Simple Harmonic Motion
Displacement, velocity, acceleration and energy in SHM with phase relationships. (Physics › Oscillations and Waves, NEET UG syllabus.)
What is Simple Harmonic Motion?
A type of oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction.
Key formula / rule: Restoring Force
Key points
- Define Simple Harmonic Motion and state its conditions.
- Derive and apply equations for displacement, velocity, and acceleration in SHM.
- Calculate time period, frequency, and angular frequency for various SHM systems.
- Analyze the energy transformations (kinetic and potential) in SHM.
Common exam trap
Confusing periodic motion with SHM; SHM requires F ∝ -x.
Definitions
- Term
Simple Harmonic Motion (SHM)
- Meaning
A type of oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction.
- Term
Amplitude (A)
- Meaning
The maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position.
- Term
Angular Frequency (ω)
- Meaning
A scalar measure of the rate of rotation, measured in radians per second (rad/s). For SHM, ω = √(k/m).
- Term
Phase (ωt + φ)
- Meaning
The argument of the sinusoidal function describing the motion, indicating the state of oscillation at a given time.
- Term
Initial Phase (φ)
- Meaning
The phase of the oscillation at time t=0, determining the starting position and direction of motion.
Learning objectives
Define Simple Harmonic Motion and state its conditions.
Derive and apply equations for displacement, velocity, and acceleration in SHM.
Calculate time period, frequency, and angular frequency for various SHM systems.
Analyze the energy transformations (kinetic and potential) in SHM.
Understand and apply phase relationships between displacement, velocity, and acceleration.
Solve problems involving SHM for mass-spring systems and simple pendulums.
Formulae
- Name
Restoring Force
- Note
k is the force constant, x is displacement from equilibrium.
- Expression
F = -kx
- Name
Acceleration in SHM
- Note
ω is angular frequency.
- Expression
a = -ω²x
- Name
Angular Frequency
- Note
m is mass, k is force constant.
- Expression
ω = √(k/m)
- Name
Time Period
- Note
Time for one complete oscillation.
- Expression
T = 2π/ω = 2π√(m/k)
- Name
Frequency
- Note
Number of oscillations per unit time.
- Expression
f = 1/T = ω/(2π)
- Name
Displacement
- Note
A is amplitude, φ is initial phase. Can also use cos.
- Expression
x(t) = A sin(ωt + φ)
- Name
Velocity
- Note
Derivative of displacement. If x is cos, v is -sin.
- Expression
v(t) = Aω cos(ωt + φ)
- Name
Velocity (in terms of x)
- Note
Useful for finding velocity at a given displacement.
- Expression
v = ±ω√(A² - x²)
- Name
Maximum Velocity
- Note
Occurs at equilibrium (x=0).
- Expression
Vmax = Aω
- Name
Maximum Acceleration
- Note
Occurs at extreme positions (x=±A).
- Expression
Amax = Aω²
- Name
Kinetic Energy
- Note
Energy due to motion.
- Expression
KE = (1/2)mv² = (1/2)mω²(A² - x²)
- Name
Potential Energy
- Note
Stored energy due to position.
- Expression
PE = (1/2)kx² = (1/2)mω²x²
- Name
Total Mechanical Energy
- Note
Sum of KE and PE, constant in ideal SHM.
- Expression
E = (1/2)kA² = (1/2)mω²A²
Prerequisites
Basic concepts of force, work, energy, and power.
Understanding of Newton's Laws of Motion.
Knowledge of periodic motion, oscillations, and waves.
Basic trigonometry and calculus (differentiation of sinusoidal functions).
Common mistakes
Confusing periodic motion with SHM; SHM requires F ∝ -x.
Incorrectly applying phase relationships between x, v, and a.
Forgetting the negative sign in F = -kx or a = -ω²x, which indicates the restoring nature.
Mixing up angular frequency (ω) with frequency (f) or time period (T).
Assuming total energy changes with position; it remains constant throughout SHM.
Keywords
SHM
Oscillation
Periodic Motion
Restoring Force
Displacement
Velocity
Acceleration
Amplitude
Angular Frequency
Time Period
Frequency
Phase
Kinetic Energy
Potential Energy
Total Energy
Hooke's Law
Practice preview
Which of the following statements is true for Simple Harmonic Motion (SHM)?…
easy
At which position in Simple Harmonic Motion (SHM) is the speed of the oscillating particle maximum?…
easy
A particle executes SHM with an amplitude of 5 cm and an angular frequency of 10 rad/s. What is its maximum speed?…
medium