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Two particles of masses M and 2M are at distance D apart. Under their mutual gravitational force they start moving towards each other. The acceleration of their centre of mass when they are D/2 apart is?
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Two particles of masses M and 2M are at distance D apart. Under their ...
Understanding the System
- We have two particles with masses M and 2M, separated by a distance D.
- They exert a mutual gravitational force on each other, causing them to accelerate toward one another.
Gravitational Force Calculation
- The gravitational force \( F \) between the two particles is given by Newton's law of gravitation:
\[
F = \frac{G \cdot M \cdot (2M)}{D^2} = \frac{2GM^2}{D^2}
\]
Acceleration of Each Mass
- The acceleration \( a_1 \) of mass \( M \) due to this force is:
\[
a_1 = \frac{F}{M} = \frac{2GM^2}{D^2 M} = \frac{2GM}{D^2}
\]
- The acceleration \( a_2 \) of mass \( 2M \) is:
\[
a_2 = \frac{F}{2M} = \frac{2GM^2}{D^2 \cdot 2M} = \frac{GM}{D^2}
\]
Center of Mass Acceleration
- The center of mass (CM) of the system can be calculated using the formula:
\[
x_{CM} = \frac{M \cdot x_1 + 2M \cdot x_2}{M + 2M} = \frac{M \cdot x_1 + 2M \cdot x_2}{3M}
\]
- The acceleration of the center of mass \( a_{CM} \) can be found using:
\[
a_{CM} = \frac{F_{net}}{M_{total}} = \frac{F}{3M}
\]
- Substituting \( F \):
\[
a_{CM} = \frac{2GM^2/D^2}{3M} = \frac{2GM}{3D^2}
\]
Final Result
- Thus, the acceleration of the center of mass when the particles are \( D/2 \) apart is:
\[
a_{CM} = \frac{2GM}{3D^2}
\]
- We have two particles with masses M and 2M, separated by a distance D.
- They exert a mutual gravitational force on each other, causing them to accelerate toward one another.
Gravitational Force Calculation
- The gravitational force \( F \) between the two particles is given by Newton's law of gravitation:
\[
F = \frac{G \cdot M \cdot (2M)}{D^2} = \frac{2GM^2}{D^2}
\]
Acceleration of Each Mass
- The acceleration \( a_1 \) of mass \( M \) due to this force is:
\[
a_1 = \frac{F}{M} = \frac{2GM^2}{D^2 M} = \frac{2GM}{D^2}
\]
- The acceleration \( a_2 \) of mass \( 2M \) is:
\[
a_2 = \frac{F}{2M} = \frac{2GM^2}{D^2 \cdot 2M} = \frac{GM}{D^2}
\]
Center of Mass Acceleration
- The center of mass (CM) of the system can be calculated using the formula:
\[
x_{CM} = \frac{M \cdot x_1 + 2M \cdot x_2}{M + 2M} = \frac{M \cdot x_1 + 2M \cdot x_2}{3M}
\]
- The acceleration of the center of mass \( a_{CM} \) can be found using:
\[
a_{CM} = \frac{F_{net}}{M_{total}} = \frac{F}{3M}
\]
- Substituting \( F \):
\[
a_{CM} = \frac{2GM^2/D^2}{3M} = \frac{2GM}{3D^2}
\]
Final Result
- Thus, the acceleration of the center of mass when the particles are \( D/2 \) apart is:
\[
a_{CM} = \frac{2GM}{3D^2}
\]
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Two particles of masses M and 2M are at distance D apart. Under their mutual gravitational force they start moving towards each other. The acceleration of their centre of mass when they are D/2 apart is?
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Two particles of masses M and 2M are at distance D apart. Under their mutual gravitational force they start moving towards each other. The acceleration of their centre of mass when they are D/2 apart is? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Two particles of masses M and 2M are at distance D apart. Under their mutual gravitational force they start moving towards each other. The acceleration of their centre of mass when they are D/2 apart is? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two particles of masses M and 2M are at distance D apart. Under their mutual gravitational force they start moving towards each other. The acceleration of their centre of mass when they are D/2 apart is?.
Two particles of masses M and 2M are at distance D apart. Under their mutual gravitational force they start moving towards each other. The acceleration of their centre of mass when they are D/2 apart is? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Two particles of masses M and 2M are at distance D apart. Under their mutual gravitational force they start moving towards each other. The acceleration of their centre of mass when they are D/2 apart is? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two particles of masses M and 2M are at distance D apart. Under their mutual gravitational force they start moving towards each other. The acceleration of their centre of mass when they are D/2 apart is?.
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