Phy
Simple Harmonic Motion
Time Period
11
⚡ Quick Summary
A particle in SHM repeats its motion after a regular time interval. One complete oscillation is the motion from one point to another and back, with both position and velocity repeating. The time taken for one complete oscillation is the time period T.
['T = 2π / ω', 'T = 2π * √(m/k) (where k is the force constant and m is the mass of the particle)']
- Time Period (T): The time taken for one complete oscillation. A complete oscillation means the particle returns to its starting position with the same velocity (both magnitude and direction).
- Mathematical Definition: If x = A sin(ωt + δ), then sin(ωt + δ) = sin[ω(t + T) + δ]. Similarly, for velocity, if v = A ω cos(ωt + δ), then cos(ωt + δ) = cos[ω(t + T) + δ].
- The conditions above hold true when ωT = 2π
Amplitude
11
⚡ Quick Summary
Amplitude is the maximum displacement of the particle from the center of oscillation.
['x = A sin(ωt + δ) (where A is the amplitude)']
- The equation x = A sin(ωt + δ) describes the displacement of a particle in SHM.
- Since sin(ωt + δ) varies between -1 and +1, the displacement x varies between -A and +A.
- A represents the maximum displacement from the center of oscillation.
Velocity in SHM
11
⚡ Quick Summary
The velocity of a particle in SHM varies with its displacement.
['v = ω √(A² - x²)', 'v = A ω cos(ωt + δ)']
- The velocity of a particle executing SHM is given by the derivative of its displacement with respect to time.