Practice Questions

Calculate the vector product $\mathbf{a} \times \mathbf{b}$ for the vectors $\mathbf{a} = 2\hat{\mathbf{i}} + 3\hat{\mathbf{j}} - \hat{\mathbf{k}}$ and $\mathbf{b} = -\hat{\mathbf{i}} + \hat{\mathbf{j}} + 4\hat{\mathbf{k}}$.

Identify the rotational analogue of force in linear motion.

A flywheel, initially at rest, starts rotating with a constant angular acceleration of $2 \text{ rad/s}^2$. Recall the kinematic equation for angular displacement and find the angle it rotates through in the first 5 seconds.

Define an ideal rigid body.

Name the physical quantity that is the rotational analogue of linear velocity.

Formulate a reason why the vector product of two non-zero parallel vectors is always a null vector.

Examine why an ice skater spins faster when she pulls her arms in. Apply the principle of conservation of angular momentum to explain this phenomenon.

Justify why a force applied directly to the axis of rotation of a rigid body cannot produce a torque about that axis.

Critique the use of a point mass model for analyzing the rolling motion of a wheel down an inclined plane. Justify why a rigid body model is necessary and what additional concepts it introduces.

Evaluate the claim: 'The centre of mass of a body must always lie within the material of the body.' Justify your conclusion with at least two examples.

List the two primary types of motion a rigid body can have when it is not pivoted or fixed in any way.

Compare the moment of inertia of a thin circular ring and a circular disc, both having the same mass $M$ and radius $R$, about an axis passing through their center and perpendicular to their plane. Which object would be more difficult to start rotating? Analyze your answer.

Propose a scenario where the total linear momentum of a system of particles is conserved, but its total kinetic energy is not. Justify your proposal.

Three particles with masses $m_1 = 1$ kg, $m_2 = 2$ kg, and $m_3 = 3$ kg are placed at the vertices of a right-angled triangle. The coordinates of the vertices are A(0, 3), B(0, 0), and C(4, 0) respectively, with units in meters. Particle $m_1$ is at A, $m_2$ is at B, and $m_3$ is at C. Calculate the coordinates of the center of mass of this system.

A solid cylinder of mass $10$ kg and radius $0.5$ m is rotating about its central axis with an angular speed of $50 \text{ rad/s}$. Calculate the rotational kinetic energy of the cylinder.

Describe the two necessary conditions for a rigid body to be in mechanical equilibrium. Provide the mathematical expression for each condition.

A flywheel rotating at an initial angular speed of $120 \text{ rpm}$ is brought to rest in $10$ seconds by a constant frictional torque. If the moment of inertia of the flywheel is $5 \text{ kg m}^2$, calculate the magnitude of the frictional torque.

Evaluate whether the angular momentum $\mathbf{L}$ and angular velocity $\boldsymbol{\omega}$ of a rigid body are always parallel. Justify your answer.

Define the centre of mass for a system of n particles using a vector equation.

Explain the right-hand screw rule used to determine the direction of the vector product of two vectors, $\mathbf{a}$ and $\mathbf{b}$.

Define moment of inertia and recall the formula for the kinetic energy of a body rotating about a fixed axis.

Explain the law of conservation of angular momentum for a system of particles. Describe a real-world example, such as a spinning skater, to illustrate this principle.

State the vector relationship between linear velocity ($\mathbf{v}$), angular velocity ($\boldsymbol{\omega}$), and the position vector ($\mathbf{r}$) for a particle in a rotating rigid body.

Analyze the motion of a rigid body under the action of a couple. Does the center of mass of the body accelerate? Explain why a couple produces pure rotational motion without translation.

A child is standing at the center of a turntable that is rotating at $20 \text{ rev/min}$. The child's moment of inertia is initially $2.5 \text{ kg m}^2$. When the child stretches out their arms, their moment of inertia increases to $4.0 \text{ kg m}^2$. Assuming no external torques, calculate the new angular speed of the turntable in rev/min.

A car engine produces a power of $30 \text{ kW}$ while rotating at a uniform angular speed of $1500 \text{ rpm}$. Calculate the torque transmitted by the engine.

