Forces On Bodiessystems Involving Strings Or Springs Topic
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