the pendulum increases with increase in temperature and therefore its time period also increases. When a bob of simple pendulum of density (ρ) oscillates in a fluid of density ρ₀ ( ρ₀ < ρ), then its time period gets increased.
  Increased time period,
• Time period of oscillations of a loaded spring is given by
  
  where, m = mass suspended with the spring
  k = force constant of the spring.

Energies in SHM

• Potential energy of a particle of mass m executing SHM is given by
  where, m = mass of the particle,
  ω = angular velocity of oscillations,
  y = displacement.
• Kinetic energy of a particle of mass m executing SHM is given by
  
• Total energy (E) = U + K
  
• A girl is swinging over a swing. If she stands up over the swing, then the effective length of the swing decreases and therefore, the time period of oscillations decreases. 
• A pendulum clock cannot be used in a space-ship.

Damped Harmonic Motion

When there is friction or any other force acting within an oscillating system, the amplitudes of the oscillation decreases over time to this damping force. This is called damped harmonic motion.

Resonant Oscillations

When a body oscillates with its own natural frequency (ν₀) with the help of an external periodic force also called forced harmonic motion. And if the frequency (ν) provided by the enternal agent is equal to the natural frequency of the body, the oscillations of the body are called resonant oscillations.