Mention the expression for velocity and acceleration of a particle executing SHM.

In Simple Harmonic Motion (SHM), the motion of the particle is characterized by periodic oscillations. The expressions for velocity and acceleration of a particle executing SHM are derived from the displacement equation.
1. Displacement Equation in SHM:
The displacement of a particle in SHM at any time t is given by: x(t) = A * cos(ωt + φ)
Where:
• x(t) is the displacement of the particle at time t.
• A is the amplitude of the motion (maximum displacement).
• ω is the angular frequency (ω = 2π/T, where T is the time period).
• t is the time.
• φ is the phase constant, which depends on the initial conditions.
2. Velocity in SHM:
Velocity is the rate of change of displacement with respect to time. To find the velocity, we differentiate the displacement equation with respect to time:
v(t) = dx/dt = -Aω * sin(ωt + φ)
Thus, the velocity of a particle in SHM is given by: v(t) = -Aω * sin(ωt + φ)
Where:
• v(t) is the velocity of the particle at time t.
• A is the amplitude of the motion.
• ω is the angular frequency.
• t is the time.
• φ is the phase constant.
3. Acceleration in SHM:
Acceleration is the rate of change of velocity with respect to time. To find the acceleration, we differentiate the velocity equation with respect to time:
a(t) = dv/dt = -Aω² * cos(ωt + φ)
Thus, the acceleration of a particle in SHM is given by: a(t) = -Aω² * cos(ωt + φ)
Where:
• a(t) is the acceleration of the particle at time t.
• A is the amplitude of the motion.
• ω is the angular frequency.
• t is the time.
• φ is the phase constant.
Summary of the Expressions:
1. Displacement: x(t) = A * cos(ωt + φ)
2. Velocity: v(t) = -Aω * sin(ωt + φ)
3. Acceleration: a(t) = -Aω² * cos(ωt + φ)
These equations describe the motion of a particle executing Simple Harmonic Motion, where the velocity and acceleration are always directed towards the mean position and are proportional to the displacement.