System of Particles and Rotational Motion Class 11 Physics Notes

The velocity and acceleration of the CM are V = dR/dt and A = d²R/dt². The net external force acting on the system equals the total mass times the acceleration of the CM: F_ext = M·A. Internal forces cannot change the motion of the centre of mass.

• For a symmetric object (rod, ring, disc, sphere), the CM lies at its geometric centre.
• The CM may lie outside the body — e.g. a hollow ring's CM is at its centre even though there is no material there.
• The CM of two particles divides the line joining them in the inverse ratio of their masses (closer to the heavier particle).

MCQ Questions

1 Mark Questions

1. 1
   Define the centre of mass of a system of particles.
2. 2
   Write the expression for the position vector of the centre of mass of two particles of masses m₁ and m₂.
3. 3
   Where is the centre of mass of a uniform ring located?

2 Mark Questions

1. 1
   State the rule for locating the centre of mass of two particles of unequal masses.
2. 2
   Show that internal forces cannot change the motion of the centre of mass.
3. 3
   Two bodies of masses 2 kg and 8 kg are placed at the ends of a light rod 1 m long. Find the position of their centre of mass from the 8 kg mass.

3 Mark Questions

1. 1
   Derive the expression for the position vector of the centre of mass of a system of n particles and show it behaves like a point mass under a net external force.
2. 2
   Three particles of masses 1 kg, 2 kg and 3 kg are placed at the vertices of an equilateral triangle of side 1 m. Find the position of the centre of mass.

Torque is a vector perpendicular to both r and F, with direction given by the right-hand rule. A body in equilibrium must have zero net force (no translation) and zero net torque about any point (no rotation): ΣF = 0 and Στ = 0. This gives the three conditions that let us analyse ladders, beams and seesaws.

• The turning effect is stronger when the force is applied farthest from the pivot.
• Forces through the pivot (moment arm = 0) produce zero torque.
• A couple is two equal, opposite, parallel forces that produce pure rotation with moment τ = F·d.

MCQ Questions

1 Mark Questions

1. 1
   Define torque. Write its SI unit.
2. 2
   State the conditions for translational equilibrium of a rigid body.
3. 3
   What is a couple?

2 Mark Questions

1. 1
   A force of 10 N acts at the end of a rod of length 0.5 m, perpendicular to it. Find the torque about the pivot at the other end.
2. 2
   Explain why the effectiveness of a wrench is increased by making its handle longer.
3. 3
   State the conditions for equilibrium of a rigid body under the action of several forces.

3 Mark Questions

1. 1
   A uniform rod of weight 40 N and length 2 m is pivoted at one end and held horizontal by a force at the other end. Find the force needed to balance the rod.
2. 2
   Derive the torque due to a force F applied at a point whose position vector is r, and show τ is perpendicular to both r and F.

The rotational analogue of Newton's second law is τ = dL/dt. When the net external torque is zero, angular momentum is conserved: L = constant. This is why an ice skater speeds up by pulling in her arms — reducing I increases ω while Iω stays the same.

• Angular momentum is conserved for a system with no external torque.
• A planet orbiting the Sun sweeps equal areas in equal times — a direct application of L conservation.
• The direction of L is along the axis of rotation (right-hand rule).

MCQ Questions

1 Mark Questions

1. 1
   Define angular momentum of a particle about a point.
2. 2
   Write the relation between torque and angular momentum.
3. 3
   State the law of conservation of angular momentum.

2 Mark Questions

1. 1
   Show that τ = dL/dt follows from Newton's second law for a particle.
2. 2
   A diver curls into a tight ball during a dive then straightens out on entry. Explain the change in angular speed using angular momentum conservation.
3. 3
   State the law of conservation of angular momentum and give two examples where it applies.

3 Mark Questions

1. 1
   Prove that the angular momentum of a particle is conserved when the net torque on it is zero, and relate it to Kepler's second law of equal areas.
2. 2
   A man stands at the centre of a rotating platform and holds weights with arms stretched out. Explain what happens when he pulls the weights in, using conservation of angular momentum.

The radius of gyration k is the distance at which the entire mass of the body may be assumed concentrated to give the same moment of inertia: I = Mk². Standard results include a thin ring (MR²), a solid disc (½MR²), a solid sphere (2/5 MR²) and a thin rod about its centre (ML²/12).

MCQ Questions

1 Mark Questions

1. 1
   Define moment of inertia. Write its SI unit.
2. 2
   State the theorem of parallel axes.
3. 3
   Define the radius of gyration.

