SYSTEMS OF PARTICLES AND ROTATIONAL MOTION-Notes

A rigid body is an idealized physical body in which the distance between any two particles remains constant, even when external forces act on it. This means that the body does not deform, stretch, compress, or bend under force.

In reality, no object is perfectly rigid, but many solid objects behave approximately like rigid bodies when their deformation is very small compared to their size. This assumption makes the study of rotational motion simple and practical.

Important Aspects of a Rigid Body

• Fixed Relative Positions:
  
  • All particles of a rigid body maintain fixed positions relative to each other.
  • Shape and size of the body remain unchanged during motion.
• Translation of a Rigid Body:
  In translational motion:
  • Every particle of the body moves with the same velocity and same acceleration.
  • The body moves without rotating.
  • Example: A book sliding straight on a table.
• Rotation of a Rigid Body:
  In rotational motion:
  • The body rotates about a fixed axis.
  • Each particle moves in a circular path around the axis.
  • All particles have the same angular velocity, but different linear speeds.
• Combined Motion:
  A rigid body can have:
  • Translation + rotation at the same time.
  • Example: A wheel rolling on the road.
• Centre of Mass:
  
  • The entire translational motion of a rigid body can be described by the motion of its centre of mass.
  • External forces determine the motion of the centre of mass.
• Angular Quantities:
  

  Rigid body motion introduces angular quantities such as:

  • Angular displacement
  • Angular velocity
  • Angular acceleration

  These describe rotational motion just as linear quantities describe straight-line motion.

• Moment of Inertia:
  
  • Moment of inertia measures a rigid body’s resistance to rotational motion.
  • It depends on mass distribution and axis of rotation.
  • More mass farther from the axis → greater moment of inertia.
• Torque and Rotation:
  
  • Torque causes rotational motion in a rigid body.
  • Greater torque produces greater angular acceleration.
  • Torque is the rotational equivalent of force.
• Conservation of Angular Momentum:
  
  • If no external torque acts on a rigid body, its angular momentum remains constant.
  • Explains phenomena like a spinning skater pulling in their arms.

Translational motion of a rigid body is the motion in which every particle of the body moves through the same displacement in the same direction during a given time interval. As a result, all points of the body have identical velocity and identical acceleration at any instant.

In this type of motion, the rigid body does not rotate—its orientation remains unchanged while it moves from one position to another.

Rotational motion is the motion in which a body turns about a fixed axis in such a way that every particle of the body moves in a circular path centered on that axis. Although the linear speeds of different particles may vary, all particles share the same angular displacement, angular velocity, and angular acceleration.

Axis of Rotation

The axis of rotation is an imaginary straight line about which a rigid body rotates. During rotational motion, every particle of the body moves in a circular path whose center lies on this axis.

The axis may pass through the body or outside it, but it always remains fixed in direction and position for pure rotational motion.

The motion of the centre of mass (COM) of a system of particles or a rigid body describes how the system moves as a whole. Regardless of the internal motions of particles within the system, the centre of mass moves as if the entire mass of the system were concentrated at that point and acted upon only by external forces.
Thus, \[M\vec{R}=\sum m_ir_i=m_1r_1 + m_2r_2 + \cdots + m_nr_n +\]

Differentiating both side

Differentiating both side again

Now, from Newton’s second law, the force acting on the first particle is given by \(F_1=ma_1\) and \(F_2=ma_2\) and so on

The net force \(F\) on each particle is the vector sum of all forces acting on it, including both external forces (from outside the system) and internal forces (between particles within the system). Newton's third law ensures internal forces come in equal-and-opposite pairs, so they cancel out \((\vec{F}_{ij}=-\vec{F}_{ji})\) and contribute zero to the total momentum equation. Only external forces affect the system's center of mass motion, therfore

where \(\vec{F}_{ext}\) represents the sum of all external forces acting on the particles of the system. This concludes that

The centre of mass of a system of particles moves as if all the mass of the system was concentrated at the centre of mass and all the external forces were applied at that point.

The linear momentum of a system of particles is defined as the vector sum of the linear momenta of all individual particles present in the system.

If a system consists of many particles moving together, its total momentum represents the motion of the system as a whole rather than the motion of any single particle.
Thus

Differentiating both side

Suppose that the sum of external forces acting on a system of particles is zero.

Thus, when the total external force acting on a system of particles is zero, the total linear momentum of the system is constant. This is the law of conservation of the total linear momentum of a system of particles.

The vector product (or cross product) of two vectors results in a vector that is perpendicular to the plane containing the two vectors.

If two vectors are \( \vec{A} \) and \( \vec{B} \), their vector product is written as:

\[ \vec{A} \times \vec{B} \]

Magnitude of Vector Product

The magnitude of the vector product is given by:

\[ \mid\vec{A} \times \vec{B}\mid = A\,B\, \sin \theta \]

where \( A \) and \( B \) are magnitudes of vectors and \( \theta \) is the angle between them.

Direction of Vector Product

The direction of \( \vec{A} \times \vec{B} \) is determined using the right-hand rule.

Important Properties

Unit Vector Relations

\[ \begin{aligned} \hat{i} \times \hat{j} &= \hat{k} \\ \hat{j} \times \hat{k} &= \hat{i} \\ \hat{k} \times \hat{i} &= \hat{j} \\\\ \hat{i} \times \hat{i} &= 0 \\ \hat{j} \times \hat{j} &= 0 \\ \hat{k} \times \hat{k} &= 0 \\ \end{aligned} \]

Physical Applications

Vector product is used to define important physical quantities such as:

• Torque: \[ \vec{\tau} = \vec{r} \times \vec{F} \]
• Angular Momentum: \[ \vec{L} = \vec{r} \times \vec{p} \]

Vector Product

Find the scalar and vector products of two vectors.
\(\vec{a}=3\hat{i}-4\hat{j}+5\hat{k}\) and
\(\vec{b}=-2\hat{i}+\hat{j}+3\hat{k}\)

Solution

Scalar Product

Vector Product

Angular Velocity

Angular velocity is defined as the rate of change of angular displacement with respect to time. It describes how fast a body rotates about a fixed axis.

\[ \omega = \frac{d\theta}{dt} \]

where \( \theta \) is angular displacement and \( \omega \) is angular velocity. The SI unit of angular velocity is \( \text{rad s}^{-1} \).

Linear Velocity

Linear velocity is the rate of change of linear displacement of a particle moving along a circular path. In rotational motion, different particles have different linear velocities depending on their distance from the axis.

Relation Between Angular Velocity and Linear Velocity

Consider a particle at a distance \( r \) from the axis of rotation. If it rotates through an angle \( \theta \), the arc length covered is:

\[ s = r\theta \]

Differentiating both sides with respect to time:

\[ \begin{aligned} \frac{ds}{dt} &= r \frac{d\theta}{dt} \\ v &= r\omega \end{aligned} \]

Thus, the relation between linear velocity and angular velocity is:

\[ v = r\omega \]

Important Observations

Physical Significance

• Particles farther from the axis have greater linear velocity.
• All particles of a rigid body have the same angular velocity.
• Angular velocity determines the rotational speed of the body.

Definition

Angular acceleration is defined as the rate of change of angular velocity with respect to time. It describes how quickly the angular velocity of a rotating body changes.

\[ \alpha = \frac{d\omega}{dt} \]

where \( \omega \) is angular velocity and \( \alpha \) is angular acceleration. The SI unit of angular acceleration is \( \text{rad s}^{-2} \).

Uniform Angular Acceleration

If the angular velocity of a body changes by equal amounts in equal intervals of time, the body is said to have uniform angular acceleration.

Relation Between Angular and Linear Acceleration

Consider a particle at a distance \( r \) from the axis of rotation. The linear velocity of the particle is given by:

\[ v = r\omega \]

Differentiating with respect to time:

\[ \begin{aligned} \frac{dv}{dt} &= r \frac{d\omega}{dt} \\ a_t &= r\alpha \end{aligned} \]

Thus, the tangential linear acceleration is related to angular acceleration by:

\[ a_t = r\alpha \]

Important Characteristics

Physical Significance

• Angular acceleration determines how fast a rotating body speeds up or slows down.
• Larger torque produces greater angular acceleration.
• All particles of a rigid body have the same angular acceleration.

Definition

The moment of force, commonly called torque, is the turning effect produced by a force about a fixed point or axis of rotation.

\[ \vec{\tau} = \vec{r} \times \vec{F} \]

where \( \vec{r} \) is the position vector from the axis to the point of application of force and \( \vec{F} \) is the applied force.

Magnitude of Torque

The magnitude of torque is given by:

\[ \tau = rF\sin\theta \]

where \( \theta \) is the angle between \( \vec{r} \) and \( \vec{F} \).

Derivation Using Perpendicular Distance

If \( l \) is the perpendicular distance (lever arm) from the axis of rotation, then torque can be written as:

\[ \begin{aligned} l &= r\sin\theta \\ \tau &= Fl \end{aligned} \]

Direction of Torque

The direction of torque is determined using the right-hand rule. Curl the fingers of the right hand in the direction of rotation; the thumb points in the direction of torque.

Important Characteristics

Physical Significance

• Torque is responsible for rotational motion.
• Greater the torque, greater the angular acceleration.
• Torque is the rotational analogue of force.

Applications

• Opening a door using the handle
• Using a spanner to tighten a nut
• Rotation of wheels and gears

Definition

The angular momentum of a particle about a fixed point or axis is defined as the moment of linear momentum of the particle about that point.

\[ \vec{L} = \vec{r} \times \vec{p} \]

where \( \vec{r} \) is the position vector of the particle from the reference point and \( \vec{p} \) is its linear momentum.

Magnitude of Angular Momentum

The magnitude of angular momentum is given by:

\[ L = rp\sin\theta \]

where \( \theta \) is the angle between \( \vec{r} \) and \( \vec{p} \).

Angular Momentum for Circular Motion

For a particle of mass \( m \) moving in a circular path of radius \( r \) with speed \( v \):

\[ \begin{aligned} p &= mv \\ L &= r(mv) \\ L &= mvr \end{aligned} \]

If angular velocity is \( \omega \), then \( v = r\omega \). Substituting:

\[ \begin{aligned} L &= mr(r\omega) \\ L &= mr^2\omega \end{aligned} \]

Direction of Angular Momentum

The direction of angular momentum is determined by the right-hand rule. Curl the fingers of the right hand in the direction of rotation; the thumb gives the direction of angular momentum.

Important Characteristics

Relation Between Torque and Angular Momentum

The rate of change of angular momentum equals the applied torque:

\[ \vec{\tau} = \frac{d\vec{L}}{dt} \]

Physical Significance

• Angular momentum describes the rotational state of a particle.
• It remains conserved if no external torque acts.
• It plays a key role in planetary motion and rotational dynamics.

For a system of particles, rotational effects are described using the concepts of torque and angular momentum. These quantities help us understand how a collection of particles behaves when subjected to forces that cause rotation.

1. Torque for a System of Particles

Torque represents the turning effect of forces acting on a system. For a system of particles, the total torque about a chosen origin is obtained by adding the torques acting on each particle due to external forces.

\[ \vec{\tau} = \sum_{i=1}^{n} \vec{r}_i \times \vec{F}_i \]

Here, \( \vec{r}_i \) is the position vector of the \( i^{\text{th}} \) particle from the origin and \( \vec{F}_i \) is the external force acting on it.

Internal forces always occur in equal and opposite pairs and hence their torques cancel out. Only external forces contribute to the net torque of the system.

2. Angular Momentum of a System of Particles

The angular momentum of a system is defined as the vector sum of the angular momenta of all its constituent particles about the same reference point.

\[ \vec{L} = \sum_{i=1}^{n} \vec{r}_i \times \vec{p}_i \]

where the linear momentum of the \( i^{\text{th}} \) particle is \( \vec{p}_i = m_i \vec{v}_i \).

3. Relation Between Torque and Angular Momentum

To establish the connection between torque and angular momentum, we differentiate the total angular momentum of the system with respect to time:

\[ \frac{d\vec{L}}{dt} = \sum_{i=1}^{n} \frac{d}{dt} \left( \vec{r}_i \times \vec{p}_i \right) \]

Applying the product rule of differentiation:

\[ \begin{aligned} \frac{d\vec{L}}{dt} &= \sum_{i=1}^{n} \left( \frac{d\vec{r}_i}{dt} \times \vec{p}_i + \vec{r}_i \times \frac{d\vec{p}_i}{dt} \right) \end{aligned} \]

Since \( \frac{d\vec{r}_i}{dt} = \vec{v}_i \) and \( \vec{p}_i = m_i \vec{v}_i \), the first term becomes zero:

\[ \vec{v}_i \times \vec{p}_i = \vec{v}_i \times (m_i \vec{v}_i) = 0 \]

Thus,

\[ \frac{d\vec{L}}{dt} = \sum_{i=1}^{n} \vec{r}_i \times \frac{d\vec{p}_i}{dt} \]

Using Newton’s second law \( \frac{d\vec{p}_i}{dt} = \vec{F}_i \), we get:

\[ \frac{d\vec{L}}{dt} = \sum_{i=1}^{n} \vec{r}_i \times \vec{F}_i \]

\[ \boxed{ \vec{\tau}_{\text{external}} = \frac{d\vec{L}}{dt} } \]

4. Important Aspects

• Internal forces do not change the angular momentum of the system.
• Only external torque can alter the angular momentum.
• If net external torque is zero, angular momentum remains conserved.
• These results apply equally to rigid bodies, which are special systems of particles.

5. Conservation of Angular Momentum

When the net external torque acting on a system is zero:

\[ \begin{aligned} \vec{\tau}_{\text{external}} &= 0\\ \Rightarrow \vec{L} &= \text{constant} \end{aligned} \]

This principle explains phenomena such as spinning skaters pulling in their arms, planetary motion, and the working of gyroscopes.

Exam Tip: The relation \( \vec{\tau} = \frac{d\vec{L}}{dt} \) is the rotational analogue of Newton’s second law for a system of particles.

Find the torque of a force \(7\hat{i}+3\hat{j}-5\hat{k}\) about the origin. The force acts on a particle whose position vector is \(\hat{i}-\hat{j}+\hat{k}\).

Solution

Show that the angular momentum about any point of a single particle moving with constant velocity remains constant throughout the motion.

Solution

Proof: Angular Momentum of a Particle Moving with Constant Velocity

We are required to show that the angular momentum about any point of a single particle moving with constant velocity remains constant throughout its motion.

Given

• The particle moves with constant velocity.
• Hence, no external force acts on the particle.

Let:

• Position vector of the particle from a fixed point \( O \) be \( \vec{r} \)
• Linear momentum of the particle be \( \vec{p} = m\vec{v} \)

Expression for Angular Momentum

The angular momentum \( \vec{L} \) of a particle about point \( O \) is defined as:

\[ \vec{L} = \vec{r} \times \vec{p} \]

Time Rate of Change of Angular Momentum

Differentiate angular momentum with respect to time:

\[ \frac{d\vec{L}}{dt} = \frac{d}{dt}(\vec{r} \times \vec{p}) \]

Using the product rule:

\[ \begin{aligned} \frac{d\vec{L}}{dt} &= \frac{d\vec{r}}{dt} \times \vec{p} + \vec{r} \times \frac{d\vec{p}}{dt} \end{aligned} \]

Simplification

Since:

• \( \frac{d\vec{r}}{dt} = \vec{v} \)
• \( \vec{p} = m\vec{v} \)

The first term becomes:

\[ \vec{v} \times \vec{p} = \vec{v} \times (m\vec{v}) = 0 \]

Also, since the velocity is constant, acceleration is zero:

\[ \frac{d\vec{p}}{dt} = m\frac{d\vec{v}}{dt} = 0 \]

Hence, the second term is also zero.

Final Result

\[ \frac{d\vec{L}}{dt} = 0 \]

This implies that the angular momentum remains constant with time.

Conclusion

Thus, the angular momentum about any point of a single particle moving with constant velocity remains constant throughout its motion, as no external force or torque acts on the particle.

A rigid body is said to be in equilibrium when it has no tendency to change either its state of rest or its state of motion under the action of external forces. In equilibrium, the body is mechanically balanced and remains free from both linear and rotational acceleration.

1. Definition of Equilibrium of a Rigid Body

A rigid body is in equilibrium when it satisfies the conditions that ensure no translational motion and no rotational motion. Unlike a particle, a rigid body can rotate as well as translate, so both effects must be controlled.

Equilibrium of a rigid body implies balance of both forces and torques.

