As shown in the figure, two spherical cavities are made in the uniform solid sphere of radius R. The boundaries of the cavities touch at the centre of the sphere. The centers of the cavities and the sphere lie on the x-axis. The mass of the solid sphere before the cavities were created was M. The gravitational force on a point mass m at a distance 'd' away from the center of the solid sphere is
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Official Solution
Correct Option: (1)
Problem Summary: A solid sphere of mass and radius has two spherical cavities (each of radius ) carved such that they touch at the center. A point mass lies on the axis of symmetry at a distance from the center of the sphere ( ). Find the net gravitational force on .
Step 1: Use Superposition Principle
The net gravitational force is:
The force due to the original full solid sphere
Minus the force due to the two missing cavity masses (as if they were present and exerting gravitational pull)
Step 2: Compute Density and Cavity Mass
Density of original sphere:
Volume of each cavity
Mass of each cavity if it were filled:
Step 3: Gravitational Forces
Force from original sphere:
Force from first cavity at :
Force from second cavity at :
Step 4: Net Force
By superposition:
Factor :
Convert into normalized form:
✅ Final Answer:
This matches option (a).
02
PYQ 2022
medium
physicsID: ap-eapce
When a ball is dropped from a height it takes sec to reach the ground. If the same experiment is done on a different planet having the mass 100 times the earth’s mass and radius 10 times the earth’s radius, then the time it will take to cover the same height is
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Official Solution
Correct Option: (1)
The time taken for a ball to fall from a height under gravity is determined using the equation of motion , where is the acceleration due to gravity and is the time. - On Earth: Let the acceleration due to gravity be , the mass of Earth be , and the radius of Earth be . The time to fall height is . Using the equation: - On the other planet: The planet has mass and radius . The acceleration due to gravity is given by , where is the gravitational constant. For the planet: The acceleration due to gravity on the planet is the same as on Earth, . Now, calculate the time to fall the same height on the planet: Since , the time is equal to . Thus, the time to fall the same height on the planet is , which matches option (1). Thus, the correct answer is (1).
03
PYQ 2022
medium
physicsID: ap-eapce
Statement (A): Two artificial satellites revolving in the same circular orbit have the same period of revolution.
Statement (B): The orbital velocity is inversely proportional to the square root of the radius of the orbit.
Statement (C): The escape velocity of a body is independent of the altitude of the point of projection.
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A, B, C are true
2
A, B true; C false
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A, C true; B false
4
B, C true; A false
Official Solution
Correct Option: (2)
Statement A is true: Satellites in the same orbit have the same radius and hence same time period, derived from . Statement B is true: Orbital velocity is , which implies . Statement C is false: Escape velocity depends on the distance from the center of the Earth, i.e., altitude affects the value. is valid only at surface.
04
PYQ 2022
medium
physicsID: ap-eapce
A projectile is thrown straight upward from the earth’s surface with an initial speed , where is a constant and is the escape speed. The projectile travels up to a height 800 km from the earth’s surface, before it comes to rest. The value of the constant is:
(Take radius of the earth )
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Official Solution
Correct Option: (1)
From energy conservation:
Initial KE = Final PE – Initial PE
Escape speed:
05
PYQ 2022
medium
physicsID: ap-eapce
A rubber band catapult has initial length and cross-sectional area . It is stretched to twice its length and then released to project a stone of mass . The velocity of the projected stone is:(Young's modulus of rubber )
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3
4
Official Solution
Correct Option: (4)
Step 1: Use elastic potential energy:
Where:
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-
-
- Step 2: Equating to kinetic energy:
06
PYQ 2022
medium
physicsID: ap-eapce
A solid sphere of radius has acceleration at surface. At what distance from center is ?
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3
4
Official Solution
Correct Option: (1)
Outside solid sphere, gravitational acceleration varies as: Wait — correction: Mistake! Correct answer is (3) . Image key is incorrect.
07
PYQ 2023
medium
physicsID: ap-eapce
Which of the following statements are true about acceleration due to gravity ?
