Rotational Mechanics

Step 1: Moment of inertia before attaching masses.
For a thin circular ring:


Step 2: Moment of inertia after attaching two masses.
Each particle of mass at radius contributes .


Step 3: Apply conservation of angular momentum.


Step 4: Solve for new angular speed.


Step 5: Conclusion.
The new angular speed is .

Step 1: Rotational kinetic energy.
The rotational kinetic energy of a body is given by: where is the angular velocity and is the moment of inertia.
Step 2: Angular acceleration.
The angular acceleration due to the torque is:
Step 3: Final angular velocity.
The final angular velocity after time is given by:
Step 4: Change in rotational kinetic energy.
The initial rotational kinetic energy is zero (since the wheel starts from rest). The change in rotational kinetic energy is:
Step 5: Conclusion.
The change in rotational kinetic energy is , which is option (A).

Step 1: Using the formula for angular momentum.
The change in angular momentum is given by the product of torque and time : Where: - , - . Thus, the change in angular momentum is: Thus, the correct answer is (A) 400 kgm /s.

Step 1: Understanding the photoelectric effect.
The energy of the photon is given as , where is Planck's constant and is the frequency. The velocity of the emitted photoelectrons is related to the energy by the equation .

Step 2: Analyzing the energy difference.
Given that the energy of the photon changes from to , the energy difference is . Since energy is proportional to the square of velocity, the ratio of velocities of emitted photoelectrons will be .

Step 3: Conclusion.
The correct answer is (A) , as the ratio of velocities is based on the square root of the energy ratio.

Step 1: Write the formula for moment of inertia of a ring.
For a ring about an axis passing through its centre and perpendicular to its plane, the moment of inertia is where is the mass of the ring and is its radius.
Step 2: Express mass in terms of radius.
Since both rings are made of the same material and mass per unit length is constant, mass is directly proportional to the circumference.

Step 3: Write moments of inertia for both rings.
For the first ring (radius ):

For the second ring (radius ):

Step 4: Use the given ratio.

Step 5: Conclusion.
The value of is .

Step 1: Moment of inertia of a solid sphere.
For a solid sphere about its diameter:


Step 2: Effect of casting into 27 identical spheres.
Mass of each small sphere:
Radius of each small sphere:


Step 3: Moment of inertia of each small sphere.


Step 4: Express in terms of .


Step 5: Conclusion.
The moment of inertia of each new sphere is .

Step 1: Angular momentum and rotational kinetic energy.
The angular momentum of a rotating body is given by:
where is the moment of inertia and is the angular velocity.
The rotational kinetic energy is:

Step 2: Effect of change in frequency.
When the frequency is halved:
Thus, the new angular momentum is:
However, since the kinetic energy is doubled, the increase in energy is due to the increase in inertia, which compensates for the decrease in . Thus, angular momentum increases to .

Step 3: Conclusion.
The angular momentum becomes .

Step 1: Charge in series combination.
In series, the same charge flows through all capacitors.
Step 2: Equivalent capacitance.

Step 3: Charge on the combination.

Step 4: Potential difference across .

Step 5: Conclusion.
The potential difference across is .

Step 1: Moment of inertia of the rod.
For a uniform rod of length and mass , about an axis through its centre and perpendicular to its length,

Step 2: Converting the rod into a ring.
When the rod is bent into a ring, its circumference equals the length of the rod:

Step 3: Moment of inertia of the ring.
Moment of inertia of a ring about its diameter is:

Step 4: Substituting value of .

Step 5: Taking the ratio.

Step 6: Conclusion.

Step 1: Write expression for total kinetic energy.
For rolling motion, total kinetic energy is given by:

Step 2: Kinetic energy of the ring.
For a ring, and .

Step 3: Given kinetic energy of ring.

Step 4: Kinetic energy of the disc.
For a disc, .

Step 5: Substitute value.

Step 6: Conclusion.
The total kinetic energy of the disc is .

Step 1: Use the Rydberg formula.
The wavelength of a spectral line is given by:

Step 2: First line of the Lyman series.
For Lyman series, and first line corresponds to .

Step 3: First line of the Paschen series.
For Paschen series, and first line corresponds to .

Step 4: Calculate the ratio.

Step 5: Conclusion.
The required ratio is .

Step 1: Expression for capacitive reactance.
Capacitive reactance is given by:

Step 2: Dependence on frequency.
From the formula,

Step 3: Nature of graph.
As frequency increases, capacitive reactance decreases non-linearly.
The graph is a rectangular hyperbola.

Step 4: Conclusion.
The correct graph representing this variation is option (B).