2. First Condition of Equilibrium (Translational Equilibrium)

For translational equilibrium, the vector sum of all external forces acting on the rigid body must be zero.

\[ \sum \vec{F} = 0 \]

Applying Newton’s second law to the centre of mass:

\[ \vec{F}_{\text{net}} = M \vec{a}_{\text{CM}} \]

For equilibrium, the acceleration of the centre of mass is zero:

\[ \begin{aligned} \vec{a}_{\text{CM}} &= 0 \\\\ \Rightarrow \sum \vec{F} &= 0 \end{aligned} \]

This condition prevents linear acceleration but does not prevent rotation.

3. Second Condition of Equilibrium (Rotational Equilibrium)

For rotational equilibrium, the algebraic sum of torques acting on the rigid body about any chosen point or axis must be zero.

\[ \sum \vec{\tau} = 0 \]

From rotational dynamics:

\[ \vec{\tau}_{\text{net}} = \frac{d\vec{L}}{dt} \]

For equilibrium:

\[ \begin{aligned} \frac{d\vec{L}}{dt} &= 0 \\\\ \Rightarrow \sum \vec{\tau} &= 0 \end{aligned} \]

This condition ensures that the body does not start rotating or change its rotational motion.

4. Complete Equilibrium of a Rigid Body

A rigid body is said to be in complete equilibrium only when both conditions are satisfied simultaneously:

\[ \begin{aligned} \sum \vec{F} &= 0 \quad \text{and} \quad\\\\ \sum \vec{\tau} &= 0 \end{aligned} \]

• First condition ensures translational equilibrium.
• Second condition ensures rotational equilibrium.

5. Types of Equilibrium

• Static Equilibrium: The rigid body remains at rest.
• Dynamic Equilibrium: The rigid body moves with constant velocity.

6. Role of Line of Action of Forces

Even if the net force on a rigid body is zero, rotation can occur if forces act at different points. The turning effect depends on the line of action of forces.

A pair of equal and opposite forces acting at different points forms a couple, producing rotation without translation.

7. Choice of Axis for Torque Calculation

For rotational equilibrium, torque can be calculated about any convenient point. Choosing a point through which unknown forces pass simplifies calculations.

8. Important Aspects

• Only external forces and torques affect equilibrium.
• Internal forces cancel out and do not disturb equilibrium.
• Torque balance is independent of force balance.
• Rigid body equilibrium extends Newton’s laws from particles to real objects.

9. Real-Life Applications

• Balancing of bridges and buildings
• Stability of ladders and beams
• Seesaws and weighing balances
• Design of mechanical structures

Exam Tip: A rigid body is in equilibrium only when both the net external force and the net external torque acting on it are zero.

The principle of moments explains how a rigid body maintains rotational balance. It connects the idea of equilibrium with the turning effects of forces acting at different points on the body.

1. Definition of the Principle of Moments

The principle of moments states that for a rigid body in equilibrium, the total clockwise moments of all the forces about any point are equal to the total anticlockwise moments about the same point.

In equilibrium, the turning effects in opposite directions exactly balance each other.

2. Moment of a Force

The moment of a force about a point is the measure of the tendency of the force to rotate the body about that point.

\[ \text{Moment of force} = F \times d \]

where \( F \) is the magnitude of the force and \( d \) is the perpendicular distance of the line of action of the force from the chosen point.

3. Mathematical Statement

If a rigid body is in equilibrium about a point \( O \), then:

\[ \sum (\text{Clockwise moments about } O) = \sum (\text{Anticlockwise moments about } O) \]

4. Derivation of the Principle of Moments

Consider a rigid body acted upon by several forces \( F_1, F_2, F_3, \dots \) at different points. Let their perpendicular distances from a fixed point \( O \) be \( d_1, d_2, d_3, \dots \).

Some forces tend to rotate the body clockwise, while others tend to rotate it anticlockwise. For rotational equilibrium, the net torque about point \( O \) must be zero:

\[ \sum \vec{\tau} = 0 \]

Since torque magnitude is \( \tau = Fd \), we may write:

\[ \sum (F d)_{\text{anticlockwise}} - \sum (F d)_{\text{clockwise}} = 0 \]

Rearranging the above expression:

\[ \sum (F d)_{\text{anticlockwise}} = \sum (F d)_{\text{clockwise}} \]

This relation is known as the principle of moments.