A uniform rod of length $1$ m and mass $0.5$ kg is pivoted at one end. A force of $10$ N is applied perpendicular to the rod at its free end. Calculate the initial angular acceleration of the rod.

A body is in translational equilibrium. Does this imply that the net torque on the body is also zero? Analyze this statement with a suitable example.

Propose a method to find the centre of gravity of a non-uniform metal bar using only a single knife-edge pivot and a set of known weights. Justify your method.

A student claims that for a body to be in mechanical equilibrium, the net force must be zero, and this condition automatically ensures the net torque is also zero. Critique this statement and provide a counterexample.

A ballet dancer spins faster by pulling her arms inward. Critique the statement: 'The dancer creates rotational kinetic energy by pulling her arms in.' Justify your answer using the principle of conservation of angular momentum and the work-energy theorem.

Formulate a proof to show that the torque of a couple is independent of the choice of the origin about which the moments are taken. Evaluate the significance of this property.

Explain the difference between the centre of mass and the centre of gravity of an extended body and state when they coincide.

Design an experiment to determine the moment of inertia of an irregularly shaped lamina about an axis perpendicular to its plane passing through its centre of gravity. Justify the procedure and the formula used.

Summarize the steps to find the coordinates of the centre of mass for a system of three particles. Then, calculate the centre of mass for a system with masses $m_1 = 1$ kg, $m_2 = 2$ kg, and $m_3 = 3$ kg located at coordinates $(1, 1)$, $(2, 2)$, and $(3, 3)$ respectively.

Demonstrate that the center of mass of a uniform semi-circular ring of radius $R$ lies at a distance of $\frac{2R}{\pi}$ from its geometric center, along the axis of symmetry.

A uniform rod of mass $M$ and length $L$ is pivoted at one end and hangs vertically. A bullet of mass $m$ moving horizontally with velocity $v$ strikes the rod at its free end and gets embedded in it. Formulate an expression for the maximum angular displacement $\theta_{max}$ of the rod after the collision.

A solid sphere and a hollow sphere of the same mass $M$ and radius $R$ are released from rest at the top of an inclined plane of height $h$. Create a derivation to determine which one reaches the bottom first. Justify your conclusion by evaluating their moments of inertia.

A stationary bomb of mass $10$ kg explodes into three fragments. Two fragments, each of mass $3$ kg, move off at right angles to each other with speeds of $5 \text{ m/s}$. Calculate the velocity of the third, larger fragment.

Recall the relationship between the torque acting on a particle and the time rate of change of its angular momentum.

A $2$ kg particle is moving with a constant velocity $\mathbf{v} = (3\hat{\mathbf{i}} + 4\hat{\mathbf{j}}) \text{ m/s}$. At time $t=0$, its position vector is $\mathbf{r} = (2\hat{\mathbf{i}}) \text{ m}$ with respect to the origin. Calculate the angular momentum of the particle about the origin.

A person sits on a frictionless swivel chair holding a spinning bicycle wheel with its axis vertical. The wheel spins clockwise with angular momentum $\mathbf{L}_{wheel}$. The person and chair are initially at rest. The person then flips the wheel over. Create a detailed explanation of what happens to the person and the chair, justifying your answer with the law of conservation of angular momentum.

A uniform ladder of length $5$ m and mass $20$ kg rests against a frictionless vertical wall. Its lower end is on rough horizontal ground, $3$ m away from the wall. Calculate the reaction forces exerted by the wall and the ground on the ladder. Assume $g = 10 \text{ m/s}^2$.

Design a system consisting of a flywheel and a falling weight to demonstrate the relationship $\tau = I\alpha$. Formulate the equations of motion and propose how to experimentally determine the moment of inertia $I$ of the flywheel.

Explain how to calculate the torque required to produce a certain angular acceleration in a rigid body. A solid cylinder with a mass of $10$ kg and radius of $0.5$ m needs to be accelerated from rest to an angular velocity of $20 \text{ rad/s}$ in $4$ seconds. Calculate the required torque. The moment of inertia of a solid cylinder is $I = \frac{1}{2}MR^2$.