SYSTEMS OF PARTICLES AND ROTATIONAL MOTION-QnA

Q1: What is meant by a system of particles?

A system of particles is a collection of two or more particles considered together to study their combined motion and physical behavior.

Q2: Define centre of mass.

The centre of mass is a point where the entire mass of a system can be assumed to be concentrated for studying translational motion.

Q3: What is torque?

Torque is the turning effect of a force about an axis and is equal to the product of force and perpendicular distance from the axis.

Q4: Write the SI unit of angular momentum.

The SI unit of angular momentum is \(\mathrm{kg\ m^2\ s^{-1}}\).

Q5: What is moment of inertia?

Moment of inertia is the rotational analogue of mass and measures resistance of a body to change in its rotational motion.

Q6: State the condition for pure rolling.

In pure rolling, the linear velocity of the centre of mass equals the product of angular velocity and radius.

Q7: What is angular velocity?

Angular velocity is the rate of change of angular displacement with time.

Q8: Define angular acceleration.

Angular acceleration is the rate of change of angular velocity with respect to time.

Q9: What does the radius of gyration represent?

It represents the distance from the axis at which the whole mass can be assumed concentrated without changing moment of inertia.

Q10: What is the direction of angular momentum?

It is directed along the axis of rotation following the right-hand thumb rule.

Q11: What is a rigid body?

A rigid body is one in which the distance between any two particles remains constant during motion.

Q12: State Newton’s second law for rotation.

The rate of change of angular momentum is equal to the applied external torque.

Q13: What is a couple?

A couple consists of two equal and opposite forces acting at different points causing rotation without translation.

Q14: What happens to torque if force passes through the axis?

Torque becomes zero because the perpendicular distance from axis is zero.

Q15: Write the dimensional formula of torque.

Torque has the dimensional formula \([M L^2 T^{-2}]\).

Q1: Explain why centre of mass may lie outside a body.

For hollow or irregular bodies like rings, the mass distribution is such that the average mass position lies outside the material region.

Q2: Distinguish between translational and rotational motion.

Translational motion involves all particles moving with same velocity, while in rotational motion particles move in circles about an axis.

Q3: Why is moment of inertia not a fixed quantity?

It depends on mass distribution and axis of rotation, so it changes with axis position.

Q4: Define angular impulse.

Angular impulse is the product of torque and time for which it acts, equal to change in angular momentum.

Q5: Why does a rolling wheel have both KE forms?

A rolling wheel has translational KE due to centre of mass motion and rotational KE due to spinning.

Q6: Write the relation between torque and angular acceleration.

Torque is equal to the product of moment of inertia and angular acceleration.

Q7: What is conservation of angular momentum?

If net external torque is zero, angular momentum of a system remains constant.

Q8: Give an example of variable moment of inertia.

A dancer pulling in arms reduces moment of inertia during spinning.

Q9: Why is a hollow cylinder easier to rotate than a solid one of same mass?

Mass in hollow cylinder lies farther from axis, increasing torque effectiveness.

Q10: State the perpendicular axis theorem.

For a planar body, moment of inertia about perpendicular axis equals sum of moments about two perpendicular in-plane axes.

Q1: Derive expression for torque on a particle.

Torque is defined as the vector product of position vector and applied force, measuring rotational effectiveness of force.

Q2: Explain the significance of centre of mass.

It simplifies motion analysis by reducing complex particle systems to single-point motion representation.

Q3: Describe angular momentum of a rigid body.

Angular momentum depends on moment of inertia and angular velocity, and remains conserved without external torque.

Q4: Explain rolling motion with example.

Rolling motion combines translation and rotation, as seen in a wheel moving on a road.

Q5: State and explain parallel axis theorem.

It relates moment of inertia about any axis to that about parallel axis through centre of mass.

Q1: Explain centre of mass and its importance in mechanics.

The centre of mass represents the average position of mass of a system. For translational motion, the entire system behaves as if all mass were concentrated at this point. This concept greatly simplifies motion analysis of multi-particle systems. In external force problems, the acceleration of centre of mass depends only on net external force, irrespective of internal forces. This makes it extremely useful in collision, projectile motion of bodies, and rigid body dynamics.

Q2: Discuss torque and rotational equilibrium.

Torque causes rotational motion. When the net torque acting on a body is zero, the body is said to be in rotational equilibrium. This condition is essential for static balance of objects such as beams and ladders. Clockwise and anticlockwise torques must cancel. Torque analysis is crucial in designing mechanical systems, tools, and structures to ensure stability and safe operation.

Q3: Explain moment of inertia and factors affecting it.

Moment of inertia measures resistance to rotational motion. It depends on mass, distribution of mass relative to axis, and position of axis. For same mass, bodies with mass farther from axis have higher moment of inertia. This explains why hollow objects rotate differently from solid ones. It plays a role similar to mass in linear motion.

Q4: Describe conservation of angular momentum with examples.

Conservation of angular momentum states that in absence of external torque, angular momentum remains constant. A spinning skater pulling arms inward increases angular speed. Similarly, planets conserve angular momentum while revolving. This principle explains many natural and mechanical rotational phenomena and is fundamental to rotational mechanics.

Q5: Explain rolling motion in detail.

Rolling motion is a combination of translational and rotational motion. In pure rolling, the point of contact with ground remains momentarily at rest. The total kinetic energy of a rolling body is the sum of translational and rotational kinetic energies. Rolling motion is observed in wheels, cylinders, and balls, and is important in transportation and machinery design.