2 Mark Questions

1. 1
   State and prove the theorem of perpendicular axes for a plane lamina.
2. 2
   A ring and a disc of the same mass and radius rotate about their central axes. Compare their moments of inertia.
3. 3
   Write the moment of inertia of a thin uniform rod of length L about an axis through its centre, perpendicular to its length, and state the parallel-axes theorem.

3 Mark Questions

1. 1
   State and prove the theorem of parallel axes, using it to find the moment of inertia of a thin rod of length L and mass M about an axis through one end.
2. 2
   Find the moment of inertia of a uniform disc of mass M and radius R about a tangent in its plane, using the theorems of moment of inertia.

For pure rolling motion (no slipping), the point of contact is instantaneously at rest, giving the rolling condition v = rω. The work done in rotation is W = τθ, and the rotational power is P = τω, matching W = Fs and P = Fv in translation.

• Rolling without slipping requires v = rω at the contact point.
• A body rolling down an incline converts potential energy into both translational and rotational KE.
• Objects with the smallest moment-of-inertia factor reach the bottom of a slope first (sphere beats cylinder beats ring).

MCQ Questions

1 Mark Questions

1. 1
   Write the expression for the rotational kinetic energy of a rigid body.
2. 2
   State the rolling condition for a body moving without slipping.
3. 3
   Give the expression for the total kinetic energy of a rolling body.

2 Mark Questions

1. 1
   A flywheel of moment of inertia 2 kg m² rotates at 300 rpm. Find its rotational kinetic energy.
2. 2
   Explain why a solid sphere rolls down an incline faster than a ring of the same mass and radius.
3. 3
   Derive the total kinetic energy of a rolling body as ½Mv² + ½Iω².

3 Mark Questions

1. 1
   A solid sphere rolls down an inclined plane of height h starting from rest. Obtain the velocity of its centre of mass at the bottom.
2. 2
   State and prove that the total kinetic energy of a rolling body equals ½Mv²(1 + k²/R²), where k is its radius of gyration.

Exercises (NCERT Q6.1–Q6.17)

• 6.1) Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body?
• 6.2) In the HCl molecule, the separation between the nuclei of the two atoms is about 1.27 Å (1 Å = 10⁻¹⁰ m). Find the approximate location of the CM of the molecule, given that a chlorine atom is about 35.5 times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.
• 6.3) A child sits stationary at one end of a long trolley moving uniformly with a speed V on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, what is the speed of the CM of the (trolley + child) system?
• 6.4) Show that the area of the triangle contained between the vectors a and b is one half of the magnitude of a × b.
• 6.5) Show that a·(b × c) is equal in magnitude to the volume of the parallelepiped formed on the three vectors a, b and c.
• 6.6) Find the components along the x, y, z axes of the angular momentum l of a particle, whose position vector is r with components x, y, z and momentum is p with components pₓ, pᵧ and p_z. Show that if the particle moves only in the x-y plane the angular momentum has only a z-component.
• 6.7) Two particles, each of mass m and speed v, travel in opposite directions along parallel lines separated by a distance d. Show that the angular momentum vector of the two particle system is the same whatever be the point about which the angular momentum is taken.
• 6.8) A non-uniform bar of weight W is suspended at rest by two strings of negligible weight as shown in Fig. 6.33. The angles made by the strings with the vertical are 36.9° and 53.1° respectively. The bar is 2 m long. Calculate the distance d of the centre of gravity of the bar from its left end.
• 6.9) A car weighs 1800 kg. The distance between its front and back axles is 1.8 m. Its centre of gravity is 1.05 m behind the front axle. Determine the force exerted by the level ground on each front wheel and each back wheel.
• 6.10) Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time?
• 6.11) A solid cylinder of mass 20 kg rotates about its axis with angular speed 100 rad s⁻¹. The radius of the cylinder is 0.25 m. What is the kinetic energy associated with the rotation of the cylinder? What is the magnitude of angular momentum of the cylinder about its axis?
• 6.12) (a) A child stands at the centre of a turntable with his two arms outstretched. The turntable is set rotating with an angular speed of 40 rev/min. How much is the angular speed of the child if he folds his hands back and thereby reduces his moment of inertia to 2/5 times the initial value? Assume that the turntable rotates without friction. (b) Show that the child's new kinetic energy of rotation is more than the initial kinetic energy of rotation. How do you account for this increase in kinetic energy?
• 6.13) A rope of negligible mass is wound round a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N? What is the linear acceleration of the rope? Assume that there is no slipping.
• 6.14) To maintain a rotor at a uniform angular speed of 200 rad s⁻¹, an engine needs to transmit a torque of 180 N m. What is the power required by the engine? (Note: uniform angular velocity in the absence of friction implies zero torque. In practice, applied torque is needed to counter frictional torque). Assume that the engine is 100% efficient.
• 6.15) From a uniform disk of radius R, a circular hole of radius R/2 is cut out. The centre of the hole is at R/2 from the centre of the original disc. Locate the centre of gravity of the resulting flat body.
• 6.16) A metre stick is balanced on a knife edge at its centre. When two coins, each of mass 5 g are put one on top of the other at the 12.0 cm mark, the stick is found to be balanced at 45.0 cm. What is the mass of the metre stick?
• 6.17) The oxygen molecule has a mass of 5.30 × 10⁻²⁶ kg and a moment of inertia of 1.94 × 10⁻⁴⁶ kg m² about an axis through its centre perpendicular to the lines joining the two atoms. Suppose the mean speed of such a molecule in a gas is 500 m/s and that its kinetic energy of rotation is two thirds of its kinetic energy of translation. Find the average angular velocity of the molecule.

