SATHEE: Gravitation 5 Question 1

Gravitation 5 Question 1

1. Four identical particles of mass $M$ are located at the corners of a square of side $a$ . What should be their speed, if each of them revolves under the influence of other’s gravitational field in a circular orbit

circumscribing the square?

(a) $1.35 \sqrt{\frac{G M}{a}}$

(b) $1.16 \sqrt{\frac{G M}{a}}$

(d) $1.41 \sqrt{\frac{G M}{a}}$

(Main 2019, 8 April I)

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Answer:

Correct Answer: 1. (b)

Solution:

In given configuration of masses, net gravitational force provides the necessary centripetal force for rotation.

Net force on mass $M$ at position $B$ towards centre of circle is $F_{B O net} = F_{B D} + F_{B A} \sin 45^{\circ} + F_{B C} \cos 45^{\circ}$

$= \frac{G M^{2}}{(\sqrt{2} a)^{2}} + \frac{G M^{2}}{a^{2}} \frac{1}{\sqrt{2}} + \frac{G M^{2}}{a^{2}} \frac{1}{\sqrt{2}}$

[where, diagonal length $B D$ is $\sqrt{2} a$ ]

$= \frac{G M^{2}}{2 a^{2}} + \frac{G M^{2}}{a^{2}} \frac{2}{\sqrt{2}} = \frac{G M^{2}}{a^{2}} \frac{1}{2} + \sqrt{2}$

This force will act as centripetal force.

Distance of particle from centre of circle is $\frac{a}{\sqrt{2}}$ .

Here, $F_{centripetal} = \frac{M v^{2}}{r} = \frac{M v^{2}}{\frac{a}{\sqrt{2}}} = \frac{\sqrt{2} M v^{2}}{a} ∵ r = \frac{a}{\sqrt{2}}$

So, for rotation about the centre,

$F_{centripetal} = F_{B O (net)}$

$\Rightarrow \sqrt{2} \frac{M v^{2}}{a} = \frac{G M^{2}}{a^{2}} \frac{1}{2} + \sqrt{2}$

$\Rightarrow v^{2} = \frac{G M}{a} 1 + \frac{1}{2 \sqrt{2}} = \frac{G M}{a} (1.35) \Rightarrow v = 1.16 \sqrt{\frac{G M}{a}}$

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Gravitation 5 Question 1

1. Four identical particles of mass $M$ are located at the corners of a square of side $a$ . What should be their speed, if each of them revolves under the influence of other’s gravitational field in a circular orbit

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

Gravitation 5 Question 1

1. Four identical particles of mass M are located at the corners of a square of side a. What should be their speed, if each of them revolves under the influence of other’s gravitational field in a circular orbit

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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1. Four identical particles of mass $M$ are located at the corners of a square of side $a$ . What should be their speed, if each of them revolves under the influence of other’s gravitational field in a circular orbit