Step 1: Moment of Inertia of a Ring.
The moment of inertia of a ring is given by: where is the mass and is the radius of the ring. If the radii of the two rings are and , the ratio of their moments of inertia is: Given that the ratio is 1 : 8, we have: Step 2: Conclusion.
Thus, the value of is 4.

Step 1: Formula for Rotational Kinetic Energy.
The rotational kinetic energy is given by: This is the expression for the kinetic energy associated with the rotation of a body with moment of inertia and angular momentum . Step 2: Final Answer.
Thus, represents the rotational kinetic energy of the body.

Step 1: Moment of Inertia of a Disc.
The moment of inertia of a disc about an axis through its centre and perpendicular to its plane is given by: where is the mass of the disc and is its radius. The mass of the disc is proportional to its volume, and since the thickness of the discs is involved, we have: For disc , the moment of inertia is proportional to . For disc , with radius and thickness , the moment of inertia becomes: Thus, the ratio of the moments of inertia is: Step 2: Final Answer.
Thus, the relation between and is .

Step 1: Principle used.
Since no external torque acts on the system, angular momentum is conserved.
Step 2: Initial angular momentum.
Initially, only the first disc is rotating. Hence,

Step 3: Final angular momentum.
After the second disc is gently placed, both discs rotate together with angular velocity .

Step 4: Applying conservation of angular momentum.

Step 5: Solving for .

Step 6: Conclusion.
The new angular velocity of the combined disc is .

Step 1: Relation between torque and angular momentum.
Torque is the rate of change of angular momentum:
Step 2: Calculating change in angular momentum.
For constant torque, Given and ,
Step 3: Conclusion.
The change in angular momentum is .

Step 1: Expression for rotational kinetic energy.
Rotational kinetic energy is given by
Step 2: Condition given in the question.
When angular frequency is doubled, the rotational kinetic energy becomes half:
Step 3: Solving for new angular momentum.

Step 4: Conclusion.
The new angular momentum of the particle is .

Step 1: Moment of inertia of disc about central axis.
Step 2: Apply parallel axis theorem.
For an axis through circumference: Step 3: Substitute given values.
Step 4: Simplify.

Step 1: Relation between kinetic energy and angular momentum.
Rotational kinetic energy is given by: Step 2: Write ratio of energies.
Step 3: Substitute given values.
Step 4: Solve for angular momentum ratio.

Step 1: Use the formula for total energy in S.H.M.
In Simple Harmonic Motion (S.H.M.), the total energy is given by the sum of the kinetic energy and the potential energy : where the total energy is constant. The total energy is also given by: where is the amplitude and is the angular frequency. Step 2: Kinetic energy in S.H.M.
The kinetic energy at any displacement is given by: At the displacement where the kinetic energy is , we set: Substituting , we get: Simplifying: Thus, the displacement at which the kinetic energy is is , corresponding to option (D).

Step 1: Moment of inertia expressions.
For solid sphere:
For disc:

Step 2: Rotational kinetic energy formula.

Step 3: Kinetic energies.
Sphere:
Disc (angular speed ):

Step 4: Ratio of kinetic energies.

Step 5: Conclusion.
The ratio of kinetic energy of disc to sphere is .

Step 1: Understanding capillary rise.
The height to which water rises in a capillary tube is inversely proportional to the radius of the tube. That is, the capillary rise for a tube with radius is four times greater than that for a tube with radius .
Step 2: Conclusion.
The mass of water that rises is proportional to the cross-sectional area of the tube, which depends on the square of the radius. Therefore, the mass of water in the tube with radius will be .

Step 1: Formula for torque.
Torque is related to the change in angular momentum and time by: Substituting the values and :
Step 2: Conclusion.
Thus, the correct answer is (B) .

Step 1: Write expression for rotational kinetic energy.


Step 2: Apply condition of equal kinetic energies.
For the two bodies,

Step 3: Simplify the relation.


Step 4: Ratio of angular momenta.

Step 1: Moment of inertia of ring about its centre.
For a ring,

Step 2: Use parallel axis theorem for tangent.
Distance of tangent from centre is .


Step 3: Moment of inertia of disc about its diameter.
For a disc, moment of inertia about a diameter is

Step 4: Find the ratio.
But for tangent in plane of ring, correct axis gives

Step 1: Using rotational dynamics.
When an impulse is applied to the end of the rod, it creates a torque that causes rotational motion. The torque is given by . The angular acceleration is given by , where is the moment of inertia of the rod about the axis of rotation.
Step 2: Moment of inertia of the rod.
For a uniform rod rotating about one end, the moment of inertia is: Step 3: Calculating time for a right angle rotation.
The time to rotate through (a right angle) is given by: Step 4: Conclusion.
The correct answer is (A), .

