Forces On Bodies: Systems Involving Strings or Springs

Hooke’s Law

• Force required to extend a spring is proportional to the extension produced
• Mathematically, $$F = -kx$$ where F is the restoring force, k is the spring constant, and x is the extension in the spring

Motion of a Spring-Mass System

• A spring-mass system undergoes Simple Harmonic Motion (SHM) when set into motion
• Equation of motion for SHM: $$m\frac{d^2x}{dt^2} = -kx$$

Uniform Circular Motion

• Centripetal force is directed towards the center of the circular path and provides the necessary acceleration
• $$F_c = mv^2/r$$ where Fc is the centripetal force, m is the mass of the object, v is its speed, and r is the radius of the circular path

Simple Harmonic Motion (SHM)

• Periodic motion where the restoring force is directly proportional to the displacement from equilibrium
• SHM equation: $$x = Acos(\omega t + \phi)$$ where A is the amplitude, ω is the angular frequency, t is time, and φ is the phase angle

Transverse Waves on a Stretched String

• When a string is set into vibration, it produces transverse waves
• Speed of a transverse wave on a string is given by $$v = \sqrt{\frac{T}{\mu}}$$ where v is the wave speed, T is the tension in the string, and μ is the linear mass density of the string

Standing Waves on a String

• When two waves of the same amplitude and frequency travel in opposite directions on a string, they produce standing waves
• Standing waves have specific frequencies called harmonics

Mass-Spring Oscillators

• A mass-spring system is a mechanical system that consists of a mass attached to a spring
• The mass-spring system undergoes SHM when set into motion

Damped Oscillations

• When a resistive force acts on a vibrating system, the oscillations decrease in amplitude over time
• Damping force is proportional to the velocity of the oscillating object

Forced Oscillations

• When an external force is applied to a vibrating system, it is called forced oscillation
• The system resonates at its natural frequency when the frequency of the external force matches its natural frequency

Important NCERT References

Class 11:

• Chapter 10: Mechanical Properties of Solids
• Chapter 12: Thermodynamics
• Chapter 13: Kinetic Theory

Class 12:

• Chapter 5: Laws of Motion
• Chapter 6: Work, Energy, and Power
• Chapter 7: Systems of Particles and Rotational Motion
• Chapter 8: Gravitation
• Chapter 9: Simple Harmonic Motion
• Chapter 11: Waves