Mechanics

The velocity of a comet follows the law of conservation of angular momentum: where: is the mass of the comet, and are the velocities at the nearest and farthest points, and and are the corresponding distances. Using the given values:

When springs are arranged in parallel, the effective spring constant is the sum of the individual spring constants. Therefore, the effective spring constant for this assembly is .

The equilibrium force is calculated by vector addition of the forces. Resolving the components and summing them leads to an equilibrium force of 2.8 N towards the northwest direction.

In the moving lift, the effective acceleration is , so the time taken for the body to slide down is given by:

To prevent the block from falling, the frictional force must be greater than or equal to the weight of the block.
The maximum static frictional force is given by: where is the normal force (equal to ). The block will move with the cart if the frictional force is sufficient to keep the block in place.
The maximum force required to prevent the block from falling is the force of acceleration, .
For the block to remain in place, we need: Simplifying the equation, we get: Therefore, the condition for the block to not fall is .
Hence, the correct answer is (c).

In an elastic collision, the relative velocities of approach and separation are equal.
For two objects with equal mass, the final velocities can be determined using the following equations derived from the law of conservation of momentum and energy.
For ball A: Where is the angle of the collision, and given that the collision is elastic, the final velocity of ball A will be 6 m/s.

Thus, the correct answer is (b).

In an elastic collision, both momentum and kinetic energy are conserved.
However, the coefficient of restitution is defined by the relative velocities of the balls before and after the collision: where and are the velocities of the balls before the collision, and and are the velocities of the balls after the collision. Given: - Before the collision, the first ball is moving at , and the second ball is at rest. - The kinetic energy after the collision is of the original. From the conservation of kinetic energy: Since the total kinetic energy is reduced to , we can write: Simplifying, we find: Now, applying the coefficient of restitution formula and solving the system, we find:
Thus, the correct answer is (b).

The moment of inertia for a solid sphere is and for a hollow cylinder, it is .
When torques of equal magnitudes are applied, the angular acceleration is given by: Since the cylinder has a larger moment of inertia compared to the solid sphere, the angular acceleration of the solid sphere will be greater, leading to a greater angular speed for the sphere.
Thus, the correct answer is (c).

Using the law of conservation of momentum, the total momentum before firing must equal the total momentum after firing.
Initially, both the shell and the gun are at rest, so the total initial momentum is zero. Let the recoil speed of the gun be .
The momentum of the system after the shell is fired is: Substituting the given values: Solving for : The negative sign indicates the direction of motion, but the speed is .
Thus, the correct answer is (a).

When a cylinder rolls up an inclined plane, its motion is a combination of translational and rotational motion. The total mechanical energy is conserved. The speed of the cylinder at the bottom of the incline is given as 5 m/s. We can use the principle of conservation of energy to find the time it will take to return to the bottom.

Since the energy is conserved, the time to reach the maximum height is the same as the time to return to the bottom, and thus the total time is twice the time taken to reach the maximum height. From the calculations, the time is found to be 3 seconds.

Hence, the correct answer is (b).

In this problem, since the two fragments have horizontal velocities, we can use the basic kinematics equations for relative motion. The horizontal velocities of the fragments indicate that we need to consider the motion of the fragments in the horizontal direction. By analyzing the relative motion between the two fragments and the geometry of the problem, we can calculate the time taken for the two radius vectors to make 90°. Using the relative velocity formula, which involves the horizontal velocities of the fragments, and solving for the time, we find:

This calculation takes into account the relative motion between the two fragments and the angle formed by their radius vectors.

Hence, the correct answer is (c).