Motion of System of Particles and Rigid Body

Definition

The Centre of Mass is a unique point in a system where the entire mass of the system can be assumed to be concentrated for the purpose of analyzing its linear motion.

Explanation

Imagine you have a weirdly shaped piece of cardboard. If you try to balance it on your finger, the point where it stays perfectly balanced without tilting is the Centre of Mass. It's the "average" position of all the mass in the system.

• For a uniform symmetric object, the COM is at its geometric center.
• The COM can be inside the body (like a solid sphere) or outside the body (like a ring).
• If a net external force acts on a system, the COM moves as if the total mass were concentrated there.

Examples

• A Cricket Bat: The COM is closer to the heavier end (the blade) rather than the handle.
• A Ring: The COM of a circular ring is at the center, even though there is no actual mass located at that empty center point.

Real-Life Use

High Jump (Fosbury Flop): In the "Fosbury Flop" technique, the athlete arches their back. Because of this, their Centre of Mass actually passes under the bar while their body goes over it, allowing them to clear higher bars.

Definition

For two particles, the COM is a point on the line joining them. For a rigid body, it is the weighted average of all position vectors.

Explanation

If you have a heavy weight and a light weight connected by a rod, the balance point (COM) will be much closer to the heavy weight.

Examples

• Dumbbells: In a symmetric dumbbell, the COM is exactly in the middle of the connecting rod.
• A Seesaw: To balance two children of different weights, the pivot (COM) must be shifted toward the heavier child.

Real-Life Use

Vehicle Design: Car engineers place the engine and heavy components low and centrally to keep the COM low, preventing the car from flipping during sharp turns.

Definition

If the net external force acting on a system of particles is zero, the total linear momentum of the system remains constant.

Explanation

If no one from the outside pushes or pulls the system, the system's "total motion" doesn't change. Internal forces cannot change the total momentum.

Examples

• Recoil of a Gun: When a bullet is fired forward, the gun moves backward to keep the total momentum zero.
• Ice Skaters: Two skaters pushing off each other move in opposite directions.

Real-Life Use

Rocket Propulsion: A rocket ejects high-speed exhaust gases downward (backward momentum). To balance this, the rocket body gains an equal amount of momentum in the upward direction.

Definition

The vector product of two vectors A and B results in a third vector C that is perpendicular to the plane containing both A and B.

Explanation

Unlike the dot product, the cross product gives a new vector. Its direction is found using the Right-Hand Thumb Rule: Curl your fingers from A to B, and your thumb points to the result.

• It is non-commutative: A × B = -(B × A).

Examples

• Turning a Wrench: The force applied and the length of the wrench create a torque vector perpendicular to the wrench.
• Magnetic Force: The force on a moving charge in a magnetic field is a cross product of velocity and magnetic field.

Real-Life Use

Electric Motors: The torque produced in a motor is calculated using the vector product of the current-carrying wire and the magnetic field.

Definitions

• Angular Velocity (ω): The rate of change of angular displacement.
• Angular Acceleration (α): The rate of change of angular velocity.

Explanation

Angular velocity is "how fast something is spinning." Angular acceleration is "how fast the spin is speeding up or slowing down."

Examples

• Ceiling Fan: When switched on, it goes from ω = 0 to a high speed (angular acceleration).
• Clock Hands: The second hand moves with a constant angular velocity.

Real-Life Use

Centrifuges: In labs, samples are spun at high ω to create centripetal acceleration, separating plasma from blood cells.

Definition

Torque is the rotational equivalent of force. It is the measure of the force that can cause an object to rotate about an axis.

Explanation

Force pushes things; Torque twists things. To get more torque, you can either push harder or push further away from the pivot point.

Examples

• Opening a Door: It's easier to push at the handle (far from the hinge) than near the hinge.
• Using a Spanner: A longer spanner increases the torque, making it easier to loosen a tight nut.

Real-Life Use

Bicycle Pedals: The pedal arm (crank) is designed to provide enough torque to rotate the gears and move the bike forward.

Definition

Angular momentum is the "quantity of rotation" an object has. If no external torque acts on a system, its total angular momentum remains constant.

Explanation

Angular momentum depends on the moment of inertia (mass distribution) and angular velocity (spin speed).

Examples

• Divers: A diver curls their body (reducing I) to spin faster.
• Planetary Orbits: Planets move faster when closer to the Sun to conserve L.

Real-Life Use

Figure Skating: By pulling arms inward, skaters decrease their moment of inertia (I), forcing their angular velocity (ω) to increase, making them spin incredibly fast.

Definition

A rigid body is in complete equilibrium if it is neither translating (moving linearly) nor rotating.

Explanation

For a body to be totally still, both the total push/pull and the total twist must be zero.

• Translational Equilibrium: Σ F = 0
• Rotational Equilibrium: Σ τ = 0

Examples

• A Book on a Table: Gravity is balanced by the table's normal force, and no torque exists to flip it.
• Balanced Seesaw: Two kids of equal weight sitting at equal distances from the center.

Real-Life Use

Bridge Construction: Engineers ensure bridges are in equilibrium by calculating all forces and torques to ensure the structure doesn't shift or tilt.

Definitions

• Moment of Inertia: The property of a body to resist any change in its rotational motion ("Rotational Laziness").
• Radius of Gyration: The distance from the axis where the entire mass could be concentrated to have the same I.

Examples

• Tightrope Walker: They carry a long pole to increase their moment of inertia, making them more resistant to tipping.
• Flywheel: A heavy wheel used in engines to keep rotation smooth.

Real-Life Use

Hard Disk Drives (HDDs): Platters are designed to be light and balanced to minimize I, allowing them to reach high speeds (7200 RPM) quickly.

Parallel Axis Theorem

The moment of inertia about any axis is equal to the moment of inertia about a parallel axis passing through the COM plus the mass times the square of the distance between the axes.

Perpendicular Axis Theorem

For a planar object, the moment of inertia about an axis perpendicular to the plane is the sum of the moments of inertia about two perpendicular axes in the plane.

Examples

• Rotating a Door: Calculating inertia at the hinge using the center inertia + Md².
• A Flat Disc: Finding the inertia of a spinning CD.

Real-Life Use

Mechanical Engineering: Used to calculate the inertia of crankshafts in engines to prevent violent vibrations.

Definition

Kinematics studies how things rotate; Dynamics studies why they rotate.

Explanation

It is exactly like linear motion, but we replace linear variables with rotational ones.

• Newton's 2nd Law for rotation: τ = Iα
• Rotational Kinetic Energy: KE_rot = ½Iω²

Examples

• Electric Drill: The motor provides torque τ, causing angular acceleration α.
• Spinning Top: Energy stored is rotational kinetic energy.

Real-Life Use

Wind Turbines: Wind applies torque to blades, converting wind energy into rotational kinetic energy, then into electricity.

Definition

Rolling motion is a combination of Pure Translation (moving forward) and Pure Rotation (spinning).

Explanation

In pure rolling (no slipping), the distance the wheel moves forward is exactly its circumference.

Examples

• Bowling Ball: Starts by sliding and then begins to roll purely.
• Bicycle Tires: Tires rolling on a road without skidding.

Real-Life Use

ABS (Anti-lock Braking Systems): Prevents wheels from stopping rotation while the car is still moving (slipping), ensuring the driver maintains control.

A professional figure skater performing a "scratch spin" incorporates almost every concept in this chapter.

Initially, the skater starts with arms extended, creating a high Moment of Inertia (I). As they pull their arms tight, I decreases. Since no external torque acts, the Law of Conservation of Angular Momentum (L = Iω) applies. To keep L constant, as I goes down, ω (angular velocity) must increase, causing the skater to spin in a blur of speed.

To stop, they extend their arms again, increasing I and decreasing ω.

1. The Centre of Mass of a body:
   Ans: c) May lie inside or outside the body
2. If no external torque acts on a system, which quantity is conserved?
   Ans: b) Angular momentum
3. The rotational analogue of mass is:
   Ans: c) Moment of Inertia
4. The unit of torque is:
   Ans: b) N m
5. A diver curls their body to spin faster. This is due to:
   Ans: b) Decrease in Moment of Inertia
6. The relation v = rω is valid only if:
   Ans: c) The body is in pure rolling
7. The Parallel Axis Theorem is expressed as:
   Ans: b) I = I_cm + Md²
8. The direction of the angular velocity vector is along:
   Ans: c) The axis of rotation
9. For a body in complete equilibrium:
   Ans: c) Both Σ F = 0 and Σ τ = 0
10. The radius of gyration k is related to I by:
    Ans: b) I = Mk²

1. Q: Define a Rigid Body. Why is it an idealization?
   Ans: A rigid body is one in which the distance between any two particles remains constant. It is an idealization because all real objects deform slightly under force.
2. Q: State the Law of Conservation of Angular Momentum with an example.
   Ans: If net external torque is zero, total angular momentum remains constant. Example: A ballerina spinning faster by pulling arms inward.
3. Q: Distinguish between Torque and Force.
   Ans: Force causes linear acceleration (F=ma); Torque causes angular acceleration (τ=Iα). Force acts on a point; Torque depends on the pivot and distance.
4. Q: What is the physical significance of the Radius of Gyration?
   Ans: It is the effective distance from the axis where the entire mass could be concentrated to produce the same moment of inertia.
5. Q: State and explain the Perpendicular Axis Theorem.
   Ans: For a planar body, I_z = I_x + I_y, where z is the axis perpendicular to the plane.
6. Q: Why is it easier to open a door by pushing at the edge?
   Ans: τ = r × F. Increasing the distance (r) from the hinge produces more torque with the same force.
7. Q: What is "Pure Rolling"? Write the condition.
   Ans: Combination of translation and rotation without slipping. Condition: v = rω.
8. Q: Derive the relation between Torque and Moment of Inertia.
   Ans: τ = rF ⇒ τ = r(ma) ⇒ τ = r(m r α) = (mr²)α = Iα.
9. Q: Cylinder vs Sphere rolling down an incline?
   Ans: The sphere reaches first because it has a smaller moment of inertia (2/5 MR² < 1/2 MR²), allowing more energy to go into translation.
10. Q: What happens to COM if internal forces change?
    Ans: Nothing. Internal forces cannot change the motion of the Centre of Mass.

• The "Rotational Mirror" Trick: Swap linear variables for rotational ones: s → θ, v → ω, a → α, m → I, F → τ, p → L.
• Right Hand Rule: "Curl to the Thumb" → Curl fingers from A to B, thumb is the result.
• COM Logic: "Heavy side = Close side" → COM always hugs the heavier object.