Acceleration

Let be the acceleration due to gravity at the earth's surface. The acceleration due to gravity at a height above the surface is given by:

The acceleration due to gravity at a depth below the surface is given by:

According to the problem, . Therefore:

We can cancel from both sides and solve the resulting equation:

For simplicity, let's approximate using binomial expansion, considering :

Thus, the equation simplifies to:

Expanding the left side using binomial approximation again:

Equating coefficients of :

Thus, the value of is , which fits within the given range of .

Step 1: Calculate the Retardation
Given initial velocity , distance , and final velocity . Using the third equation of motion:

The negative sign indicates retardation.

Step 2: Calculate the Velocity at Half the Distance
Now, the brakes are applied at half the distance, so . The initial velocity is still . We need to find the final velocity ( ) when the train crosses the station. Using the third equation of motion:

Step 3: Find the Value of
The velocity is given as . We have found that . Therefore,

Conclusion: The value of is 200.

• Statement I: The acceleration due to gravity (g') decreases when moving away from the earth's surface either upwards or downwards. The formulae for variations are:

(for height h above surface)

(for depth d below surface)

Both variations indicate a decrease in g.

• Statement II: At equal height h and depth d (h = d), the values of g' are not the same. This is because g decreases faster as we go up (due to the inverse square law) compared to going down (due to linear variation inside the earth).

Thus, Statement I is correct, and Statement II is incorrect.

Understanding the effect of Earth's rotation on gravity.
Due to the rotation of the Earth, the effective acceleration due to gravity is given by: Where is the angle made with the equator.
At the poles, where , the change in gravity is zero because . This shows no effect on the poles. For the equator, where , the change in gravity is maximum: Thus, the change in gravity is maximum at the equator and zero at the poles.
This contradicts Statement II, but Statement I is correct.

To solve the problem, we need to understand how acceleration due to gravity, , depends on the dimensions and mass of the Earth. The formula for gravitational acceleration at the surface of a planet is given by:

where:

• is the gravitational constant.
• is the mass of the Earth.
• is the radius of the Earth.

According to the problem, the diameter of the Earth is reduced to half, which means the radius will also be reduced to half. So, the new radius .

Since the mass of the Earth remains constant, we can substitute into the formula for gravity:

This simplifies to:

The above expression shows that the new acceleration due to gravity becomes:

Therefore, if the diameter of the Earth is reduced to half and the mass remains constant, the acceleration due to gravity on the surface of the Earth would be , which corresponds to the correct answer: 4g.

To find the linear acceleration of the cylinder axis as it rolls down the inclined plane, we need to analyze the forces acting on the cylinder and apply Newton's second law of motion along with the rotational dynamics. We are given that the inclined plane is rough, allowing the cylinder to roll without slipping.

Given:

• Angle of inclination,
• Gravitational acceleration,

Concept: For a cylinder rolling without slipping, the relationship between linear acceleration and angular acceleration is given by:

The moment of inertia for a solid cylinder about its axis is:

Applying Newton's second law along the incline, we have:

where is the frictional force.

For rotational motion:

Substituting gives:

Substituting the expression for in the linear motion equation:

Simplifying this equation, we get:

Canceling and solving for :

Substituting :

Hence, the linear acceleration is:

This is the correct answer.

Conclusion: The linear acceleration of the cylinder axis as it rolls down the inclined plane is .