Oscillations of a Spring-Mass System
Applying SHM principles to a spring-mass system, including calculation of its time period and frequency.
What is Oscillations of a Spring-Mass System?
A measure of the stiffness of a spring; it is the ratio of the force applied to the spring to the displacement it produces.
Key formula / rule: Restoring Force
Key points
- To understand the conditions for SHM in a spring-mass system.
- To derive and apply the formulas for time period and frequency.
- To analyze the factors affecting the oscillation of a spring-mass system.
- To differentiate between horizontal and vertical oscillations.
Common exam trap
Confusing angular frequency (ω) with frequency (f).
Definitions
- Term
Spring Constant (k)
- Meaning
A measure of the stiffness of a spring; it is the ratio of the force applied to the spring to the displacement it produces.
- Term
Simple Harmonic Motion (SHM)
- Meaning
A type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
- Term
Time Period (T)
- Meaning
The time taken for an object to complete one full cycle of oscillation.
Learning objectives
To understand the conditions for SHM in a spring-mass system.
To derive and apply the formulas for time period and frequency.
To analyze the factors affecting the oscillation of a spring-mass system.
To differentiate between horizontal and vertical oscillations.
Formulae
- Name
Restoring Force
- Note
k is the spring constant, x is the displacement from equilibrium.
- Expression
F = -kx
- Name
Angular Frequency
- Note
m is the mass, k is the spring constant.
- Expression
ω = \sqrt{\frac{k}{m}}
- Name
Time Period
- Note
Time taken for one complete oscillation.
- Expression
T = \frac{2\π}{\ω} = 2\π \sqrt{\frac{m}{k}}
- Name
Frequency
- Note
Number of oscillations per unit time.
- Expression
f = \frac{1}{T} = \frac{\ω}{2\π} = \frac{1}{2\π} \sqrt{\frac{k}{m}}
Prerequisites
Newton's Laws of Motion
Hooke's Law
Concept of Simple Harmonic Motion (SHM)
Basic calculus (differentiation and integration)
Common mistakes
Confusing angular frequency (ω) with frequency (f).
Forgetting the negative sign in the restoring force equation, which indicates the direction.
Including the effect of gravity in the time period calculation for horizontal oscillations.
Using incorrect units for mass or spring constant.
Keywords
Spring-Mass System
Simple Harmonic Motion
SHM
Time Period
Frequency
Angular Frequency
Spring Constant
Restoring Force
Oscillation
Practice preview
If the mass attached to a spring-mass system is quadrupled, how does the time period of oscillation change?…
easy
A block of mass 0.5 kg is attached to a spring with a spring constant of 200 N/m. Calculate the time period of its oscillation.…
medium
A mass of 2 kg is suspended from a spring and oscillates with a time period of 2 seconds. What is the spring constant of the spring?…
medium