Question

A particle having mass 10 g oscillates according to the equation $x=\left(2.0cm\right)sin\left[\right(100s^{-1})t+\pi /6]$. Find (a) the amplitude, the time period and the force constant (b) the position, the velocity and the acceleration at t=0.

Answer 1

Standard equation of SHM $x=A\sin \left(\omega t\right)+\varphi )$ where,

$A=$ amplitude of oscillation

$\omega =$Natural frequency

$\varphi =$phase angle

(a)Comparing equation, $A=2cm$or $2\times {10}^{-2}m$

$\omega =100s^{-1}$

$\omega =\sqrt{\frac{k}{m}}$

$100=\sqrt{\frac{k}{10\times {10}^3}}$

$100=\frac{\sqrt{k}}{{10}^{-1}}$

$\sqrt{k}=1000$

$k={10}^6$

(b) At $t=0$

$x=\left(2cm\right)\sin \left(\frac{\pi }{6}\right)=1cm$ (position at t=0)

$\frac{dx}{dt}=\left(200cm/s\right)\cos ({100}^{-1}t+\frac{\pi }{6})$

$\frac{dx}{dt}\underset{t=0}{|}=200\frac{\sqrt{3}}{2}cm/sv=m/s\frac{d^{2}x}{d{t}^2}\underset{t=0}{|}=-(2\times {10}^{-2}\times 100\times 100)m/s^2)\sin \frac{\pi }{6}a=-100m/s^2$