5. Proof of the Principle of Moments

The principle of moments is a direct consequence of the second condition of equilibrium. A rigid body in equilibrium must have zero net external torque.

Zero net torque means that the algebraic sum of all turning effects is zero, which is possible only when clockwise and anticlockwise moments are equal in magnitude.

Hence, the principle of moments follows directly from rotational equilibrium.

6. Important Aspects and Observations

• The principle is valid only when the rigid body is in equilibrium.
• Moments can be taken about any convenient point.
• Correct identification of clockwise and anticlockwise moments is essential.
• Internal forces do not affect the moment balance.
• The principle is a scalar form of torque equilibrium.

7. Physical Significance

The principle of moments explains how balance can be achieved even when forces are unequal. A smaller force can balance a larger force if it acts at a greater distance from the pivot.

8. Applications

• Balancing of a seesaw
• Working of beam balances
• Stability of ladders and beams
• Design of levers and mechanical supports

Exam Tip: The principle of moments states that for equilibrium, clockwise moments must equal anticlockwise moments about the same point.

The centre of gravity of a body is a concept used to describe how the force of gravity acts on an extended object. It allows us to treat the gravitational effect on a body as if the entire weight were concentrated at a single point.

1. Definition of Centre of Gravity

The centre of gravity (C.G.) of a body is the point through which the resultant gravitational force (weight) of the body acts, irrespective of the orientation of the body.

If a body is supported at its centre of gravity, it experiences no tendency to rotate due to its own weight.

2. Explanation of the Concept

Every small particle of a body experiences a gravitational force directed vertically downward. The centre of gravity represents the single point where the combined effect of all these forces can be considered to act.

For uniform gravitational fields (near Earth’s surface), the centre of gravity coincides with the centre of mass. However, conceptually, centre of gravity depends on gravity, while centre of mass depends only on mass distribution.

3. Mathematical Expression for Centre of Gravity

Consider a rigid body made up of particles of weights \( W_1, W_2, W_3, \dots \) located at position vectors \( \vec{r}_1, \vec{r}_2, \vec{r}_3, \dots \).

The position vector of the centre of gravity \( \vec{r}_{\text{CG}} \) is given by:

\[ \vec{r}_{\text{CG}} = \frac{\sum W_i \vec{r}_i}{\sum W_i} \]

4. Derivation Using Principle of Moments

Let the body be placed in a uniform gravitational field. Each particle produces a moment about a chosen reference point due to its weight.

For equilibrium, the algebraic sum of moments of all weights about the centre of gravity must be zero.

\[ \begin{aligned} \sum (W_i \times d_i) &= W \times d_{\text{CG}} \\ \sum W_i d_i &= \left( \sum W_i \right) d_{\text{CG}} \end{aligned} \]

Hence, the centre of gravity lies at a position where the total moment of individual weights is equal to the moment of the resultant weight.

5. Proof: Centre of Gravity Lies on the Line of Action of Weight

If the entire weight of a body is assumed to act at a single point and produces the same turning effect as the actual distributed weights, that point must lie on the line of action of the resultant gravitational force.

This proves that the centre of gravity is always located along the direction of gravity.

6. Important Properties of Centre of Gravity

• The centre of gravity may lie inside or outside the physical body.
• Its position depends on the shape and distribution of mass.
• For symmetrical bodies, the centre of gravity lies along the axis of symmetry.
• Changing the orientation of a body does not change the position of its centre of gravity relative to the body.

7. Difference Between Centre of Mass and Centre of Gravity

The centre of mass depends only on mass distribution, whereas the centre of gravity depends on the gravitational field. In uniform gravity, both points coincide, but in non-uniform gravity, they may differ.

8. Physical Significance

The concept of centre of gravity helps in understanding balance and stability. A body remains stable as long as the vertical line passing through its centre of gravity falls within its base of support.

9. Real-Life Applications

• Design of vehicles to prevent toppling
• Balancing of objects and structures
• Stability of human posture and motion
• Construction of tall buildings and towers

Exam Tip: In a uniform gravitational field, the centre of gravity of a body coincides with its centre of mass.

