Simple Harmonic Motion: Velocity and Acceleration Explained

What are the expressions for velocity and acceleration in Simple Harmonic Motion?

Given the equation of motion x=0.20sin(30t) m, what are the correct expressions for velocity and acceleration of the mass when the object is 5 cm from its equilibrium position?

Answer:

Correct expressions for the velocity and acceleration of the mass in Simple Harmonic Motion are:

Simple Harmonic Motion (SHM) is a fundamental concept in Physics that describes the repetitive motion of a mass around an equilibrium position. The equation of motion for a mass in SHM is often represented as x = A sin(ωt), where x is the displacement from equilibrium, A is the amplitude, ω is the angular frequency, and t is time.

When the object is 5 cm from its equilibrium position or x = 0.05 m, we can calculate the velocity and acceleration by differentiating the equation of motion with respect to time. The velocity v(t) is found by taking the derivative of x with respect to time, v(t) = dx/dt. Similarly, the acceleration a(t) is found by taking the derivative of velocity with respect to time, a(t) = dv/dt.

For the given equation x = 0.20sin(30t) m, the velocity v(t) is calculated as 0.20 × 30cos(30t) m/s. This expression represents the rate of change of displacement over time, with the amplitude multiplied by the angular frequency and cosined to maintain the sinusoidal relationship.

The acceleration a(t) is then calculated as -0.20 × 30²cos(30t) m/s². This expression shows the rate of change of velocity over time, with an additional negative sign and the squared angular frequency due to the differentiation process.

Therefore, the correct expressions for the velocity and acceleration in Simple Harmonic Motion when the object is 5 cm from its equilibrium position are as follows: