Name: Superposition of Two or More SHMs
Uploaded: 2019-07-19
Duration: 9 min 11 s
Description: What happens when two or more simple harmonic motions are superimposed? Watch this video to learn what would be the result of the superposition of SHMs.

Question

A block of mass $m$ hangs from a smooth fixed pulley $P,$ by an inextensible string filled with two springs of stiffness $k_{1}$ and $k_{2} .$ The string passes over a movable smooth light pulley $P_{2}$ which is connected with another ideal spring of stiffness $k_{3}$ Find the period of oscillation of the block.

Accepted Answer

Let the block is displaced by a distance $' x_{0}'$ beyond its initial equilibrium position by applying a force $F$ so that the system remains in equilibrium. This will increase in the tension in the string connecting block and spring and this increased tension is equal to the force $k_{2} .$ Due to this increase in tension in the springs $k_{1}, k_{2}$ and $k_{3}$ further elongate by $x_{1}, x_{2}$ and $x_{3}$ respectively.

The constraint relation should be: $x_{1} + x_{2} + 2 x_{3} = x_{0} \dots (i)$

The spring $k_{1}$ and $k_{2}$ are connected with string, hence

$K_{1} x_{1} = K_{2} x_{2} = F \dots (ii)$

It gives $x_{1} = \frac{F}{K_{1}}$ and $x_{2} = \frac{F}{K_{2}}$ as pulleys are light, $2 F = K_{3} x_{3}$

or $x_{3} = \frac{2 F}{K_{3}}$

Substituting the values of $x_{1}, x_{2}$ and $x_{3}$ in Eq. (i)

$(\frac{F}{K_{1}} + \frac{F}{K_{2}} + 4 \frac{F}{K_{3}}) = x_{0}$ or $F = \frac{1}{(\frac{1}{K_{1}} + \frac{1}{K_{2}} + \frac{4}{K_{3}})} \cdot x_{0}$

Hence $\frac{F}{x_{0}} = \frac{1}{(\frac{1}{K_{1}} + \frac{1}{K_{2}} + \frac{4}{K_{3}})} = K_{eff}$

$\Rightarrow ω^{2} = \frac{K_{eff}}{m} = \frac{1}{m (\frac{1}{K_{1}} + \frac{1}{K_{2}} + \frac{4}{K_{3}})}$

or $T = 2 π \sqrt{m (\frac{1}{K_{1}} + \frac{1}{K_{2}} + \frac{4}{K_{3}})}$

Important Questions on Linear and Angular Simple Harmonic Motion

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A block of mass m hangs from a smooth fixed pulley P, by an inextensible string filled with two springs of stiffness k1 and k2. The string passes over a movable smooth light pulley P2 which is connected with another ideal spring of stiffness k3 Find the period of oscillation of the block.

Important Questions on Linear and Angular Simple Harmonic Motion

Learn and Prepare for any exam you want !