Question

A particle is executing simple harmonic motion with time period T. Let x, v an a denote the displacement, velocity and acceleration of the particle, respectively- at time t. Then,

• A. \(\frac{aT}{x}\) does not change with time
• B. (αT + 2πv) does not change with time
• C. x and v are related by an equation of a straight line
• D. v and a are related by an equation of an ellipse

Answer 1

The Correct Option is A, D

To solve this problem, we need to understand the concepts of simple harmonic motion (SHM) and how displacement (\(x\)), velocity (\(v\)), and acceleration (\(a\)) relate to each other.

1. Understanding Simple Harmonic Motion Equations:
   • The displacement in SHM is given by: \(x(t) = A\cos(\omega t + \phi)\)
   • The velocity is the derivative of the displacement: \(v(t) = -A\omega\sin(\omega t + \phi)\)
   • The acceleration is the derivative of the velocity: \(a(t) = -A\omega^2\cos(\omega t + \phi)\)
2. Relating Acceleration and Displacement:
   • From the equations, acceleration can also be expressed as: \(a = -\omega^2 x\)
   • This implies \(\frac{a}{x} = -\omega^2\), a constant.
   • Given the period \(T = \frac{2\pi}{\omega}\), we have \(\omega = \frac{2\pi}{T}\).
   • Thus, \(\frac{aT}{x} = - (2\pi)^2\), which does not change with time.
3. Relating Velocity and Acceleration:
   • Using earlier expressions: \(v = -\omega^2 x\) and \(a = -\omega^2 x\).
   • Substitute \(x = \frac{v}{-\omega}\sin(\omega t + \phi)\) into the acceleration equation.
   • It can be shown that the equation connecting \(v\) and \(a\) forms an ellipse equation upon squaring and adding.
4. Conclusion:
   • The correct answers are:
     1. \(\frac{aT}{x}\) does not change with time.
     2. \(v\) and \(a\) are related by an equation of an ellipse.