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Infinite number of bodies, each of mass 2 kg are situated on x-axis at distances 1 m, 2 m, 4 m, 8 m ,..., respectively, from the origin. The resulting gravitational potential due to this system at the origin will be
- a)−4/3G
- b)−4G
- c)−G
- d)−8/3G
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
Infinite number of bodies, each of mass 2 kg are situated on x-axis a...
Gravitational potential is the amount of work done in bringing a unit mass from infinity to a point in the gravitational field. In this case, we need to calculate the gravitational potential at the origin due to an infinite number of bodies located at specific distances along the x-axis.
Let's analyze the problem step by step:
1. Calculation of the gravitational potential due to a single body:
The gravitational potential due to a single body of mass M at a distance r from the origin is given by the equation:
V = -GM/r
where G is the universal gravitational constant.
2. Calculation of the gravitational potential due to the infinite number of bodies:
To calculate the total gravitational potential at the origin due to the infinite number of bodies, we need to sum up the contributions from each individual body.
The first body is located at a distance of 1 meter and has a mass of 2 kg. The gravitational potential due to this body at the origin is:
V1 = -GM/1
The second body is located at a distance of 2 meters and has a mass of 2 kg. The gravitational potential due to this body at the origin is:
V2 = -GM/2
The third body is located at a distance of 4 meters and has a mass of 2 kg. The gravitational potential due to this body at the origin is:
V3 = -GM/4
And so on...
To find the total gravitational potential at the origin, we need to sum up all these individual potentials:
Vtotal = V1 + V2 + V3 + ...
3. Calculation of the total gravitational potential:
Since the distances between the bodies are doubling each time, we can write the sum of the potentials as a geometric series:
Vtotal = -GM(1/1 + 1/2 + 1/4 + ...)
Using the formula for the sum of an infinite geometric series, we can simplify this expression:
Vtotal = -GM(1/(1 - 1/2))
Vtotal = -GM(2)
Vtotal = -2GM
Therefore, the total gravitational potential at the origin due to the infinite number of bodies is -2GM.
Comparing this with the given answer options, we see that the correct answer is option B: -4G.
Let's analyze the problem step by step:
1. Calculation of the gravitational potential due to a single body:
The gravitational potential due to a single body of mass M at a distance r from the origin is given by the equation:
V = -GM/r
where G is the universal gravitational constant.
2. Calculation of the gravitational potential due to the infinite number of bodies:
To calculate the total gravitational potential at the origin due to the infinite number of bodies, we need to sum up the contributions from each individual body.
The first body is located at a distance of 1 meter and has a mass of 2 kg. The gravitational potential due to this body at the origin is:
V1 = -GM/1
The second body is located at a distance of 2 meters and has a mass of 2 kg. The gravitational potential due to this body at the origin is:
V2 = -GM/2
The third body is located at a distance of 4 meters and has a mass of 2 kg. The gravitational potential due to this body at the origin is:
V3 = -GM/4
And so on...
To find the total gravitational potential at the origin, we need to sum up all these individual potentials:
Vtotal = V1 + V2 + V3 + ...
3. Calculation of the total gravitational potential:
Since the distances between the bodies are doubling each time, we can write the sum of the potentials as a geometric series:
Vtotal = -GM(1/1 + 1/2 + 1/4 + ...)
Using the formula for the sum of an infinite geometric series, we can simplify this expression:
Vtotal = -GM(1/(1 - 1/2))
Vtotal = -GM(2)
Vtotal = -2GM
Therefore, the total gravitational potential at the origin due to the infinite number of bodies is -2GM.
Comparing this with the given answer options, we see that the correct answer is option B: -4G.
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Community Answer
Infinite number of bodies, each of mass 2 kg are situated on x-axis a...
The resulting gravitational potential at the origin O due to each of mass 2 kg located at positions as shown in figure is
= 
= -4G
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Infinite number of bodies, each of mass 2 kg are situated on x-axis at distances 1 m, 2 m, 4 m, 8 m ,..., respectively, from the origin. The resulting gravitational potential due to this system at the origin will bea)−4/3Gb)−4Gc)−Gd)−8/3GCorrect answer is option 'B'. Can you explain this answer?
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