Gravitation

According to the law of conservation of angular momentum, ......(i) From the law of conservation of total mechanical energy. ......(ii) From Eqs. (i) and (ii), we get Angular momentum,

Step 1: Formula for Acceleration Due to Gravity at a Height
The acceleration due to gravity at a height from the surface of the Earth is given by: where is the acceleration due to gravity at height , is the acceleration due to gravity on the Earth's surface, and is the Earth's radius.
Step 2: Given Ratio of Gravity Values
It is given that: and , so using the formula:
Step 3: Finding
Let , then: Using the given ratio:
Step 4: Conclusion
Thus, , so the correct answer is option (B).

Step 1: Analyze each statement about the gravitational force.
(A) Conservative force: A force is conservative if the work done by it in moving a particle between two points is independent of the path taken. The gravitational force is a classic example of a conservative force. The work done depends only on the change in gravitational potential energy. So, this statement is correct.
(B) Attractive force: The gravitational force between any two masses is always attractive; it pulls the objects toward each other. This statement is correct.
(C) Central force: A central force is a force that is always directed towards or away from a single point (a center) and whose magnitude depends only on the distance from that center. The gravitational force between two point masses acts along the line joining them and depends only on the distance between them. Therefore, it is a central force. The statement "Not a central force" is incorrect.
(D) Contact force: Contact forces are forces that act at the point of contact between two objects (like friction, normal force). The gravitational force acts between objects even when they are separated by a large distance (action-at-a-distance). Therefore, it is not a contact force. This statement is correct.

Step 2: Identify the incorrect statement.
Based on the analysis, the only incorrect statement is (C). The gravitational force is a central force.

The orbital speed ( ) of a satellite revolving around the Earth at a distance from the center of the Earth is given by the formula:
, where G is the gravitational constant and M is the mass of the Earth.
This formula shows that .
Let's consider two cases.
Case 1: Orbit near the surface of the Earth.
The distance from the center is the radius of the Earth, km.
The orbital speed is given as km/s.
Case 2: Orbit at a height km from the surface.
The distance from the center is km.
The orbital speed is , which we need to find.
Using the proportionality, we can write a ratio:
.
.
.
Solving for :
km/s.

Step 1: Understanding the Concept:
This problem can be solved using the principle of conservation of total mechanical energy. The total energy (kinetic + potential) of the body remains constant throughout its journey. We will compare the total energy at the Earth's surface with the total energy at an infinite distance (where it has "escaped" the gravitational influence).
Step 2: Key Formulae:
Let M and R be the mass and radius of the Earth.
Kinetic Energy:
Gravitational Potential Energy:
Escape Speed: , which implies .
Conservation of Energy: .
Step 3: Detailed Explanation:
Let the initial projection speed be . We are given .
Let the final speed at infinity be .
Initial Energy (at Earth's surface, r=R):
Final Energy (at infinity, r= ):
At an infinite distance, the gravitational potential energy is zero.
Applying Conservation of Energy:
Divide the entire equation by :
Now, substitute and :
Multiply the entire equation by 2:
Taking the square root of both sides:
Step 4: Final Answer:
The speed of the body when it escapes the Earth's gravitational influence is . Option (B) is correct.

• Weight at height above the surface: • For , N.
• For , • Hence, the weight at height is 360 N.

1. By the shell theorem, a particle outside a spherical shell experiences a gravitational force as if the shell's entire mass were concentrated at its center.
2. Force due to shell 1:
3. Force due to shell 2:
4. Net force: