Motion of System of Particles and Rigid Body

Hello everyone! Welcome to today's physics class. If you have ever watched a gymnast spinning gracefully in the air, a spinning top, or even a car turning around a corner, you have witnessed complex motion. Until now, we mostly studied point masses where an object's size and shape do not matter. But in the real world, objects are made of countless tiny particles, and they can not only move from one place to another, they can also spin and change shape. Today, we are going to unlock how to study the motion of these complex systems of particles and rigid bodies. Grab your notebooks, and let us dive right in!

Definition

Imagine trying to balance a long stick on your finger. There is one exact spot where the stick rests in perfect balance. That magical point where all the mass of a system seems to be concentrated is called the Centre of Mass.

According to Halliday, Resnick, and Walker: The centre of mass of a system of particles is the point that moves as though all the mass were concentrated there and all external forces were applied there.

According to Sears and Zemansky: The centre of mass is a uniquely defined point in a physical system or object where the weighted relative position of the distributed mass sums to zero.

Explanation

When we look at a system of two particles of masses m1 and m2 located at positions x1 and x2 along a straight line, their centre of mass Xcm is the weighted average of their positions.

• The centre of mass depends only on the masses and their relative positions, not on external forces.
• If no external force acts on the system, the centre of mass moves with a constant velocity.
• It can lie inside or even outside the physical body, like in the case of a ring or a horseshoe.

Examples

• A dumbbell: With two heavy iron balls connected by a light rod, the centre of mass lies closer to the heavier ball.
• The Earth-Moon system: Their common centre of mass lies beneath the surface of the Earth, around 1,700 kilometers below the crust, because the Earth is much more massive than the Moon.

Real-Life Uses & Misconceptions

Real-Life Uses: High jumpers use the technique called the Fosbury Flop. By arching their back over the high jump bar, their centre of mass actually passes *under* the bar while their body goes *over* it. Automotive designers keep the centre of mass as low as possible to prevent sports cars from flipping over during sharp turns.

Misconception & Correction: Misconception: The centre of mass and the centre of gravity are always at the exact same point. Correction: While usually identical on Earth, the centre of gravity takes into account variations in a gravitational field over very large objects (like skyscrapers), making it fractionally higher.

Definition

Momentum conservation is a fundamental law stating that the total momentum of an isolated system remains constant if no external forces act upon it.

According to David Halliday: If the net external force acting on a system is zero, the total linear momentum of the system remains constant.

According to Richard Feynman: The conservation of momentum is a profound consequence of the homogeneity of space, meaning the laws of physics do not change from one place to another.

Explanation

Think about a skateboarder standing still on a skateboard and throwing a heavy medicine ball forward. The skateboarder rolls backward. Why? Because before the throw, the total momentum was zero. After the throw, the ball has forward momentum, so the skateboarder must gain equal backward momentum to keep the total momentum at zero.

• Total initial momentum equals total final momentum.
• It applies to collisions, explosions, and recoil.
• Internal forces do not change the total momentum of the system.

Examples

• Recoil of a gun: When a bullet is fired forward, the gun kicks backward.
• Ice skaters: Two ice skaters pushing away from each other on a frictionless rink move apart with momenta that balance out.

Real-Life Uses & Misconceptions

Real-Life Uses: Space rockets rely entirely on momentum conservation, expelling hot exhaust gases downwards to shoot upwards. Police use momentum conservation laws to calculate pre-crash vehicle speeds from skid marks.

Misconception & Correction: Misconception: Momentum is conserved even if external friction is present. Correction: Friction is an external force; momentum is only conserved for an isolated system where net external forces are zero.

Definition

A rigid body is an idealized solid object in which the distance between any two given points remains fixed and invariable, no matter how hard you push or pull it.

According to Goldstein: A rigid body is a system of material particles subject to the holonomic constraint that the distance between any pair of particles remains constant over time.

According to Kleppner and Kolenkow: The centre of mass of a continuous rigid body is found by integrating position weighted by differential mass elements over the entire volume.

Explanation

Unlike a two-particle system, a rigid body like a baseball bat or a solid cylinder has continuous mass. We find its centre of mass by using calculus, integrating over every tiny mass particle ($dm$) across the body's geometry.

• For symmetrical objects with uniform density (like a sphere, cylinder, or uniform rod), the centre of mass lies at their geometric center.
• The mass can be treated as concentrated at the centre of mass for translational motion analysis.

Examples

• Uniform meter scale: Its centre of mass is at the 50-centimeter mark.
• Solid cone: Its centre of mass lies on its axis of symmetry at a height of one-fourth of its total height from the base.

Real-Life Uses & Misconceptions

Real-Life Uses: Architects and civil engineers need to know the centre of mass of complex structural components (like curved bridge arches or asymmetrical roofs) to ensure stability against wind loads and earthquakes.

Misconception & Correction: Misconception: The centre of mass of a rigid body must always be located inside the material. Correction: Not at all! Rings, doughnuts, hollow pipes, and boomerangs have their centres of mass located in empty space outside their physical structure.

Definition

The vector product, or cross product, of two vectors is a mathematical operation that produces a third vector which is perpendicular to both of the original vectors.

Mathematical Convention: $\vec{A} \times \vec{B} = AB \sin(\theta) \hat{n}$, where $\theta$ is the angle between them and $\hat{n}$ is a unit vector perpendicular to both, given by the right-hand rule.

According to Gibbs: Vector products provide the natural mathematical language for describing rotational quantities like torque and angular momentum in three-dimensional space.

Explanation

When you multiply two vectors using a cross product, you are finding out how much they act perpendicularly to each other.

• The magnitude represents the area of the parallelogram formed by the two vectors.
• The direction is given by the Right-Hand Rule: curl the fingers of your right hand from the first vector to the second vector, and your thumb points in the direction of the cross product.
• The cross product is anti-commutative: $\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$.

Examples

• Calculating torque: $\vec{\tau} = \vec{r} \times \vec{F}$
• Calculating angular momentum: $\vec{L} = \vec{r} \times \vec{p}$

Real-Life Uses & Misconceptions

Real-Life Uses: In computer graphics and game development, vector cross products are used extensively to calculate surface normals for rendering shadows and lighting correctly.

Misconception & Correction: Misconception: The cross product of two vectors is a scalar number. Correction: That is the dot product. The cross product always yields a new *vector* pointing in a perpendicular direction.

Definition

Angular velocity tells us how fast an object is rotating around an axis, while angular acceleration tells us how fast that rotational speed is changing.

According to Halliday and Resnick: Angular velocity $\omega$ is the rate of change of angular displacement with respect to time, measured in radians per second.

According to Tipler: Angular acceleration $\alpha$ is the time rate of change of angular velocity, measured in radians per second squared.

Explanation

Just as linear velocity tells you how many meters you cover per second, angular velocity tells you how many radians you sweep through per second.

• Angular displacement $\theta$, angular velocity $\omega = \frac{d\theta}{dt}$, and angular acceleration $\alpha = \frac{d\omega}{dt}$ are direct rotational analogs to linear quantities.
• Direction of angular velocity is along the axis of rotation, given by the right-hand screw rule.

Examples

• Ceiling fan: Spinning at a constant 120 RPM has a fixed angular velocity and zero angular acceleration.
• Accelerating wheels: A car starting from rest and spinning its wheels faster exhibits angular acceleration.

Real-Life Uses & Misconceptions

Real-Life Uses: Hard disk drives and Blu-ray disc readers require precise control of angular velocity to read data accurately off spinning platters.

Misconception & Correction: Misconception: All parts of a spinning wheel have different angular velocities. Correction: Every particle on a rigid rotating wheel has the exact *same* angular velocity $\omega$ and angular acceleration $\alpha$. What differs is linear speed ($v = r\omega$).

Definition

Torque is the rotational equivalent of a straight-line force. It is the measure of how much a force acting on an object causes that object to rotate.

According to Serway and Jewett: Torque $\tau$ is defined as the cross product of the position vector from the axis of rotation to the point of application of force and the force vector: $\tau = r F \sin(\theta)$.

Explanation

Think about opening a heavy door. You push it at the handle, far away from the hinges, because that extra distance gives you more torque with less force.

• Torque depends on three things: magnitude of force, distance from the pivot point (lever arm), and the angle of application.
• Maximum torque is achieved when force is applied perpendicularly ($90^\circ$) to the lever arm.
• SI unit is Newton-meter ($\text{N m}$).

Examples

• Using a long wrench to loosen a rusted lug nut.
• Pushing a merry-go-round at the outer edge versus near the center.

Real-Life Uses & Misconceptions

Real-Life Uses: Mechanics use torque wrenches to tighten engine cylinder head bolts to precise values. Electric vehicles deliver high torque instantly from zero RPM for rapid acceleration.

Misconception & Correction: Misconception: Applying any force will always make an object rotate. Correction: If you push a box directly toward its centre of mass, the line of force passes through the pivot, producing zero torque and sliding it without spinning.

Definition

Angular momentum is the rotational counterpart of linear momentum, representing the quantity of rotation of a body, accounting for mass, shape, and spinning speed.

According to Feynman: Angular momentum is a measure of the amount of rotational motion, defined for a single particle as $\vec{L} = \vec{r} \times \vec{p}$.

According to Giancoli: For a rigid body rotating about a fixed axis, it is the product of moment of inertia and angular velocity ($L = I\omega$).

Explanation

An object spinning around an axis wants to keep spinning unless stopped. That persistence in spinning is its angular momentum.

• It is a vector quantity directed along the axis of rotation.
• For a rigid body, $L = I\omega$.
• SI unit is kilogram meter squared per second ($\text{kg m}^2/\text{s}$).

Examples

• A spinning bicycle wheel held by handles resists being tilted.
• Planet Earth orbiting the Sun and spinning on its axis possesses massive angular momentum.

Real-Life Uses & Misconceptions

Real-Life Uses: Helicopters use a tail rotor to counteract the angular momentum generated by main overhead blades. Satellites use reaction wheels to adjust pointing angles in space without wasting fuel.

Misconception & Correction: Misconception: Only heavy objects have high angular momentum. Correction: A lightweight object spinning extremely fast (like a dentist's drill) can have massive angular momentum.

Definition

The law of conservation of angular momentum states that if the net external torque acting on a system is zero, the total angular momentum of the system remains constant.

According to Halliday and Resnick: When net external torque is zero, total angular momentum $\vec{L}$ is conserved: $I_i \omega_i = I_f \omega_f$.

Explanation

Think of an ice skater spinning with arms outstretched. When she pulls her arms inward, her mass is closer to the axis, reducing her moment of inertia ($I$). Since angular momentum ($L = I\omega$) must stay constant, her angular velocity ($\omega$) shoots up!

• $I_1 \omega_1 = I_2 \omega_2$ is our golden conservation equation.
• Decreased moment of inertia increases angular velocity, and vice versa.

Examples

• A ballet dancer pulling in arms during a pirouette.
• A high diver tucking their body into a tight ball for multiple somersaults.

Real-Life Uses & Misconceptions

Real-Life Uses: Ice skating competitions and acrobatic diving rely entirely on this principle. In astrophysics, collapsing spinning gas clouds form high-speed pulsars due to this conservation law.

Misconception & Correction: Misconception: A skater can speed up their spin from nothing without pulling arms in or getting a push. Correction: The skater must do internal muscular work drawing on stored chemical energy to change their moment of inertia.

Definition

A rigid body is in equilibrium when both its linear acceleration and angular acceleration are zero, meaning it is neither speeding up in any direction nor starting to spin.

According to Hibbeler: A rigid body is in complete mechanical equilibrium when the resultant force is zero and the resultant torque about any point is also zero.

Explanation

For an object to be completely at rest (or moving with steady, unaccelerating velocity and rotation), two conditions must be met simultaneously:

• Forces pulling left equal forces pulling right; forces up equal forces down ($\sum F_x = 0$, $\sum F_y = 0$).
• Torques trying to spin it clockwise equal torques trying to spin it counter-clockwise ($\sum \tau = 0$).
• Equilibrium means zero *acceleration*, not necessarily zero velocity (e.g., a car cruising at steady speed is in dynamic equilibrium).

Examples

• A heavy picture frame hanging quietly on a wall nail.
• A suspension bridge holding up heavy traffic.

Real-Life Uses & Misconceptions

Real-Life Uses: Civil engineers apply static equilibrium equations when designing bridges, cranes, and skyscrapers.

Misconception & Correction: Misconception: If net force is zero, an object is guaranteed to be in static equilibrium. Correction: Not necessarily! Two equal and opposite forces acting along *different* lines create a couple, causing the object to spin.

Definition

Moment of inertia (rotational inertia) is the resistance of a rigid body to changes in its rotational motion, depending on mass and how that mass is distributed relative to the rotation axis.

According to Halliday and Resnick: $I = \sum m_i r_i^2$, where $m_i$ is particle mass and $r_i$ is perpendicular distance from the rotation axis.

Explanation

Holding a long heavy pole in the middle is easy to spin; holding it at one extreme end makes it feel extremely heavy to twist. That difficulty is the moment of inertia.

• Further mass distribution from the rotation axis drastically increases the moment of inertia ($r^2$ dependence).
• SI unit is $\text{kg m}^2$.

Examples

• Solid cylinder versus hollow cylinder of equal mass and radius: The hollow cylinder has a larger moment of inertia because all mass is concentrated at the rim.

Real-Life Uses & Misconceptions

Real-Life Uses: Flywheels store mechanical energy smoothly in industrial engines and regenerative braking systems due to their high rotational inertia.

Misconception & Correction: Misconception: Moment of inertia is a fixed property. Correction: It completely changes depending on *where* you choose the axis of rotation!

Definition

The radius of gyration is a geometric property representing the distance from the axis of rotation to a single imaginary point where all the mass could be concentrated without changing the actual moment of inertia.

Standard Relation: $I = M k^2$, where $k$ is the radius of gyration.

Explanation

Instead of calculating complex integrals for every machine part, engineers use the radius of gyration as an effective "average" distance of mass from the spin axis.

• $k = \sqrt{\frac{I}{M}}$
• SI unit is meters (m).

Examples

• Uniform ring of radius $R$: $k = R$.
• Uniform disc of radius $R$: $k = \frac{R}{\sqrt{2}} \approx 0.707R$.

Real-Life Uses & Misconceptions

Real-Life Uses: Civil and mechanical engineers use it to calculate how slender columns, steel beams, and bridge pillars behave under compressive loads before buckling.

Misconception & Correction: Misconception: Radius of gyration is the actual physical radius. Correction: Only for a thin ring is $k$ equal to the physical radius; for solid bodies, $k$ is always smaller.

Definition

These theorems are brilliant shortcuts allowing us to calculate moments of inertia without re-integrating from scratch.

Perpendicular Axis Theorem (Planar bodies): $I_z = I_x + I_y$ (moment of inertia about an axis perpendicular to a plane lamina equals sum of moments about two mutually perpendicular axes in the plane).

Parallel Axis Theorem (Any rigid body): $I = I_{cm} + Md^2$ (moment of inertia about any axis equals moment of inertia about a parallel axis through the centre of mass plus $Md^2$).

Explanation

• Perpendicular Axis Theorem works *only* for flat, two-dimensional objects (lamina).
• Parallel Axis Theorem works for *any* 3D rigid body.

Examples

• Finding the moment of inertia of a circular ring about its diameter using polar moment of inertia.
• Finding rod moment of inertia about its end using $I_{cm} = \frac{1}{12}ML^2$ and the parallel axis theorem.

Real-Life Uses & Misconceptions

Real-Life Uses: Mechanical designers use the Parallel Axis Theorem when analyzing rotating shafts, gears, and off-center flywheels.

Misconception & Correction: Misconception: You can apply the Perpendicular Axis Theorem to a solid 3D football or brick. Correction: Absolutely not! It is strictly restricted to planar, 2D flat objects.

Definition

Rotational kinematics describes *how* objects rotate, while rotational dynamics explains *why* they rotate.

Kinematic Analogy: Swap variables $x \rightarrow \theta$, $v \rightarrow \omega$, $a \rightarrow \alpha$.

Dynamics: Newton's Second Law for rotation: $\tau_{net} = I\alpha$.

Explanation

• $\omega = \omega_0 + \alpha t$
• $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$
• $\omega^2 = \omega_0^2 + 2\alpha \theta$

Examples

• Calculating revolutions a ceiling fan makes before stopping when switched off.

Real-Life Uses & Misconceptions

Real-Life Uses: Robotics engineers use rotational kinematics and dynamics algorithms to program joint motors in robotic arms.

Misconception & Correction: Misconception: Angles in kinematic formulas can be plugged in using degrees. Correction: Never! Angles must *always* be expressed in radians.

Definition

Rolling motion is a combination of pure translational motion and pure rotational motion.

According to Resnick, Halliday, and Krane: Rolling without slipping occurs when the point of contact has zero instantaneous velocity relative to the surface.

Explanation

• Condition for rolling without slipping: $v = R\omega$.
• Total Kinetic Energy = $\frac{1}{2} M v^2 (1 + \frac{k^2}{R^2})$.
• Objects with smaller $k^2/R^2$ ratios (solid spheres) roll down inclined planes faster than rings.

Examples

• A bowling ball rolling down a wooden alley.
• Car tires gripping asphalt smoothly.

Real-Life Uses & Misconceptions

Real-Life Uses: Automotive traction control and anti-lock braking systems (ABS) maintain rolling without slipping.

Misconception & Correction: Misconception: The topmost point of a rolling wheel moves at the same speed as the bicycle. Correction: The top point moves forward at *twice* the speed ($v_{top} = 2v_{cm}$) because translational and rotational tangential speeds add up!

• Centre of Mass (Two Particles): $X_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$
• Vector Product Magnitude: $|\vec{A} \times \vec{B}| = AB \sin(\theta)$
• Angular Velocity: $\omega = \frac{d\theta}{dt}$
• Angular Acceleration: $\alpha = \frac{d\omega}{dt}$
• Torque: $\vec{\tau} = \vec{r} \times \vec{F}$
• Angular Momentum: $\vec{L} = \vec{r} \times \vec{p} = I\omega$
• Newton's Second Law for Rotation: $\tau_{net} = I\alpha$
• Parallel Axis Theorem: $I = I_{cm} + Md^2$
• Perpendicular Axis Theorem: $I_z = I_x + I_y$
• Rolling Kinetic Energy: $K = \frac{1}{2}Mv^2 + \frac{1}{2}I\omega^2$

Problem 1

A flywheel of moment of inertia $2.0 \text{ kg m}^2$ is accelerated from rest to an angular velocity of $50 \text{ rad/s}$ in $5 \text{ seconds}$. Calculate the applied constant torque.

Step 1: Identify given variables.
- Moment of inertia (I) = 2.0 kg m²
- Initial angular velocity (ω₀) = 0 rad/s
- Final angular velocity (ω) = 50 rad/s
- Time (t) = 5 s

Step 2: Find angular acceleration (α).
Using kinematic equation ω = ω₀ + αt:
50 = 0 + α(5)  =>  α = 10 rad/s².

Step 3: Apply rotational dynamics equation (τ = Iα).
τ = 2.0 kg m² × 10 rad/s² = 20 N m.

Answer: The constant torque applied is 20 N m.
        

Problem 2

Find the moment of inertia of a uniform thin rod of mass $3 \text{ kg}$ and length $2 \text{ m}$ about an axis perpendicular to its length passing through one of its ends.

Step 1: Identify formula for moment of inertia about center of mass (I_cm).
I_cm = (1/12) M L²
Given M = 3 kg and L = 2 m:
I_cm = (1/12)(3)(2)² = 12/12 = 1 kg m².

Step 2: Use Parallel Axis Theorem to find moment of inertia about the end.
Distance d from center of mass to end is L/2 = 1 m.
Formula: I = I_cm + Md²
I = 1 + (3)(1)² = 1 + 3 = 4 kg m².

Answer: The moment of inertia about the end is 4 kg m².
        

Wind energy is one of the fastest-growing renewable power sources on Earth. To truly understand how modern wind turbines operate, engineers must master every single concept from the motion of systems of particles and rigid body dynamics that we have studied in this chapter.

Consider a massive utility-scale wind turbine with three giant composite blades sweeping across a rotor circle of over 100 meters in diameter. Each blade acts as an asymmetrical rigid body subjected to distributed aerodynamic lift forces along its length. Because the wind blows harder at the top of the sweep than near the ground, the torque generated across the rotor disc changes constantly. Engineers must carefully calculate the Centre of Mass of the rotating blade assembly. If the blades are unevenly balanced, dangerous vibrations occur, leading to structural fatigue and catastrophic mechanical failure.

To prevent overspeeding during heavy storm gales, turbine blades feature a pitch control mechanism. Hydraulic motors rotate the blades along their long axis. By changing the blade angle relative to the incoming wind, the aerodynamic torque is reduced. This is a brilliant practical application of Torque and Angular Momentum Conservation. As wind gusts strike, the turbine generator acts as a resistive load, creating an electromagnetic counter-torque. If the wind speed fluctuates, the massive Moment of Inertia of the spinning blades acts as a rotational flywheel stabilizer, preventing sudden spikes in generator rotational speed. Furthermore, engineers use the Parallel Axis Theorem when designing the steel tower housing. The heavy generator nacelle sits dozens of meters off-center from the tower base, generating massive bending torques that must satisfy strict Static Equilibrium conditions ($\sum \tau = 0$ and $\sum F = 0$) under extreme hurricane-force winds.

From vector cross products used in computational fluid dynamics simulations of wind flow, to angular acceleration profiles during start-up, wind turbines are a living testament to the power of rigid body mechanics. Without precise calculations of rolling components in the gearbox, conservation of angular momentum, and rotational kinetic energy, harvesting clean energy from moving air would be impossible.

1. The centre of mass of a system of two particles of equal mass is located:
   A) Closer to the heavier particle
   B) Exactly at the midpoint between them
   C) Outside the system
   D) At the position of the second particle
   Answer: B
   Explanation: For two equal masses, the centre of mass lies precisely at the geometric midpoint, dividing the distance equally.


2. Which of the following quantities is a vector?
   A) Moment of inertia
   B) Radius of gyration
   C) Torque
   D) Kinetic energy
   Answer: C
   Explanation: Torque is defined as a cross product of position and force vectors ($\vec{\tau} = \vec{r} \times \vec{F}$), making it a vector quantity.


3. The cross product of two parallel vectors is:
   A) Maximum
   B) Equal to one
   C) Zero
   D) Undefined
   Answer: C
   Explanation: The magnitude of a cross product involves $\sin(\theta)$. Since the angle $\theta$ between parallel vectors is $0^\circ$ and $\sin(0) = 0$, the cross product is zero.


4. If the net external torque acting on a system is zero, which of the following is conserved?
   A) Total linear momentum
   B) Total angular momentum
   C) Moment of inertia
   D) Angular acceleration
   Answer: B
   Explanation: Just as net external force conservation yields linear momentum, zero net external torque guarantees the conservation of total angular momentum.


5. The moment of inertia of a solid cylinder of mass $M$ and radius $R$ about its central geometric axis is:
   A) $MR^2$
   B) $\frac{1}{2}MR^2$
   C) $\frac{2}{5}MR^2$
   D) $\frac{2}{3}MR^2$
   Answer: B
   Explanation: Integration of mass distribution for a solid cylinder or disc about its central longitudinal axis gives $I = \frac{1}{2}MR^2$.


6. The Perpendicular Axis Theorem is applicable only to:
   A) Spherical bodies
   B) Planar (two-dimensional) bodies
   C) Long cylindrical rods
   D) Irregular 3D masses
   Answer: B
   Explanation: The Perpendicular Axis Theorem ($I_z = I_x + I_y$) is strictly valid only for flat, 2D lamina sheets lying in the xy-plane.


7. When an ice skater pulls her outstretched arms inward during a spin, her angular velocity increases because:
   A) Her mass increases
   B) Her moment of inertia decreases
   C) An external motor pushes her
   D) Gravity pulls her faster
   Answer: B
   Explanation: By pulling her arms in, her moment of inertia ($I$) decreases. Since angular momentum ($L = I\omega$) must remain conserved, her angular velocity ($\omega$) must increase.


8. The condition for rolling without slipping for a wheel of radius $R$ moving with linear velocity $v$ and angular velocity $\omega$ is:
   A) $v = \omega / R$
   B) $v = R^2 \omega$
   C) $v = R\omega$
   D) $v = \omega$
   Answer: C
   Explanation: For the point of contact to remain instantaneously at rest relative to the ground, the forward linear speed must equal the tangential speed $R\omega$.


9. The SI unit of torque is:
   A) $\text{kg m/s}$
   B) $\text{N m}$
   C) $\text{Joule/second}$
   D) $\text{rad/s}^2$
   Answer: B
   Explanation: Torque is force multiplied by distance, measured in Newtons times meters ($\text{N m}$).