The moment of inertia is a fundamental quantity in rotational motion that measures how strongly a body resists changes in its rotational state. It plays the same role in rotation as mass does in linear motion.

1. Definition of Moment of Inertia

The moment of inertia of a rigid body about a given axis is defined as the sum of the products of the mass of each particle and the square of its perpendicular distance from the axis of rotation.

For a particle at a distance \(r_i\) from the axis, the linear velocity is \(v_i=r_1\omega\). The kinetic energy of motion of this particle is

where \(m_i\) is the mass of the particle. The total kinetic energy K of the body is then given by the sum of the kinetic energies of individual particles,

Moment of Inertia \(I\) is given by

Rotational kinematics deals with the description of rotational motion without considering the forces that cause it. When a rigid body rotates about a fixed axis, every particle of the body moves in a circular path centered on that axis, and the motion can be described using angular quantities.

1. Fixed Axis of Rotation

In rotational motion about a fixed axis, the axis remains stationary in space. All points of the rigid body rotate in planes perpendicular to this axis, maintaining a constant distance from it.

Although particles have different linear speeds, all particles share the same angular motion.

2. Angular Displacement

Angular displacement is defined as the angle through which a rigid body rotates about the fixed axis in a given time interval.

\[ \theta = \frac{\text{arc length}}{\text{radius}} \]

It is measured in radians and is independent of the distance of the particle from the axis.

3. Angular Velocity

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

\[ \omega = \frac{d\theta}{dt} \]

For a rigid body rotating about a fixed axis, angular velocity is the same for all particles.

4. Angular Acceleration

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

\[ \alpha = \frac{d\omega}{dt} \]

It describes how quickly the rotational speed of the body changes.

5. Relation Between Linear and Angular Quantities

For a particle at a distance \( r \) from the axis of rotation:

\[ \begin{aligned} s &= r\theta \\ v &= r\omega \\ a_t &= r\alpha \end{aligned} \]

Thus, linear quantities depend on both angular quantities and distance from the axis.

6. Kinematic Equations for Rotational Motion

For rotational motion with constant angular acceleration, equations analogous to linear kinematics apply:

\[ \begin{aligned} \omega &= \omega_0 + \alpha t \\ \theta &= \omega_0 t + \frac{1}{2}\alpha t^2 \\ \omega^2 &= \omega_0^2 + 2\alpha\theta \end{aligned} \]

Here, \( \omega_0 \) is the initial angular velocity and \( t \) is time.

7. Comparison with Linear Motion

8. Important Aspects and Observations

• All particles of a rigid body have the same angular displacement, velocity, and acceleration.
• Linear velocity and acceleration vary with distance from the axis.
• Angular quantities fully describe the rotational motion of the rigid body.
• These equations are valid only for rotation about a fixed axis.

9. Physical Significance

Rotational kinematics provides the foundation for understanding rotational dynamics. It allows prediction of rotational motion in machines, wheels, gears, and rotating tools before considering the forces responsible for the motion.

Exam Tip: Rotational kinematic equations are directly analogous to linear motion equations, with angular quantities replacing linear quantities.

The angular speed of a motor wheel is increased from 1200 rpm to 3120 rpm in 16 seconds.
(i) What is its angular acceleration, assuming the acceleration to be uniform?
(ii) How many revolutions does the engine make during this time?

Solution

Angular acceleration

Angular Displacement

Number of revolutions

Rotational dynamics deals with the causes of rotational motion. While kinematics describes how a body rotates, dynamics explains why it rotates by relating angular motion to the applied forces and torques. For a rigid body rotating about a fixed axis, the analysis becomes systematic and clear.

1. Fixed Axis of Rotation

In rotational motion about a fixed axis, the axis remains stationary in space. All particles of the rigid body move in circular paths with centers lying on this axis, maintaining constant distances from it.

Although particles have different linear velocities, the entire body shares the same angular motion.

2. Torque and Its Role in Rotation

Torque is the turning effect of a force that tends to rotate a body about an axis. It plays the same role in rotational motion as force does in translational motion.

\[ \vec{\tau} = \vec{r} \times \vec{F} \]

where \( \vec{r} \) is the position vector of the point of application of force from the axis and \( \vec{F} \) is the applied force.

