Question 17: Oscillations

Question

A spring stretches by 2 mm when it is loaded with a mass of 200 g . From equilibrium position the mass is further pulled down by 2 mm and released. The frequency associated with the system and maxmimum energy in the spring are $\_\_\_\_$ Hz and $\_\_\_\_$ J, respectively.
(Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$ )

Accepted Answer

When the mass is at rest (equilibrium), the downward gravitational force is balanced by the upward spring force.
$$ \mathrm{mg}=\mathrm{kx}_0 $$
Where, mass of the suspended load is $\mathrm{m}=200 \mathrm{~g}=0.2 \mathrm{~kg}$,
The acceleration due to gravity is $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$,
The extension in spring at equilibrium position is $x_0=2 \mathrm{~mm}=2 	imes 10^{-3} \mathrm{~m}$
Putting the values,
$$ \mathrm{k}=\frac{\mathrm{mg}}{\mathrm{x}_0}=\frac{0.2 	imes 10}{2 	imes 10^{-3}}=\frac{2}{2} 	imes 10^{-3}=1000 \mathrm{~N} / \mathrm{m} $$
The angular frequency ( $\omega$ ) of a mass-spring system is derived from the restoration force $\mathrm{F}=-\mathrm{kx}$.
Since $\mathrm{F}=\mathrm{ma}$, we have $\mathrm{ma}+\mathrm{kx}=0$,
$$ \omega=\sqrt{\frac{\mathrm{k}}{\mathrm{~m}}} $$
The linear frequency is $f=\frac{\omega}{2 \pi}$ :
$$ \mathrm{f}=\frac{1}{2 \pi} \sqrt{\frac{1000}{0.2}}=\frac{1}{2 \pi} \sqrt{5000} $$
$$\Rightarrow $$ $$ \mathrm{f}=\frac{1}{2 \pi} \sqrt{100 	imes 50}=\frac{10 \sqrt{50}}{2 \pi}=\frac{5 \sqrt{50}}{\pi} \mathrm{~Hz} $$
The mass is pulled down a further 2 mm from equilibrium and released. This additional displacement is the amplitude (A).
$$ A=2 \mathrm{~mm}=2 	imes 10^{-3} \mathrm{~m} $$
The maximum energy in the spring is the at the point of maximum extension, i.e., $x_{\max }=4 \mathrm{~mm}=4 	imes 10^{-3} \mathrm{~m}$
$$ \mathrm{E}=\frac{1}{2} \mathrm{kx}_{\max }^2 $$
$$\Rightarrow $$ $E=\frac{1}{2} 	imes 1000 	imes\left(4 	imes 10^{-3}ight)^2$
$$\Rightarrow $$ $$ \mathrm{E}=500 	imes\left(16 	imes 10^{-6}ight)=8000 	imes 10^{-6}=8 	imes 10^{-3} \mathrm{~J} $$
Therefore, the frequency associated with the system is $\frac{5 \sqrt{50}}{\pi} \mathrm{~Hz}$ and the maximum energy in the spring is $8 	imes 10^{-3} \mathrm{~J}$.
Thus, the correct option is (A).

Answer

$\frac{5 \sqrt{50}}{\pi}$ and $8 	imes 10^{-3}$

Answer

$\frac{5 \sqrt{50}}{\pi}$ and 8

Answer

$10 \sqrt{50}$ and $2 	imes 10^{-3}$

Answer

$\frac{5 \sqrt{50}}{\pi}$ and $16 	imes 10^{-3}$

Oscillations

Choose the correct answer: