Gravitation

Step 1: In SHM, points where velocity is zero are the extreme positions. Hence, and are the two extremes.
Step 2: Since both and lie on the same side of ,
Step 3: The midpoint between and is at distance: from the origin measured from the nearer extreme, so it is the mean position of motion between and .
Step 4: Maximum speed in SHM occurs at the mean position and is given by: Here,
Step 5: Hence,
Step 6: Time period of SHM: But motion from one extreme to the other corresponds to half a cycle, hence effective period for the given configuration is:

Step 1: Understand the physical meaning. To launch the spaceship from the earth's surface into free space, the minimum energy required is equal to the gravitational potential energy needed to escape Earth's gravitational field.
Step 2: Write the expression for escape energy. The energy required to take a mass from the earth's surface to infinity is: Using the relation , we get:
Step 3: Substitute the given values. Radius of earth:
Step 4: Account for total energy required. To completely escape Earth's gravitational field, twice this potential energy is required when starting from rest: Final Answer:

Step 1: Near Earth surface, P.E. change:
Step 2: Height .
Step 3: But the question states object lifted from surface to point above surface, total height from centre becomes . Using inverse field formula:
Step 4: Exact formula:
Step 5: Replace : Closest numeric → (C).

Step 1: Recall the definition of centre of mass. The centre of mass is the point at which the entire mass of a body may be considered to be concentrated for the purpose of analyzing motion. Step 2: Understand possible locations. Depending on the shape and mass distribution of the body:

It may lie inside the body (e.g., solid sphere)
It may lie outside the body (e.g., ring, hollow sphere)
Step 3: Evaluate the options.

(A) Only inside — incorrect
(B) Only outside — incorrect
(C) Neither — incorrect
(D) Either inside or outside — correct ✔
Step 4: Final conclusion. The centre of mass of a rigid body may lie either inside or outside the body.