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Two spherical bodies of mass M and 5M and radii R and 2R, respectively, are released in free space with initial separation between their centres equal to 12R. If they attract each other due to gravitational force only, then the distance covered by the smaller body, just before collision, is
- a)2.5R
- b)4.5R
- c)7.5R
- d)1.5R
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Two spherical bodies of mass M and 5M and radii R and 2R, respectivel...
Given information:
- Mass of smaller body (m1) = M
- Mass of larger body (m2) = 5M
- Radius of smaller body (r1) = R
- Radius of larger body (r2) = 2R
- Initial separation between their centers (d) = 12R
To find: Distance covered by the smaller body just before collision.
Gravitational force between two bodies:
The gravitational force (F) between two bodies is given by the equation:
F = G * (m1 * m2) / d^2
Where,
G is the universal gravitational constant.
Step 1: Finding the gravitational force between the bodies at initial separation
Using the equation mentioned above, we can find the gravitational force (F_initial) between the bodies at the initial separation (d).
F_initial = G * (M * 5M) / (12R)^2
= G * 5M^2 / (144R^2)
= 5GM^2 / (144R^2) -- Equation (1)
Step 2: Finding the gravitational force between the bodies just before collision
When the bodies are about to collide, the separation between their centers becomes the sum of their radii.
Separation just before collision = r1 + r2 = R + 2R = 3R
Using the equation mentioned above, we can find the gravitational force (F_collision) between the bodies just before collision.
F_collision = G * (M * 5M) / (3R)^2
= G * 5M^2 / (9R^2)
= 5GM^2 / (9R^2) -- Equation (2)
Step 3: Finding the distance covered by the smaller body just before collision
As the bodies are only under the influence of gravitational force, the work done by the gravitational force is given by the equation:
Work = Force * Distance
The work done by the gravitational force is equal to the change in potential energy.
Potential energy before collision = -G * (m1 * m2) / d
Potential energy just before collision = -G * (m1 * m2) / (r1 + r2)
The change in potential energy can be calculated as:
Change in potential energy = Potential energy before collision - Potential energy just before collision
Change in potential energy = [-G * (M * 5M) / d] - [-G * (M * 5M) / (r1 + r2)]
= -G * (M * 5M) * [1/d - 1/(r1 + r2)]
Change in potential energy = -G * (M * 5M) * [(r1 + r2 - d) / (d * (r1 + r2))]
Change in potential energy = -G * (M * 5M) * [(3R - 12R) / (12R * (3R))]
Change in potential energy = -G * (M * 5M) * [(-9R) / (36R^2)]
Change in potential energy = G * (M * 5M) * [9R / (36R^2)]
Change in potential energy = G * (M * 5M) * (1 / 4
- Mass of smaller body (m1) = M
- Mass of larger body (m2) = 5M
- Radius of smaller body (r1) = R
- Radius of larger body (r2) = 2R
- Initial separation between their centers (d) = 12R
To find: Distance covered by the smaller body just before collision.
Gravitational force between two bodies:
The gravitational force (F) between two bodies is given by the equation:
F = G * (m1 * m2) / d^2
Where,
G is the universal gravitational constant.
Step 1: Finding the gravitational force between the bodies at initial separation
Using the equation mentioned above, we can find the gravitational force (F_initial) between the bodies at the initial separation (d).
F_initial = G * (M * 5M) / (12R)^2
= G * 5M^2 / (144R^2)
= 5GM^2 / (144R^2) -- Equation (1)
Step 2: Finding the gravitational force between the bodies just before collision
When the bodies are about to collide, the separation between their centers becomes the sum of their radii.
Separation just before collision = r1 + r2 = R + 2R = 3R
Using the equation mentioned above, we can find the gravitational force (F_collision) between the bodies just before collision.
F_collision = G * (M * 5M) / (3R)^2
= G * 5M^2 / (9R^2)
= 5GM^2 / (9R^2) -- Equation (2)
Step 3: Finding the distance covered by the smaller body just before collision
As the bodies are only under the influence of gravitational force, the work done by the gravitational force is given by the equation:
Work = Force * Distance
The work done by the gravitational force is equal to the change in potential energy.
Potential energy before collision = -G * (m1 * m2) / d
Potential energy just before collision = -G * (m1 * m2) / (r1 + r2)
The change in potential energy can be calculated as:
Change in potential energy = Potential energy before collision - Potential energy just before collision
Change in potential energy = [-G * (M * 5M) / d] - [-G * (M * 5M) / (r1 + r2)]
= -G * (M * 5M) * [1/d - 1/(r1 + r2)]
Change in potential energy = -G * (M * 5M) * [(r1 + r2 - d) / (d * (r1 + r2))]
Change in potential energy = -G * (M * 5M) * [(3R - 12R) / (12R * (3R))]
Change in potential energy = -G * (M * 5M) * [(-9R) / (36R^2)]
Change in potential energy = G * (M * 5M) * [9R / (36R^2)]
Change in potential energy = G * (M * 5M) * (1 / 4
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Community Answer
Two spherical bodies of mass M and 5M and radii R and 2R, respectivel...
Let the spheres collide after time t, when the smaller sphere covered distance x1 and bigger sphere covered distance x2.
The gravitational force acting between two spheres depends on the distance (x), which is a variable quantity.
The gravitational force, F(x) = 
Acceleration of smaller body, a1(x) = 
Acceleration of bigger body, a2(x) = 
From equation of motion,
x1 = 1/2a1(x)t2 and x2 = 1/2a2(x)t2
= 5x1 = 5x2
We know that x1 + x2 = 9R
x1 + x1 / 5 = 9R
x1 = 45R/6 = 7.5R
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Two spherical bodies of mass M and 5M and radii R and 2R, respectively, are released in free space with initial separation between their centres equal to 12R. If they attract each other due to gravitational force only, then the distance covered by the smaller body, just before collision, isa)2.5Rb)4.5Rc)7.5Rd)1.5RCorrect answer is option 'C'. Can you explain this answer?
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