Oscillating Systems
Spring–mass system, simple pendulum, and damped, forced and resonant oscillations. (Physics › Oscillations and Waves, NEET UG syllabus.)
What is Oscillating Systems?
A repetitive variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states.
Key formula / rule: Restoring Force in Spring-Mass System (Hooke's Law)
Key points
- Define and identify Simple Harmonic Motion (SHM) and its characteristics.
- Derive and apply formulas for the time period of spring-mass systems and simple pendulums.
- Explain the concept of damping and its effect on oscillations.
- Describe forced oscillations and the conditions leading to resonance.
Common exam trap
Confusing natural frequency with driving frequency in forced oscillations.
Definitions
- Term
Oscillation
- Meaning
A repetitive variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states.
- Term
Simple Harmonic Motion (SHM)
- Meaning
A special type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the direction opposite to the displacement.
- Term
Period (T)
- Meaning
The time taken for one complete oscillation.
- Term
Frequency (f)
- Meaning
The number of oscillations per unit time (f = 1/T).
- Term
Angular Frequency (ω)
- Meaning
Related to frequency by ω = 2πf. It represents the rate of change of angular displacement.
- Term
Damping
- Meaning
The process where the amplitude of an oscillation decreases over time due to energy dissipation caused by resistive forces.
- Term
Natural Frequency (ω₀)
- Meaning
The frequency at which a system oscillates when disturbed from its equilibrium position and then left to oscillate freely without any external driving force or damping.
- Term
Forced Oscillation
- Meaning
Oscillations of a system under the continuous influence of an external periodic driving force.
- Term
Resonance
- Meaning
A phenomenon where a system responds with maximum amplitude when the frequency of the external driving force matches its natural frequency.
Learning objectives
Define and identify Simple Harmonic Motion (SHM) and its characteristics.
Derive and apply formulas for the time period of spring-mass systems and simple pendulums.
Explain the concept of damping and its effect on oscillations.
Describe forced oscillations and the conditions leading to resonance.
Solve problems involving various types of oscillating systems, including damped and forced oscillations.
Formulae
- Name
Restoring Force in Spring-Mass System (Hooke's Law)
- Note
k is the spring constant, x is the displacement from equilibrium. The negative sign indicates the force is restoring.
- Expression
F = -kx
- Name
Angular Frequency of Spring-Mass System
- Note
m is the mass attached to the spring.
- Expression
ω = √(k/m)
- Name
Time Period of Spring-Mass System
- Note
Time taken for one complete oscillation.
- Expression
T = 2π√(m/k) = 2π/ω
- Name
Time Period of Simple Pendulum (for small angles)
- Note
L is the length of the pendulum, g is the acceleration due to gravity. Valid for angles typically less than 10-15 degrees.
- Expression
T = 2π√(L/g)
- Name
Damping Force
- Note
b is the damping constant, v is the velocity of the oscillating object. The negative sign indicates it opposes motion.
- Expression
Fd = -bv
- Name
Amplitude of Damped Oscillations
- Note
A₀ is the initial amplitude, t is time. Amplitude decreases exponentially.
- Expression
A(t) = A₀e^(-bt/2m)
- Name
Angular Frequency of Damped Oscillations
- Note
ω₀ is the natural angular frequency without damping. Damping slightly reduces the frequency.
- Expression
ω' = √(ω₀² - (b/2m)²)
- Name
Condition for Resonance
- Note
Resonance occurs when the frequency of the external driving force (ωdriving) matches the natural frequency (ω₀) of the system.
- Expression
ωdriving = ω₀
Prerequisites
Basic concepts of force, work, energy, and power.
Newton's Laws of Motion.
Understanding of periodic motion, frequency, and time period.
Basic calculus (differentiation for velocity/acceleration from displacement, integration for energy).
Knowledge of vectors and scalars.
Common mistakes
Confusing natural frequency with driving frequency in forced oscillations.
Applying the simple pendulum formula T = 2π√(L/g) for large angles.
Ignoring the negative sign in Hooke's Law (F = -kx), which indicates the restoring nature of the force.
Assuming damping always stops oscillations immediately; it gradually reduces amplitude.
Not understanding that resonance requires the driving frequency to match the *natural* frequency, not just any frequency.
Keywords
Oscillation
SHM
Spring-mass
Pendulum
Damping
Forced Oscillation
Resonance
Period
Frequency
Angular Frequency
Restoring Force
Practice preview
What happens to the amplitude of an oscillating system undergoing damped oscillations?…
easy
If a simple pendulum and a spring-mass system are taken to the Moon, where acceleration due to gravity is less than on Earth, what will be the effect on their respective periods of oscillation?…
medium
A particle executes SHM with an amplitude A. At what displacement from the equilibrium position is its kinetic energy equal to its potential energy?…
hard