System Of Particles Rotational Motion

Step 1: Concept
Rocket propulsion involves the high-speed ejection of exhaust gases.
Step 2: Analysis
The backward momentum of the gases must be balanced by an equal forward momentum of the rocket.
Step 3: Conclusion
This is a direct application of the conservation of linear momentum.
Final Answer: (C)

Step 1: Understanding angular momentum.
The angular momentum of a particle is given by the formula: where: - is the position vector of the particle, - is the linear momentum, and - is the velocity of the particle.

Step 2: Analysis of the situation.
In this problem, the mass is moving with a constant velocity on a line parallel to the -axis. Since the motion is in a straight line and there are no external forces acting on the mass, the angular momentum will remain constant. The velocity is constant, and the position vector relative to the origin does not change in direction with time.

Step 3: Conclusion.
The angular momentum remains constant, which is option (2).

Step 1: Formula for Angular Momentum.
The orbital angular momentum of a satellite moving in a circular orbit is given by: where is the mass of the satellite, is its velocity, and is the distance from the center of the orbit.

Step 2: Relationship Between Angular Momentum and Distance.
The orbital velocity of the satellite is related to the gravitational force acting on it. The velocity of a satellite moving in a circular orbit is given by: where is the gravitational constant and is the mass of the central body (e.g., Earth). Substituting this into the expression for : Simplifying the expression:

Step 3: New Angular Momentum with Increased Distance.
Now, if the distance is increased to , the new angular momentum will be: Thus, the new angular momentum .

Step 4: Conclusion.
Therefore, the new angular momentum is .

Step 1: Understand the motion of the scale.
The scale falls without slipping, meaning it rotates about the point of contact with the ground. The motion is a combination of both rotational and translational motion. The velocity of the free end of the scale can be calculated using the energy conservation principle.

Step 2: Energy conservation approach.
Initially, the potential energy of the scale is given by , where (height of the scale). As the scale falls, this potential energy is converted into both translational and rotational kinetic energy. The total energy at the moment just before it strikes the ground is: where is the velocity of the center of mass, is the moment of inertia, and is the angular velocity. Using the relationship between the linear velocity and angular velocity for rolling motion , the equation simplifies.

Step 3: Final velocity calculation.
Using the rotational dynamics equations and energy conservation, we can find that the velocity of the free end when it strikes the ground is approximately:

Step 4: Conclusion.
The velocity with which the free end of the scale strikes the earth is , which is option (2).

Step 1: Conservation of momentum in explosion.
In an explosion, the total momentum of the system is conserved. If a body breaks into parts of unequal masses, the momentum of each part depends on its mass and velocity.

Step 2: Kinetic energy and mass relationship.
Kinetic energy is given by: where is the mass and is the velocity of the part. For unequal masses, the velocities of the parts may differ, but they will have equal kinetic energy.

Step 3: Conclusion.
Thus, the explosion ensures that both parts will have equal kinetic energy, despite having unequal masses and velocities.

Step 1: Understanding the Concept:
Moment of inertia of thin spherical shell about diameter: .
Moment of inertia of solid sphere about diameter: .
Step 2: Detailed Explanation:
Given .
.
Thus, .
Step 3: Final Answer:
Thus, ratio of radii = .

Step 1: Understanding the Concept:
Acceleration due to gravity at depth below surface: .
Step 2: Detailed Explanation:
Given weight on surface = . Halfway below surface means .
.
New weight = .
Step 3: Final Answer:
Thus, weight = .

Step 1: Understanding the Concept:
For diatomic gas, , . At constant pressure, . .
Step 2: Detailed Explanation:
Fraction = .
Step 3: Final Answer:
Thus, fraction = .

Step 1: Understanding the Concept:
Coulomb force is attractive, directed from electron to nucleus. . Here , .
Step 2: Detailed Explanation:
Magnitude . Direction: towards nucleus, i.e., opposite to position vector (if origin at nucleus).
. But , so .
Step 3: Final Answer:
Thus, .

Step 1: Understanding the Concept:
Centre of mass formula: .
Step 2: Detailed Explanation:



Step 3: Final Answer:
Carbon is 0.7 \AA from centre of mass.

Concept: Moment of inertia of a rod about its centre:

Step 1: Treat as composite rod.
Total mass:

Step 2: Apply standard formula.
Even though densities differ, symmetry about centre remains.

Concept: Total energy = translational + rotational: For solid sphere:

Step 1: Total energy.


Step 2: Convert to potential energy.

Concept: From Kepler’s second law:

Step 1: Use angular momentum conservation.


Step 2: Substitute in areal velocity.


Step 3: Interpretation.
Thus areal velocity is independent of , but among given options closest proportional dependence comes from relation involving velocity: Hence effective dependence corresponds to:

Concept: Moment of inertia of a rod about its centre: Use parallel axis theorem for each side.

Step 1: Hexagon properties.
Total length , so each side: Mass of each side:

Step 2: MOI of one side about its centre.


Step 3: Distance from centre.
For regular hexagon, distance from centre to midpoint: Using parallel axis theorem:

Step 4: Total MOI.
(Considering full geometry correction factor)

Concept: Angular momentum: If no external torque acts, angular momentum remains constant.

Step 1: Direction of motion.
Velocity is constant and parallel to x-axis.

Step 2: Torque about origin.
Since no force is acting (constant velocity), net torque is zero.

Step 3: Conclusion.

Concept: Moment of inertia:

Step 1: Given relation.


Step 2: Shell moment of inertia.

Concept: Angular momentum about a point: At highest point, velocity is horizontal.

Step 1: Velocity at highest point.


Step 2: Coordinates of highest point.
Time to reach highest point: Horizontal distance: Height:

Step 3: Angular momentum.
Perpendicular distance from origin to velocity line =

Concept: For rolling motion: Using : At maximum height:

Step 1: Compare with given height.
So,

Step 2: Solve for moment of inertia term.


Step 3: Identify object.
This corresponds to a solid sphere.

Concept: Total kinetic energy of a rolling body: For a disc:

Step 1:

Step 2:

Step 3: