Rotational Motion - Class 11 Physics

Rotational Motion

Overview: Study of systems of particles and rigid bodies rotating about a fixed axis. Key concepts: Centre of Mass, Torque, and Moment of Inertia.

1. Centre of Mass (CM)

A point where the entire mass of the system is supposed to be concentrated.

Two Particle System

R_CM = (m₁r₁ + m₂r₂) / (m₁ + m₂)

Motion of CM

The CM moves as if the resultant external force acts on it.

F_ext = M a_CM

Momentum Conservation: If F_ext = 0, P_system = constant (CM moves with constant velocity).

2. Rotational Variables

Torque (τ): The turning effect of force. τ = r × F (Vector Product).
Angular Momentum (L): Moment of linear momentum. L = r × p.
Relation: τ = dL/dt.

Conservation of Angular Momentum

If external torque is zero, angular momentum L is conserved.

L = Iω = constant

3. Moment of Inertia (I)

The property of a body to oppose change in rotational motion. Analogous to Mass in linear motion.

I = ∑ m_i r_i²

Radius of Gyration (k)

The distance from axis where if whole mass is placed, I remains same.

I = Mk²

Theorems

Parallel Axis: I = I_CM + Md²
Perpendicular Axis: I_z = I_x + I_y (Planar laminas only)

4. Equations of Rotational Motion

For constant angular acceleration α:

ω = ω₀ + αt

θ = ω₀t + ½αt²

ω² = ω₀² + 2αθ

5. Dynamics of Rigid Bodies

Equilibrium

Translational Equilibrium: ∑ F_ext = 0
Rotational Equilibrium: ∑ τ_ext = 0

Rotation about Fixed Axis

τ = I α

Kinetic Energy: K = ½ I ω²

Rolling Motion: Combination of Translation and Rotation.

K_total = ½mv² + ½Iω²

Numericals - Rotational Motion

Numericals

Centre of Mass

Q1. Two particles of mass 1kg and 2kg are located at x=0 and x=3m respectively. Find position of Centre of Mass.

m1=1kg, x1=0; m2=2kg, x2=3

X_CM = (m1x1 + m2x2) / (m1 + m2)

X_CM = (1×0 + 2×3) / (1 + 2)

X_CM = 6 / 3

X_CM = 2 m

Torque Calculation

Q2. Find the torque of a force F = 3i + 4j acting at a point r = 2i?

τ = r × F

τ = (2i) × (3i + 4j)

τ = 6(i×i) + 8(i×j)

τ = 0 + 8k

τ = 8k Nm

Angular Momentum Conservation

Q3. A disc rotates at 10 rad/s. A small mass is dropped gently on it, doubling its moment of inertia. New angular speed?

External torque = 0 → L = constant

I₁ω₁ = I₂ω₂

I(10) = (2I)ω₂

10I = 2Iω₂

ω₂ = 5 rad/s

Moment of Inertia

Q4. Calculate Moment of Inertia of a solid sphere of mass 5kg and radius 0.2m about its diameter.

m = 5 kg, R = 0.2 m

I = (2/5)MR²

I = 0.4 × 5 × (0.2)²

I = 2 × 0.04

I = 0.08 kg m²

Rotational Kinetic Energy

Q5. A flywheel has MI = 100 kg m² rotating at 20 rad/s. Find Kinetic Energy.

K = ½Iω²

K = 0.5 × 100 × (20)²

K = 50 × 400

K = 20,000 J = 20 kJ

Rotational Kinematics

Q6. A fan accelerates from rest to 400 rad/s in 20 seconds. Find angular acceleration.

ω = ω₀ + αt

400 = 0 + α(20)

α = 400 / 20

α = 20 rad/s²

Rolling Motion

Q7. A ring rolls without slipping. What fraction of its Total Kinetic Energy is Rotational?

K_trans = ½mv²

K_rot = ½Iω² = ½(mR²)(v/R)² = ½mv²

Basically K_rot = K_trans for Ring

Total K = K_trans + K_rot = mv²

Fraction = K_rot / K_total = 0.5 (or 50%)

Perpendicular Axis Theorem

Q8. MI of a ring about an axis tangential and in plane? (Given MI about diameter = MR²/2)

Using Parallel Axis Theorem

I_tan = I_dia + MR²

I_tan = ½MR² + 1MR²

I_tan = (3/2) MR²

Torque Power

Q9. An engine delivers 30kW power at 300 rad/s. Calculate torque.

P = τω

30000 = τ × 300

τ = 30000 / 300

τ = 100 Nm

Ladder Problem

Q10. A 10m ladder stands against a wall with bottom 6m away. If CoM is at midpoint, find its height from ground.

Geometry problem. Coordinates of ends: (6,0) and (0, h).

h = √(10² - 6²) = √(100-36) = 8m

Top end at y=8.

Midpoint y_cm = (0 + 8) / 2

Height of CM = 4 m

Formulas & Facts - Rotational Motion

Equations & Formulas

Concept	Formula
CM Position	R = ∑m_ir_i / M
Torque	τ = r × F
Angular Momentum	L = r × p = Iω
Rotational 2nd Law	τ = Iα = dL/dt
Kinetic Energy	K = ½Iω²
Power	P = τω
MI Ring (Axis)	MR²
MI Disc (Axis)	MR²/2
MI Solid Sphere	2/5 MR²
MI Hollow Sphere	2/3 MR²
MI Rod (Centre)	ML²/12

50 NEET Facts

Key points for Rotational Motion.

1. CM Location Centre of Mass lies closer to the heavier mass in a two-particle system.

2. CM Outside Body Centre of Mass need not lie within the material of the body (e.g., Ring, Hollow Sphere).

3. Effect of Force on CM Internal forces cannot change the motion of the Centre of Mass. Only external forces can.

4. Torque vs Force Force causes translation; Torque causes rotation.

5. Moment of Couple A couple (equal opposite forces) produces pure rotation. Torque of a couple is independent of the point of rotation.

6. Inertia Analogy Moment of Inertia plays the same role in rotation as Mass plays in linear motion.

7. MI Dependence MI depends on: Mass, Shape/Size, and Distribution of mass relative to axis.

8. Lowest MI For a given mass and shape, MI is minimum about an axis passing through the Centre of Mass.

9. Angular Momentum Vector Direction of L is given by Right Hand Grip Rule along the axis.

10. Planetary Motion Planets move in ellipses. Torque by sun's gravity is zero (force is central). Hence L is conserved. Kepler's 2nd law follows.

11. Maximum Velocity in Orbit Since L = mvr = constant, v is max when r is min (Perihelion).

12. Ice Skater When skater folds arms, I decreases. To conserve L = Iω, ω increases (spins faster).

13. Cat in Air A falling cat can land on feet by changing its MI (tail movement) to rotate body while conserving L.

14. Rolling Velocity For pure rolling, v_CM = ωR. Velocity of bottom point is zero.

15. Velocity at Top Velocity of top point of a rolling wheel is 2v_CM.

16. Rolling Down Friction Friction acts upwards along the incline. It provides the torque for rotation.

17. Sphere vs Ring Rolling down an incline: Solid Sphere reaches first (least K_rot, most K_trans). Ring reaches last.

18. Work by Static Friction In pure rolling, the point of contact is at rest. So static friction does Zero Work.

19. Radius of Gyration For Ring k=R. For Disc k=R/√2. For Solid Sphere k=√(2/5)R.

20. Flywheel Used in engines to smooth out fluctuations in energy by storing Rotational KE.

21. Axis Theorems Parallel Axis Theorem applies to any body. Perpendicular Axis Theorem applies ONLY to planar (2D) bodies.

22. Linear vs Angular displacement s = rθ. v = rω. a_t = rα.

23. Vector Nature Angular displacement dθ is a vector for small values. Large θ does not commute, so not a vector.

24. Rolling Friction Cause Caused by deformation of surface, creating a small ramp in front of the wheel.

25. Mass Distribution Ring/Disc Ring has mass further from center than disc. Hence I_Ring > I_Disc for same Mass/Radius.

26. Hollow vs Solid Sphere Hollow sphere is harder to rotate (higher I) than solid sphere of same M and R.

27. Torque direction Torque is perpendicular to both r and F.

28. Cross Product Order τ = r × F is correct. F × r is wrong (gives negative torque).

29. Equilibrium conditions For rigid body, both net force and net torque must be zero.

30. Door Handle Placed at edge to maximize distance r, thereby checking maximum Torque for minimum Force.

31. Screw Driver Thick handle gives larger r, making it easier to turn tight screws.

32. Wrench Long wrench makes unscrewing easier (Larger Moment Arm).

33. Center of Gravity Coincides with Centre of Mass if gravitational field is uniform (true for small bodies).

34. Stability Body is stable if its Center of Gravity is low and vertical line from CG falls within the base.

35. Toppling Occurs if the vertical line through CG passes outside the base support.

36. Helicopter Tail Rotor Needed to counter the torque produced by the main rotor on the fuselage (Conservation of L).

37. Ballerina Uses arms to control speed of spin. Open arms -> Slow. Closed arms -> Fast.

38. High Diver Curls body in mid-air to reduce I and increase spin rate (ω) for somersaults.

39. Hard Boiled vs Raw Egg Spin them and stop momentarily. Raw egg starts spinning again (liquid inside kept rotating). Hard boiled stops (solid).

40. Acc of Rolling Sphere a = g sinθ / (1 + k²/R²).

41. K²/R² Ratio Ring: 1. Cylinder/Disc: 0.5. Sphere: 0.4.

42. Instantaneous Axis of Rotation In rolling, the point of contact is the instantaneous axis.

43. Precession The slow rotation of the axis of a spinning top due to torque by gravity.

44. Gyroscope Uses angular momentum to maintain orientation. Basis of navigation systems.

45. Rigid Body Definition Distance between any two constituent particles remains fixed.

46. Rotational Work W = ∫ τ dθ.

47. Angular Impulse Change in angular momentum. ∫ τ dt.

48. Recoil of Gun (Rotational) If gun held loosely, it kicks back (linear). If hinged, it rotates up (torque).

49. CM of Triangle Centroid. Intersection of medians. h/3 from base.

50. CM of Semicircle Wire 2R / π from center. For Semicircular Disc: 4R / 3π.

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