A. is greater at poles.
B. decreases with height.
C. is same all over Earth.
D. is maximum at centre of Earth.
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A and B
2
A and D
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B and C
4
C and D
Official Solution
Correct Option: (1)
- At poles, is more due to lesser centrifugal force.
- decreases with height as
- C is false: varies over Earth. D is false: at centre of Earth.
08
PYQ 2023
medium
physicsID: ap-eapce
The time period of a simple pendulum on the surface of the Earth is . At what height above the surface will the time period become ?
(Radius of Earth = )
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Official Solution
Correct Option: (2)
Time period
At height ,
Given:
09
PYQ 2023
medium
physicsID: ap-eapce
A satellite is plad in a circular orbit around the earth at an altitude of 1000 km. The time period of the satellite in minutes is approximately (mass of the earth kg, radius of the earth m, Nm kg )
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Official Solution
Correct Option: (1)
Total orbital radius m.
Kepler’s 3rd law: . Compute .
.
Then, .
s minutes. Hence, answer is 105 minutes.
10
PYQ 2023
medium
physicsID: ap-eapce
The ratio of inertial mass to the gravitational mass of a body is
1
1 : 1
2
1 : 2
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2 : 1
4
1 : 4
Official Solution
Correct Option: (1)
According to the principle of equivalence in classical physics, the inertial mass and gravitational mass of a body are equal in magnitude. Hence, their ratio is 1:1. This fundamental equivalence was confirmed by various experiments and is also a cornerstone in Einstein’s theory of General Relativity.
11
PYQ 2023
medium
physicsID: ap-eapce
The gravitational potential energy of a system of three masses , , and placed at the three vertices of an equilateral triangle of side isOptions:
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Official Solution
Correct Option: (3)
The gravitational potential energy between two masses and separated by a distance is given by:
For the three masses , , and placed at the vertices of an equilateral triangle of side , the potential energy is the sum of the potential energies between each pair of masses. Step 1: Calculate potential energy between each pair of masses. Between and : Between and : Between and : Step 2: Total gravitational potential energy.
The total potential energy is the sum of the individual potential energies:
Final Answer:
12
PYQ 2023
medium
physicsID: ap-eapce
Two objects separated by a distance r gravitationally attract each other with force F. If the distance between them is tripled, the force of attraction between them is
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Official Solution
Correct Option: (2)
According to Newton's law of universal gravitation, the force of attraction is inversely proportional to the square of the distance between two objects:
If the distance between the objects is tripled, the new force becomes:
Thus, the new force is .
13
PYQ 2023
medium
physicsID: ap-eapce
A hole is drilled from one end to the other end of Earth and an object of mass is dropped down the hole. The gravitational force acting on the object as a function of distance from the center of Earth is (Assume mass of Earth = , radius = , and uniform density)
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Official Solution
Correct Option: (4)
Inside the Earth, the gravitational force varies linearly with distance from the center due to the shell theorem.
The force is given by:
Here, is the mass of Earth, is the radius of Earth, and is the distance from the center. The direction is radial, hence .
14
PYQ 2024
medium
physicsID: ap-eapce
A particle is projected from the surface of the Earth with a velocity equal to twice the escape velocity. When the particle is far from the Earth, its speed will be:
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Official Solution
Correct Option: (3)
Step 1: Using Energy Conservation The total mechanical energy of the particle is: For escape velocity: Step 2: Applying Given Condition The initial velocity given is : Substituting : At infinity, kinetic energy remains: Solving for : Thus, the correct answer is option (3).
15
PYQ 2025
medium
physicsID: ap-eapce
A satellite is launched into a circular orbit of radius R around the earth. A second satellite is launched into another circular orbit of radius 1.01 R around the earth. The period of revolution of the second satellite is larger than that of the first one by a percent of (approximately)
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0.5
2
1.0
3
1.5
4
3.0
Official Solution
Correct Option: (3)
Step 1: Apply Kepler's Third Law of Planetary Motion.
Kepler's Third Law states that for any satellite orbiting a central body (like the Earth), the square of its orbital period ( ) is directly proportional to the cube of the semi-major axis (which is the radius for a circular orbit).
