Notes from Toppers
Introduction to Simple Harmonic Motion
Definition and Concept
- SHM is the periodic motion where the restoring force is proportional to the displacement from the equilibrium position.
- The force is always directed towards the equilibrium position.
Properties of SHM
- Time period (
T) : Time taken to complete one oscillation. - Frequency (
f) : Number of oscillations per second. - Amplitude (
A) : Maximum displacement from the equilibrium position. - Angular frequency (
ω) : Rate at which the phase of harmonic motion changes, measured in radians per second. - Phase (
φ) : The position of a point in a cycle of oscillation relative to a reference point, often measured in radians or degrees.
Equations of SHM
- Displacement:
x(t) = Acosωt + φ - Velocity:
v(t) = -Aωsinωt - Acceleration:
a(t) = -Aω²cosωt
Graphical Representation
Graphs of (x(t), v(t), a(t)) versus time (t) for a particle executing simple harmonic motion.
Energy in SHM
- Potential Energy:
PE = ½kA²cos²ωt - Kinetic Energy:
KE = ½kA²sin²ωt - Total Energy:
TE = ½kA²
Relationship with Uniform Circular Motion
- SHM can be viewed as the projection of uniform circular motion on a diameter of the circle.
- The radius of the circle is equal to the amplitude of the SHM.
Applications
- Springs
- Pendulums
- Oscillating Systems
Problem-Solving Techniques
- Understand the question and identify the relevant concepts.
- Choose an appropriate equation(s) to solve the problem.
- Substitute the given values into the equation(s).
- Solve for the unknown variable.
References:
- NCERT Physics Class 11, chapter 15.
- NCERT Physics Class 12, chapters 1, 2.
