Question

A particle of mass m = 100 g is attached on both sides to a pair of light springs along each of the Cartesian coordinate axis; x-, y- and z-axis. The spring each has a different spring constant kx, ky and kz along the x-, y- and z-axis respectively. In all cases, neglect the effects of weight and resistance on the particle. a) At some instant of time, the particle is located at position r = x^ i +y ^ j+z ^k from the equilibrium point. i. Write the expression for the net force F (x ,y , z) acting on the particle. ii. Write the components of the equation of motion (EOM) for the particle. iii. Write the general solutions to the equations in part ii. above.

Answer 1

Given,

$r = x\hat{i} +y \hat{j}+z\hat{k}$

$v=\frac{dr}{dt}=\frac{dx}{dt}\hat{i}+\frac{dy}{dt}\hat{j}+\frac{dz}{dt}\hat{k}$

$a=\frac{dv}{dt}=\frac{d^2x}{dt^2}\hat{i}+\frac{d^2y}{dt^2}\hat{i}+\frac{d^2z}{dt^2}\hat{k}$

We know that,

$F=ma$

$\Rightarrow F=0.1(\frac{d^2x}{dt^2}\hat{i}+\frac{d^2y}{dt^2}\hat{i}+\frac{d^2z}{dt^2}\hat{k})$

As per the situation given in the question, if one spring will elongate then other spring will get compress.

$F=(K_1+K_2)\Delta x$

$K_{eq}=K_1+K_2$

$\Rightarrow a=\frac{F}{m}=\frac{K_1 +K_2}{m}\Delta x$

Hence, the required time period of the oscillation

$T=\frac{2\pi}{f}$ $=2\pi \sqrt{\frac{m}{K_1+K_2}}$