Question

1) The angular velocity of a vibrating loaded spring is 20rads^-1. Calculate it's new angular velocity when the force constant is quadrapled

2) A circular circus weighing 6MN moves round with a force of 40KN in a merry-go-round. If it covers an area of 36πm^2, calculate the time taken to make 12 complete revolutions

Answer 1

1)

$m\omega_0^2R=k_0x$

If $k=4k_0$ , then:

$\omega=\omega_0\sqrt{4}=2\omega_0=2\cdot20=40\ rad/s$

2)

$ma=mv^2/R=m\omega^2R=40\cdot10^3\ N$

$R=\sqrt{36\pi/\pi}=6\ m$

Time taken to make 1 complete revolution:

$T=2\pi/\omega=\sqrt{\frac{mR}{40\cdot10^3}}$

Time taken to make 12 complete revolutions:

$t=12\cdot2\pi/\omega=12\cdot2\pi\sqrt{\frac{mR}{40\cdot10^3}}=12\cdot2\pi\sqrt{\frac{6\cdot10^6\cdot6}{40\cdot10^3}}=2262\ s=37\ min\ 42\ sec$