10. An object is in complete mechanical equilibrium if:
    A) Only its net force is zero
    B) Only its net torque is zero
    C) Both net force and net torque are zero
    D) Its angular velocity is maximum
    Answer: C
    Explanation: Complete equilibrium requires zero translational acceleration ($\sum \vec{F} = 0$) and zero rotational acceleration ($\sum \vec{\tau} = 0$).

1. State the law of conservation of angular momentum and give two real-life examples.
   Answer: When the net external torque acting on a system is zero, the total angular momentum of the system remains constant. Examples: (1) A ballet dancer pulling in their arms to spin faster during a pirouette. (2) A high diver tucking their body into a tight ball during a somersault.


2. Explain why a wrench with a longer handle is much easier to use when loosening a tight bolt.
   Answer: Torque is given by $\tau = r F \sin(\theta)$. A longer handle increases the distance ($r$) from the pivot (the bolt). Thus, for the same amount of pushing force ($F$), a longer lever arm produces greater torque, making it much easier to turn the bolt.


3. Distinguish clearly between moment of inertia and mass.
   Answer: Mass is the measure of an object's resistance to linear acceleration (translational inertia), whereas moment of inertia is the measure of an object's resistance to angular acceleration (rotational inertia), which depends on both the total mass and how that mass is distributed relative to the axis of rotation.


4. State the Parallel Axis Theorem and write its mathematical equation.
   Answer: The Parallel Axis Theorem states that the moment of inertia ($I$) of a rigid body about any axis is equal to the sum of the moment of inertia about a parallel axis passing through its centre of mass ($I_{cm}$) and the product of the mass ($M$) and the square of the perpendicular distance ($d$) between the two axes. Equation: $I = I_{cm} + Md^2$.


5. Can the centre of mass of a solid body lie outside the body? Give an example.
   Answer: Yes, the centre of mass can lie entirely in empty space outside the physical material of an object. Example: A circular ring, a horseshoe, or a boomerang.


6. Why is it more difficult to open a door by pushing near the hinges than by pushing near the outer edge?
   Answer: Pushing near the hinges makes the lever arm ($r$) very small, resulting in negligible torque. Pushing near the outer edge maximizes $r$, providing maximum torque for the same applied force.


7. What is meant by rolling without slipping? What is the velocity of the lowest point of a rolling wheel relative to the ground?
   Answer: Rolling without slipping occurs when there is no relative motion (no sliding or skidding) between the contact point of the rolling object and the surface. The velocity of the lowest point relative to the ground is exactly zero.


8. Explain why a helicopter requires a small vertical tail rotor in addition to its large overhead main rotor.
   Answer: As the main overhead rotor spins, by conservation of angular momentum, the helicopter body tends to spin in the opposite direction. The tail rotor creates a counter-torque to keep the cabin stable and prevent unwanted spinning.


9. Define radius of gyration. How is it related to moment of inertia?
   Answer: The radius of gyration is the distance from the axis of rotation to a point where the entire mass of the body can be imagined to be concentrated so that its moment of inertia remains unchanged. Relation: $I = M k^2$, where $k$ is the radius of gyration.


10. A solid sphere, a disc, and a ring all roll down an inclined plane from the same height without slipping. Which one reaches the bottom first and why?
    Answer: The solid sphere reaches the bottom first. Objects with smaller moments of inertia relative to their mass ($k^2/R^2$ ratio) convert potential energy into translational kinetic energy more efficiently rather than wasting it in rotational spin energy. A solid sphere has the smallest ratio, followed by the disc, and lastly the ring.

• T-I-A: Torque equals Inertia times Acceleration ($\tau = I\alpha$), analogous to $F = ma$ in linear motion.
• C-M Position: Centre of Mass can lie inside or float in thin air outside physical bodies (e.g., rings or boomerangs).
• Torque vs Force: Force causes linear sliding; torque causes rotational twisting.
• Moment of Inertia vs Mass: Mass is heaviness; moment of inertia accounts for mass distribution relative to the spin axis ($r^2$).

Mnemonic for Rotational Dynamics: Remember T-I-A for torque, and keep in mind that rolling without slipping means the contact point has zero relative velocity!