Answers at a glance

1. 1
   Q6.1 → All four lie at the centre of symmetry: the geometric centre (cylinder: mid-point of the axis). No, the CM need not lie inside the body.
2. 2
   Q6.2 → CM ≈ 1.24 Å from the hydrogen nucleus (about 0.035 Å from the chlorine nucleus).
3. 3
   Q6.3 → The CM of the system continues to move with speed V.
4. 4
   Q6.4 → Area = ½ |a||b| sin θ = ½ |a × b|.
5. 5
   Q6.5 → Volume = base area × height = |b × c| × [a·(b × c)/|b × c|] = a·(b × c).
6. 6
   Q6.6 → lₓ = y p_z − z pᵧ; lᵧ = z pₓ − x p_z; l_z = x pᵧ − y pₓ. In the x-y plane only l_z remains.
7. 7
   Q6.7 → L = m v d, fixed in magnitude and direction (⊥ to the plane), independent of the reference point.
8. 8
   Q6.8 → The centre of gravity is 0.72 m from the left end.
9. 9
   Q6.9 → Each front wheel: 3675 N; each back wheel: 5145 N.
10. 10
    Q6.10 → The solid sphere acquires the greater angular speed (J = (2/5)MR² is smaller than MR²).
11. 11
    Q6.11 → KE = 3125 J; L = 62.5 kg m² s⁻¹.
12. 12
    Q6.12 → (a) ω₂ = 100 rev/min. (b) KE rises to 2.5 times — the increase comes from the muscular work the child does in folding his arms.
13. 13
    Q6.13 → α = 25 rad s⁻²; linear acceleration of the rope = 10 m s⁻².
14. 14
    Q6.14 → P = 36 kW.
15. 15
    Q6.15 → CG is at distance R/6 from the centre of the disk, on the side opposite the hole.
16. 16
    Q6.16 → Mass of the metre stick = 66 g.
17. 17
    Q6.17 → Average angular velocity ≈ 6.75 × 10¹² rad s⁻¹.

MCQ Questions

1 Mark Questions

1. 1
   Define the centre of mass of a system of particles.
2. 2
   Define torque and give its SI unit.
3. 3
   State the unit of angular momentum.
4. 4
   Define the radius of gyration.

2 Mark Questions

1. 1
   Two particles of masses 2 kg and 3 kg are 1 m apart. Find the position of their centre of mass. (Ans: 0.4 m from the 2 kg body)
2. 2
   A force of 10 N is applied tangentially to the rim of a wheel of radius 0.5 m. Find the torque about the centre. (Ans: τ = Fr = 5 N m)
3. 3
   Find the moment of inertia of a disc of mass 2 kg and radius 0.1 m about its own axis. (Ans: I = ½MR² = 0.01 kg m²)

3 Mark Questions

1. 1
   Derive the expression for the moment of inertia of a rod of mass M and length L about an axis through its centre perpendicular to its length. (Ans: I = ML²/12)
2. 2
   State and prove the theorem of parallel axes for the moment of inertia of a rigid body.
3. 3
   Obtain the relation L = Iω for the angular momentum of a rigid body rotating about a fixed axis. (Ans: sum of mᵢ rᵢ² × ω)

5 Mark Questions

1. 1
   State and prove the theorem of parallel axes and apply it to find the moment of inertia of a thin rod about an axis through one end perpendicular to its length. (Ans: I = ML²/3)
2. 2
   Explain the equilibrium conditions of a rigid body (translational and rotational), and derive the work–energy theorem for rotation using torque and angular displacement.