Introduction
Simple harmonic motion (SHM) is a type of oscillatory motion in which the restoring force is directly proportional to the displacement and acts in the opposite direction. The motion is sinusoidal in time and exhibits characteristics such as constant amplitude and periodicity.
Displacement Equation
Given the equation of SHM:
x(t) = a cos(ωt)
where:
- x(t) is the displacement at time t
- a is the amplitude
- ω is the angular frequency
1. Expression for Velocity
Velocity is the first derivative of displacement with respect to time:
v(t) = dx/dt = -aω sin(ωt)
2. Expression for Acceleration
Acceleration is the derivative of velocity or the second derivative of displacement:
a(t) = dv/dt = d²x/dt² = -aω² cos(ωt)
So, acceleration is directly proportional to the displacement but opposite in direction, confirming the SHM nature.
3. Time Variation Graphs
Let us describe the nature of graphs without actual plotting:
- Displacement (x): A cosine wave starting at maximum amplitude a when t = 0.
- Velocity (v): A sine wave (inverted) starting at 0, going negative, crossing zero at π, and so on.
- Acceleration (a): A cosine wave but inverted, i.e., starts at -aω² when t = 0.
These curves are all sinusoidal, with displacement and acceleration being in phase but opposite in direction, and velocity lagging behind displacement by π/2 radians.
Conclusion
In SHM described by x(t) = a cos(ωt), the velocity and acceleration are given by:
- Velocity: v(t) = -aω sin(ωt)
- Acceleration: a(t) = -aω² cos(ωt)
These relationships reveal that SHM is governed by harmonic oscillation with acceleration directly proportional and opposite to displacement, and velocity leading or lagging by π/2. The plots of x(t), v(t), and a(t) against time are sinusoidal with phase differences.