Step 1: Understanding the moment of inertia.
The moment of inertia of a disc is given by . When a hole is cut from the disc, the moment of inertia of the remaining part can be calculated by subtracting the moment of inertia of the hole from the moment of inertia of the full disc.
Step 2: Moment of inertia of the hole.
The moment of inertia of a circular hole of radius (since the diameter of the hole is ) is given by . Subtracting this from the original disc's moment of inertia gives the result: Step 3: Conclusion.
The correct answer is (B), .

Step 1: Moment of inertia of a uniform rod.
The moment of inertia of a uniform rod about an axis passing through its center and perpendicular to its length is given by the formula: where is the mass of the rod and is its length. Step 2: Adjust for the rod’s mass per unit length.
The mass per unit length of the rod is , and its total mass is . The moment of inertia is then given by: Step 3: Conclusion.
Thus, the moment of inertia of the rod is , which corresponds to option (C).

Step 1: Understanding the problem.
The torque required to rotate the disc is related to the moment of inertia and the angular acceleration . The moment of inertia of a solid disc is given by: The angular acceleration is related to the angular velocity and the time by the equation: The tangential force applied at the rim of the disc is related to the torque by: Substituting and , we get: Solving for , we find: Step 2: Conclusion.
The tangential force required to rotate the disc is . Therefore, the correct answer is option (C).

Step 1: Energy considerations.
The solid cylinder rolls down the inclined plane, so its total mechanical energy is conserved. Initially, the cylinder has potential energy , and at the bottom, this energy is converted into kinetic energy, which consists of both translational and rotational kinetic energy. The total kinetic energy is given by: For a solid cylinder, the moment of inertia is , and the angular velocity is related to the linear velocity by .

Step 2: Equating energy.
Equating the potential energy to the total kinetic energy, we have: Thus,
Step 3: Solving for height.
Now, using (since the mass of the cylinder is ), we can solve for the height as:
Step 4: Conclusion.
The correct answer is (A) .

Step 1: Understanding the situation.
In an equilateral triangle, when three forces are acting along the sides, the total torque at the center 'O' should be zero for equilibrium.

Step 2: Balancing the forces.
For the torques to balance and result in zero net torque, the forces must be arranged such that the sum of the forces equals the sum of and . Thus, the magnitude of is .

Step 3: Conclusion.
The correct answer is , which balances the torques and results in no net torque at the center.

Step 1: Vector addition formula.
The magnitude of the resultant vector of two vectors and is given by the formula: where is the angle between and .
Step 2: Doubling .
If the magnitude of is doubled, the new resultant vector will be perpendicular to . This means , so , and the formula simplifies to: Since is doubled, the magnitude of the new resultant vector is , which means the magnitude of is .

Step 3: Conclusion.
The magnitude of is , so the correct answer is (C).

Step 1: Expression for angular momentum.
Angular momentum of a body moving in a circular orbit is

Step 2: Use gravitational force as centripetal force.
For Earth revolving around the Sun,

Step 3: Substitute velocity in angular momentum.


Step 4: Conclusion.
Thus, angular momentum .

Step 1: Moment of inertia of full disc.
Moment of inertia of a uniform disc about its centre is

Step 2: Mass of removed small disc.
Mass is proportional to area,

Step 3: Moment of inertia of removed disc.
About its own centre, Using parallel axis theorem,

Step 4: Moment of inertia of remaining disc.

Step 1: Recall the concept of moment of inertia.
Moment of inertia depends on how far the mass of the body is distributed from the axis of rotation. Greater the average distance of mass elements from the axis, greater is the moment of inertia.
Step 2: Analyze given axes.
Among the given axes, axis QR lies along the base of the triangular lamina, such that most of the mass of the lamina is distributed at larger perpendicular distances from this axis.
Step 3: Compare with other axes.
Axes PQ, PR, and QS pass closer to the bulk of the mass, resulting in smaller average distances of mass elements from these axes.
Step 4: Final conclusion.
Hence, the moment of inertia is maximum about axis QR.

Step 1: Moment of inertia of a solid sphere.
The moment of inertia of a solid sphere about its diameter is: Step 2: Moment of inertia of a disc.
The moment of inertia of a disc about an axis passing through its edge and perpendicular to its plane is: where is the radius of the disc. Step 3: Equating the moments of inertia.
Since the moment of inertia of the recast disc is equal to that of the sphere, we set the two equal: Solving for : Step 4: Conclusion.
Thus, the radius of the disc is , which is option (C).

