SATHEE: Centre of Mass 3 Question 32

Centre of Mass 3 Question 32

35. A simple pendulum is suspended from a peg on a vertical wall. The wall to a horizontal position (see fig.) and released. The ball hits the wall, the coefficient of restitution being $\frac{2}{\sqrt{5}}$ . What is the minimum number of

collisions after which the amplitude of oscillations becomes less than 60 degrees?

$(1987, 7$ M)

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Answer:

Correct Answer: 35. 4

Solution:

As shown in figure initially when the bob is at $A$ , its potential energy is $m g l$ . When the bob is released and it strikes the wall at $B$ , its potential energy $m g l$ is converted into its kinetic energy. If $v$ be the velocity with which the bob strikes the wall, then

Speed of the bob after rebounding (first time)

$v_{1} = e \sqrt{(2 g l)}$

The speed after second rebound is $v_{2} = e^{2} \sqrt{(2 g l)}$

In general after $n$ rebounds, the speed of the bob is

$v_{n} = e^{n} \sqrt{(2 g l)}$

Let the bob rises to a height $h$ after $n$ rebounds. Applying the law of conservation of energy, we have

$\frac{1}{2} m v_{n}^{2} = m g h$

$∴ h = \frac{v_{n}^{2}}{2 g} = \frac{e^{2 n} \cdot 2 g l}{2 g} = e^{2 n} \cdot l = {\frac{2}{\sqrt{5}}}^{2 n} \cdot l = {\frac{4}{5}}^{n} l$

If $θ_{n}$ be the angle after $n$ collisions, then

…(v)

$h = l - l \cos θ_{n} = l (1 - \cos θ_{n})$

From Eqs. (iv) and (v), we have

${\frac{4}{5}}^{n} l = l (1 - \cos θ_{n}) or {\frac{4}{5}}^{n} = (1 - \cos θ_{n})$

For $θ_{n}$ to be less than $60^{\circ}$ , i.e. $\cos θ_{n}$ is greater than $1 / 2$ , i.e. $(1 - \cos θ_{n})$ is less than $1 / 2$ , we have

${\frac{4}{5}}^{n} < \frac{1}{2}$

This condition is satisfied for $n = 4$ .

$∴$ Required number of collisions $= 4$ .

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Centre of Mass 3 Question 32

35. A simple pendulum is suspended from a peg on a vertical wall. The wall to a horizontal position (see fig.) and released. The ball hits the wall, the coefficient of restitution being $\frac{2}{\sqrt{5}}$ . What is the minimum number of

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

Centre of Mass 3 Question 32

35. A simple pendulum is suspended from a peg on a vertical wall. The wall to a horizontal position (see fig.) and released. The ball hits the wall, the coefficient of restitution being 25. What is the minimum number of

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

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35. A simple pendulum is suspended from a peg on a vertical wall. The wall to a horizontal position (see fig.) and released. The ball hits the wall, the coefficient of restitution being $\frac{2}{\sqrt{5}}$ . What is the minimum number of