SATHEE: Simple Harmonic Motion 4 Question 3

Simple Harmonic Motion 4 Question 3

3. Two light identical springs of spring constant $k$ are attached horizontally at the two ends of an uniform horizontal $rod A B$ of length $l$ and mass $m$ . The rod is pivoted at its centre ’ $O$ ’ and can rotate freely in horizontal plane. The other ends of the two springs are fixed to rigid supports as shown in figure.

The rod is gently pushed through a small angle and released. The frequency of resulting oscillation is (2019 Main, 12 Jan I)

(a) $\frac{1}{2 π} \sqrt{\frac{2 k}{m}}$

(b) $\frac{1}{2 π} \sqrt{\frac{3 k}{m}}$

(d) $\frac{1}{2 π} \sqrt{\frac{k}{m}}$

Show Answer

Answer:

Correct Answer: 3. (c)

Solution:

When a system oscillates, the magnitude of restoring torque of system is given by

$τ = C θ$

where, $C =$ constant that depends on system.

$Also, τ = I α$

where, $I =$ moment of inertia

and $α =$ angular acceleration

From Eqs. (i) and (ii),

$α = \frac{C}{I} \cdot θ$

and time period of oscillation of system will be

$T = 2 π \sqrt{\frac{I}{C}}$

In given case, magnitude of torque is

$τ =$ Force $\times$ perpendicular distance

$τ = 2 k x \times \frac{l}{2} \cos θ$

For small deflection,

$τ = \frac{k l^{2}}{2} θ$

$∵$ For small deflections, $\sin θ = \frac{x}{(l / 2)} \approx θ$

$\Rightarrow x = \frac{l θ}{2}$

Also,

$\cos θ \approx 1$

comparing Eqs. (iv) and (i), we get

$\begin{aligned} C & = \frac{k l^{2}}{2} \Rightarrow α = \frac{(k l^{2} / 2)}{\frac{1}{12} m l^{2}} \cdot θ \\ \Rightarrow α & = \frac{6 k}{m} \cdot θ \end{aligned}$

Hence, time period of oscillation is

$T = 2 π \sqrt{\frac{m}{6 k}}$

Frequency of oscillation is given by

$f = \frac{1}{T} = \frac{1}{2 π} \sqrt{\frac{6 k}{m}}$

पिछला

अगला

Simple Harmonic Motion 4 Question 3

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

Simple Harmonic Motion 4 Question 3

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

Menu