Mathematically, this can be written as:
Or, , where is a constant. Step 2: Set up the ratio of periods for the two satellites.
Let be the period of the first satellite and its orbital radius.
Let be the period of the second satellite and its orbital radius. From the problem statement:
Using Kepler's Third Law for both satellites:
Now, take the ratio of the squares of the periods:
Substitute the given values for and :
Step 3: Calculate the ratio of the periods.
Take the square root of both sides:
Since , we can use the binomial approximation for small . Here, and .
Step 4: Calculate the percentage increase in the period.
The percentage increase is given by:
Substitute the calculated ratio:
The final answer is .
16
PYQ 2025
easy
physicsID: ap-eapce
Two objects of masses 5 kg and 10 kg are placed 2 meters apart. What is the gravitational force between them? (Use )
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N
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N
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N
4
Official Solution
Correct Option: (1)
- Given: Mass , Mass , Distance , Gravitational constant
- Gravitational force is given by Newton’s law of gravitation:
- Substitute values:
- Rounded to N (check options carefully): Actually, the above result is N, which matches none exactly, but the closest option is (A) N if there is a typo in options.
- Rechecking calculation carefully:
So the force is approximately N, closest to none of the given options exactly. Maybe the options have a typo or intended to be scale. If the distance is 5 m instead of 2, force reduces.
Assuming options typo, the answer is:
If we consider options as is, option (B) N is 10 times less than the calculation. So please verify question data.
Answer:
17
PYQ 2025
medium
physicsID: ap-eapce
If an object of mass 1 kg is taken to a height which is equal to three times the radius of the earth, then the change in its potential energy is
(Radius of the earth , acceleration due to gravity )
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Official Solution
Correct Option: (1)
Step 1: Gravitational potential energy at a distance :Step 2: Potential energy difference:Step 3: Use
18
PYQ 2025
medium
physicsID: ap-eapce
Two solid spheres each of radius made of same material are placed in contact with each other. If the gravitational force acting between them is , then
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Official Solution
Correct Option: (1)
Step 1: Mass of a solid sphere
Each sphere is of radius and made of the same material, so their mass is proportional to the volume:
Step 2: Gravitational force between two masses
Since they are in contact, the distance between centers
19
PYQ 2025
medium
physicsID: ap-eapce
The acceleration due to gravity at a height of from the surface of the earth is (where and is the radius of the earth)
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4
Official Solution
Correct Option: (3)
Acceleration due to gravity at height is given by: Given: Calculate:
20
PYQ 2025
medium
physicsID: ap-eapce
If the escape velocity of a body from the surface of the earth is 11.2 km/s, then the orbital velocity of a satellite in an orbit which is at a height equal to the radius of the earth is
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11.2 km/s
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2.8 km/s
3
22.4 km/s
4
5.6 km/s
Official Solution
Correct Option: (4)
Escape velocity , where is the gravitational constant, is the mass of the earth, and is the radius of the earth. Orbital velocity at a height is given by . Given , . Since km/s, km/s. . Since then , but the closest option given is 5.6. , and . so . so .
21
PYQ 2025
medium
physicsID: ap-eapce
An artificial satellite is revolving around a planet of radius in a circular orbit of radius . If the time period of revolution of the satellite, , then the values of and are respectively:
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2
3
4
Official Solution
Correct Option: (4)
Step 1: Start with Kepler’s Third Law But Step 2: Substitute in the original formulaStep 3: Compare with given form Given:
22
PYQ 2025
medium
physicsID: ap-eapce
The time period of a simple pendulum on the surface of the earth is T. If the pendulum is taken to a height equal to half of the radius of the earth, then its time period is
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T/2
2
3T/2
3
2T
4
3T
Official Solution
Correct Option: (3)
The time period of a simple pendulum is given by , where is the length of the pendulum and is the acceleration due to gravity. At the surface of the earth, . At a height above the surface, the acceleration due to gravity is given by . The new time period . However, the given options are incorrect and does not contain the term. At height , . If then . Then If the pendulum is taken to a height , where is the Earth's radius. The acceleration due to gravity at a height is given by . If we use . The new time period will be . If we consider then .
