Question

The displacement (x) of a particle as a function of time (t) is given by
X = a sin (bt + c)
Where a, b and c are constants of motion. Choose the correct statement(s) from the following

• A. The motion repeats itself in a time interval $\frac{2\pi }{b}$
• B. The energy of the particle remains constant
• C. The velocity of the particle is zero at x = $\pm$ a
• D. The acceleration of the particle is zero at x =$\pm$ a

Answer 1

The correct option is A The motion repeats itself in a time interval $\frac{2\pi }{b}$

The motion of the particle is simple harmonic. The displacement at time t is X = a sin $(bt+c)$ $\therefore$ Displacement at time $(t+\frac{2\pi }{b})$ is $x(att+\frac{2\pi }{b})=asin[b(t+\frac{2\pi }{b})+c]$ $=asin(bt+c+2\pi )$ $=asin(bt+c)$ $=xattimet$ Hence statement (a) is correct. Statement (b) is also correct since the same displacement is recovered after a time interval of $\frac{2\pi }{b}$ .Statement (c) is correct because the velocity is zero when the displacement = $\pm$ the amplitude, i.e. at the extreme ends of the motion. Statement (d)is incorrect, the acceleration is maximum (in magnitude) at x = $\pm$ a.