Gravitation Chapter-Wise Test 10

Correct answer Carries: 4.

Wrong Answer Carries: -1.

What is the minimum speed required to escape from a point \( R_E \) above Earth’s surface? (\( R_E = 6.4 \times 10^6 \, \text{m}, g = 9.8 \, \text{m/s}^2 \))

\( v_e = \sqrt{\frac{2 G M_E}{R_E + h}} = \sqrt{\frac{2 g R_E^2}{2 R_E}} = \sqrt{\frac{g R_E}{2}} \).

\( v_e = \sqrt{\frac{9.8 \times 6.4 \times 10^6}{2}} = \sqrt{3.136 \times 10^7} \).

\( v_e \approx 5.6 \times 10^3 \, \text{m/s} = 5.6 \, \text{km/s} \).

5.0 km/s
5.6 km/s
6.0 km/s
6.5 km/s
2

Why does the gravitational potential energy of two masses become more negative as they get closer?

\( V = -\frac{G M m}{r} \). As \( r \) decreases, the term \( -\frac{G M m}{r} \) becomes larger in magnitude (more negative), reflecting increased binding energy due to stronger attraction.

Force decreases
Distance decreases
Mass increases
Energy becomes positive
2

Three masses of \( 15 \, \text{kg} \) each form an equilateral triangle with side \( 10 \, \text{m} \). What is the net force on one mass? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Force between two masses: \( F = G \frac{m^2}{r^2} = 6.67 \times 10^{-11} \frac{15 \times 15}{10^2} = 1.50 \times 10^{-10} \, \text{N} \).

Two forces at 60°: \( F_R = \sqrt{F^2 + F^2 + 2 F^2 \cos 60^\circ} \).

\( F_R = \sqrt{(1.50 \times 10^{-10})^2 (1 + 1 + 1)} = 1.50 \times 10^{-10} \sqrt{3} \).

\( F_R \approx 2.60 \times 10^{-10} \, \text{N} \).

2.5 × 10⁻¹⁰ N
2.6 × 10⁻¹⁰ N
2.7 × 10⁻¹⁰ N
2.8 × 10⁻¹⁰ N
2

What happens to the gravitational force on a point mass inside a hollow spherical shell?

The gravitational force on a point mass inside a hollow spherical shell of uniform density is zero. This is because forces from all parts of the shell cancel out due to symmetry, leaving no net force.

It increases with distance from the center
It remains zero
It acts toward the center
It depends on the shell’s thickness
2

A satellite orbits a planet at \( 4 \times 10^7 \, \text{m} \) from its center with a period of 8 hours. What is the planet’s mass? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

\( M = \frac{4\pi^2 r^3}{G T^2} \).

\( T = 8 \times 3600 = 28800 \, \text{s} \), \( T^2 = 8.2944 \times 10^8 \, \text{s}^2 \).

\( r^3 = (4 \times 10^7)^3 = 6.4 \times 10^{22} \, \text{m}^3 \).

\( M = \frac{4 \times (3.14)^2 \times 6.4 \times 10^{22}}{6.67 \times 10^{-11} \times 8.2944 \times 10^8} \).

\( M = \frac{2.523 \times 10^{24}}{5.532 \times 10^{-2}} \approx 4.56 \times 10^{25} \, \text{kg} \).

4.4 × 10²⁵ kg
4.5 × 10²⁵ kg
4.6 × 10²⁵ kg
4.7 × 10²⁵ kg
3

Why can’t the gravitational force on a point mass outside a spherical shell be zero?

Outside a spherical shell, the gravitational force acts as if all the mass is concentrated at the center. Since the mass is non-zero and the distance is finite, the force (\( F = \frac{G M m}{r^2} \)) cannot be zero.

The shell has no mass
The force acts as if from the center
The shell’s thickness cancels the force
The force is repulsive
2

Which of the following statements is incorrect about escape speed?

Escape speed (\( v_e = \sqrt{\frac{2 G M}{R}} \)) is independent of the object’s mass (option 1 correct), depends on \( M \) and \( R \) (option 2 correct), and decreases with altitude (option 4 correct). Option 3 is incorrect as it does depend on the planet.

It is independent of the object’s mass
It depends on the planet’s mass
It is independent of the planet
It decreases with altitude
3

What does Kepler’s third law reveal about the motion of planets farther from the Sun?

Kepler’s third law (\( T^2 \propto a^3 \)) shows that planets farther from the Sun (larger semi-major axis \( a \)) have longer orbital periods (\( T \)), as the period increases with the distance cubed.

They move faster
They have longer periods
They have circular orbits
They have constant speed
2

Why can’t a satellite in a circular orbit have zero total energy?

\( E = -\frac{G M m}{2 r} \) for a circular orbit, which is negative due to the bound state. Zero total energy implies the satellite could escape to infinity, which contradicts a stable orbit.

It would fall to Earth
It is bound with negative energy
Kinetic energy is zero
Potential energy is zero
2

Which of the following statements is incorrect about acceleration due to gravity?

\( g \) decreases above (option 1 correct) and below (option 3 correct) the surface, and is zero at the center (option 2 correct). Option 4 is incorrect as \( g \) is maximum at the surface, not below.

It decreases with altitude
It is zero at Earth’s center
It decreases with depth
It is maximum below the surface
4

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