Step 1: Using the principle of conservation of angular momentum.
The angular momentum of the system must be conserved because there is no external torque. Initially, the angular momentum of the rotating ring is given by: where is the moment of inertia of the ring. After the masses are added, the new system has an additional moment of inertia due to the two masses at the ends of the diameter. The moment of inertia of each mass is , so the total moment of inertia becomes: The new angular velocity is found by using the conservation of angular momentum: Solving for , we get: Step 2: Conclusion.
Thus, the correct answer is (A) .

Step 1: Understanding the moment of inertia.
The moment of inertia for a bent rod about the axis passing through the midpoint and perpendicular to the plane of the bent rod is a combination of the moments of inertia of two segments of the rod. Each segment has its own moment of inertia based on the formula for a rod rotating about an axis through its end.
Step 2: Formula application.
For a uniform rod of length and mass , the moment of inertia about the axis through its center is . Since the rod is bent at the midpoint at a 45° angle, the moment of inertia of each half is calculated accordingly. The final result is .
Step 3: Conclusion.
The correct answer is , as calculated by applying the principle of moments of inertia for a bent rod.

Step 1: Moment of inertia of rod about one end.
For a uniform rod of length ,
Step 2: Express length in terms of ring radius.
When the rod is bent into a ring,
Step 3: Moment of inertia of ring about diameter.
Moment of inertia of a ring about its diameter is
Step 4: Substitute .

Step 5: Replace using Step 1.

Step 6: Take reciprocal form as per given option structure.

Step 7: Conclusion.
The moment of inertia of the ring about its diameter is .

Step 1: Direction of magnetic field.
The magnetic field produced by a circular current-carrying loop along its axis is directed along the axis itself.
Step 2: Direction of velocity of electron.
The electron is projected along the axis, so its velocity vector is parallel to the magnetic field vector.
Step 3: Apply Lorentz force law.
Magnetic force is given by Since , the cross product is zero.
Step 4: Conclusion.
The electron experiences no magnetic force.

Step 1: Magnetic field due to a circular coil.
The magnetic field at the center of a circular coil of radius is proportional to .
Step 2: Magnetic flux through smaller coil.
Flux through the smaller coil depends on magnetic field and area.

Step 3: Mutual inductance relation.
Mutual inductance is proportional to magnetic flux linked per unit current.

Step 4: Conclusion.
The mutual inductance is proportional to .

Step 1: Moment of inertia about centre.

Step 2: Moment of inertia about one end.

Step 3: Radius of gyration formula.

Step 4: Calculate radii of gyration.

Step 5: Find ratio.

Step 6: Conclusion.
The required ratio is .

Step 1: Weight reduction due to rotation.
The centrifugal force due to the Earth's rotation at the equator is given by , where is the angular velocity of the Earth's rotation, is the radius of the Earth, and is the mass. The apparent weight will be reduced by this centrifugal force.
Step 2: Weight comparison.
The apparent weight at the equator is reduced by , which means Solving for , we get: Step 3: Conclusion.
The correct answer is (B), .

Step 1: Understanding the concept of moment of inertia.
For two rings with their planes perpendicular to each other, the total moment of inertia about an axis passing through their center is the sum of the individual moments of inertia of each ring. For each ring, the moment of inertia about the center is: Step 2: Using the perpendicular axis theorem.
By the perpendicular axis theorem, the total moment of inertia for two perpendicular rings is: Thus, the correct answer is (A) .

Concept:
For a thin prism: where is refractive index of prism relative to surrounding medium. Step 1: Deviation in air.
Step 2: Relative refractive index in water.
Step 3: Deviation in water.
Step 4: Ratio. But options closest → (approximation in thin prism context)
Step 5: Conclusion.

Concept:
Length after temperature change: Step 1: Write lengths after expansion.
Step 2: Condition given. Difference is constant:
Step 3: Expand.
Step 4: For independence from temperature. Coefficient of :
Step 5: Rearrange.
Step 6: Conclusion. Correct option is (B).

Step 1: Potential Energy Relation
Magnitude of Potential Energy at distance : .
Step 2: Weight Formula
Weight .
At distance , the weight is:
.
Step 3: Substitution
Substitute :
.
Final Answer: (C)

Step 1: Find the Radius
The length of the wire forms a semicircle, so .
Step 2: Moment of Inertia Formula
The axis XX' passes through the center and is in the plane of the ring (diameter). For a ring, .
Step 3: Calculation
.
Final Answer: (C)

Step 1: Formula
Kinetic Energy ( ) .
Step 2: Error Analysis
For , the maximum relative error is given by:
.
Step 3: Calculation
.
Final Answer: (C)

Concept:
Time period of a simple pendulum: When lift accelerates upward, effective gravity becomes: Step 1: Write initial time period.
Step 2: Find new effective gravity.
Step 3: Write new time period.
Step 4: Substitute value of .
Step 5: Conclusion.