Motion In A Plane

Question

Four particles, each of mass M and equidistant from each other, move along a circle of radius R under the action of their mutual gravitational attraction. The speed of each particle is

$square root of GM over straight R end root$
$square root of 2 square root of 2 GM over straight R end root$
$square root of GM over straight R left parenthesis 1 plus 2 square root of 2 right parenthesis end root$
$1 half square root of GM over straight R left parenthesis 1 plus 2 square root of 2 right parenthesis end root$

Solution

1 half square root of GM over straight R left parenthesis 1 plus 2 square root of 2 right parenthesis end root

Net force acting on any one particle M,
= $fraction numerator GM squared over denominator left parenthesis 2 straight R right parenthesis squared end fraction space plus space fraction numerator GM squared over denominator left parenthesis straight R square root of 2 right parenthesis squared end fraction cos space 45 to the power of straight o space plus space fraction numerator GM squared over denominator left parenthesis straight R square root of 2 right parenthesis squared end fraction Cos space 45 to the power of straight o space equals space GM squared over straight R squared open parentheses 1 fourth plus fraction numerator 1 over denominator square root of 2 end fraction close parentheses$

This force will equal to centripetal force
So, $Mv squared over straight R space equals space GM squared over straight R squared open parentheses 1 fourth plus fraction numerator 1 over denominator square root of 2 end fraction close parentheses$
$straight v space equals space square root of fraction numerator GM over denominator 4 straight R end fraction left parenthesis 1 plus 2 square root of 2 right parenthesis end root space equals space 1 half square root of GM over straight R left parenthesis 2 square root of 2 plus 1 right parenthesis end root$

Some More Questions From Motion in A Plane Chapter

Pick out the only vector quantities in the following list:
Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction and charge.

Pick out the two scalar quantities in the following list:
force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, reaction as per Newton’s third law, relative velocity.

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Motion In A Plane

Some More Questions From Motion in A Plane Chapter

Give three examples of vector quantities.

What are the basic characteristics that a quantity must possess so that it may be a vector quantity?

Is a physical quantity, having a direction, necessarily a vector?

Name a quantity which is both scalar as well as vector.

Which of the followings is/are vector quantities: work, power, acceleration, mass, momentum?

Pick up the odd one from the following: force, linear momentum, mass, impulse.

Pick out the only vector quantities in the following list:
Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction and charge.

Pick out the two scalar quantities in the following list:
force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, reaction as per Newton’s third law, relative velocity.

What are polar vectors?

Give two examples of polar vectors.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

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Motion In A Plane

Four particles, each of mass M and equidistant from each other, move along a circle of radius R under the action of their mutual gravitational attraction. The speed of each particle is

Some More Questions From Motion in A Plane Chapter

Give three examples of vector quantities.

What are the basic characteristics that a quantity must possess so that it may be a vector quantity?

Is a physical quantity, having a direction, necessarily a vector?

Name a quantity which is both scalar as well as vector.

Which of the followings is/are vector quantities: work, power, acceleration, mass, momentum?

Pick up the odd one from the following: force, linear momentum, mass, impulse.

Pick out the only vector quantities in the following list:Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction and charge.

Pick out the two scalar quantities in the following list: force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, reaction as per Newton’s third law, relative velocity.

What are polar vectors?

Give two examples of polar vectors.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Pick out the only vector quantities in the following list:
Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction and charge.

Pick out the two scalar quantities in the following list:
force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, reaction as per Newton’s third law, relative velocity.