Question 181: Oscillations

Question

The potential energy of a particle of mass $$4 \mathrm{~kg}$$ in motion along the x-axis is given by $$\mathrm{U}=4(1-\cos 4 x)$$ J. The time period of the particle for small oscillation $$(\sin 	heta \simeq 	heta)$$ is $$\left(\frac{\pi}{K}ight) s$$. The value of $$\mathrm{K}$$ is _________.

Accepted Answer

$$U = 4(1 - \cos 4x)$$
$$ \Rightarrow F =  - {{dU} \over {dx}} =  - (4)(4\sin 4x)$$
$$ =  - 16\sin 4x$$
as small x
$$F =  - 16(4x) =  - 64x \equiv  - kx$$
$$T = 2\pi \sqrt {{m \over k}}  = 2\pi \sqrt {{4 \over {64}}}  = {\pi  \over 2}$$
$$ \Rightarrow K = 2$$

Oscillations

The potential energy of a particle of mass $4 \mathrm{~kg} in motion along the x-axis is given by \mathrm{U}=4(1-\cos 4 x) J. The time period of the particle for small oscillation (\sin \theta \simeq \theta) is \left(\frac{\pi}{K}\right) s. The value of \mathrm{K}$ is _________.

Integer / Numeric Answer Type