Time Period of SHM

I wanna know how to solve this problem using Energy method , i have got answer using Torque method .

#Mechanics

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Steven Chase sir , @Mark Hennings sir pls help

Aryan Yadav - 1 year, 8 months ago

I just wrote the energy equation,but on differetiating wrt to time , i am not getting time period as such . Maybe something more needed . @Steven Chase sir what u think ?

Kudo Shinichi - 1 year, 8 months ago

I derived some equations for the dynamics, but they don't straightforwardly indicate an oscillation rate. They form a system of coupled differential equations in two variables.

Steven Chase - 1 year, 8 months ago

@Steven Chase sir , in the energy equation at an intermediate state, by diff wrt to time , there is w term coming up , what should be the other equation be like as such equilibrium condition dont give any equation . Can u show the coupled differential equation , maybe it can help in getting the required answer . ( Torque analysis really solves the problem by considering torque about hinge point and one force equation for block M)

Kudo Shinichi - 1 year, 8 months ago

@Kudo Shinichi – Let $\theta$ be the clockwise angle of the rod with respect to the vertical. Let $x_b$ be the horizontal position of the block relative to its starting position. I used Lagrangian mechanics to derive the following equations. By inspection, these would also result from force / torque analysis.

For small angles $\theta$ :

$m \ddot{x_b} = -k_2 (x_b - (b+c) \theta) \\ I \ddot{\theta} = -k_1 c^2 \theta + k_2 (b+c) (x_b - (b + c) \theta)$

Steven Chase - 1 year, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$