3. Angular Momentum About a Fixed Axis

For a particle of mass \( m \) moving in a circular path of radius \( r \) with angular velocity \( \omega \), the angular momentum about the fixed axis is:

\[ \begin{aligned} v &= r\omega \\ L &= mvr \\ L &= mr^2\omega \end{aligned} \]

For a rigid body made up of many particles:

\[ L = I\omega \]

where \( I \) is the moment of inertia of the body about the axis.

4. Fundamental Equation of Rotational Dynamics

The rate of change of angular momentum equals the net external torque acting on the body:

\[ \vec{\tau}_{\text{net}} = \frac{d\vec{L}}{dt} \]

For rotation about a fixed axis, the moment of inertia \( I \) remains constant. Hence:

\[ \begin{aligned} \vec{\tau}_{\text{net}} &= \frac{d}{dt}(I\vec{\omega}) \\ &= I\frac{d\vec{\omega}}{dt} \\ &= I\vec{\alpha} \end{aligned} \]

\[ \boxed{\vec{\tau}_{\text{net}} = I\vec{\alpha}} \]

This is the rotational analogue of Newton’s second law.

5. Comparison with Translational Dynamics

6. Work Done and Rotational Kinetic Energy

When a torque produces angular displacement, work is done on the rotating body. This work appears as rotational kinetic energy.

\[ K_{\text{rot}} = \frac{1}{2} I \omega^2 \]

This expression is directly analogous to translational kinetic energy \( \frac{1}{2}mv^2 \).

7. Power in Rotational Motion

The rate at which work is done in rotational motion is called rotational power.

\[ P = \tau \omega \]

This relation shows that power depends on both the applied torque and the angular velocity.

8. Important Aspects and Observations

• Only external torques can change the angular momentum of a rigid body.
• Moment of inertia plays the role of mass in rotational dynamics.
• Torque determines how quickly angular velocity changes.
• Rotational dynamics applies strictly when the axis of rotation is fixed.

9. Physical Significance

Dynamics of rotational motion explains the working of wheels, gears, turbines, flywheels, and all rotating machinery. It provides a complete cause-and-effect description of rotation, linking forces to angular motion.

Exam Tip: The equation \( \tau = I\alpha \) is the most important result of rotational dynamics and should be clearly stated and explained in examinations.

When a rigid body rotates about a fixed axis, its rotational motion can be described collectively using the concept of angular momentum. Instead of analyzing individual particles separately, angular momentum allows us to represent the entire rotational state of the body with a single physical quantity.

1. Definition of Angular Momentum About a Fixed Axis

The angular momentum of a rigid body rotating about a fixed axis is defined as the sum of angular momenta of all its constituent particles about that axis.

For rotation about a fixed axis, angular momentum is always directed along the axis of rotation.

2. Angular Momentum of a Single Particle

Consider a particle of mass \( m \) moving in a circular path of radius \( r \) with angular velocity \( \omega \) about a fixed axis.

\[ \begin{aligned} v &= r\omega \\ p &= mv \end{aligned} \]

The angular momentum of the particle about the axis is:

\[ \begin{aligned} L &= r \times p \\ L &= r(mr\omega) \\ L &= mr^2\omega \end{aligned} \]

3. Angular Momentum of a Rigid Body About a Fixed Axis

A rigid body consists of a large number of particles. If the body rotates with angular velocity \( \omega \), every particle has the same angular velocity but different distances from the axis.

The total angular momentum of the rigid body is the sum of angular momenta of all particles:

\[ \begin{aligned} L &= \sum m_i r_i^2 \omega \\\\ L &= \omega \sum m_i r_i^2 \end{aligned} \]

The term \( \sum m_i r_i^2 \) is defined as the moment of inertia \( I \) of the body about the fixed axis.

\[ \boxed{\;L = I\omega\;} \]

4. Proof of the Relation \( L = I\omega \)

Since each particle contributes \( m_i r_i^2 \omega \) to angular momentum, and angular velocity is common for all particles in rigid rotation, it can be taken outside the summation.

This proves that angular momentum of a rigid body about a fixed axis is directly proportional to its angular velocity, with moment of inertia as the proportionality constant.

5. Direction of Angular Momentum

The direction of angular momentum is determined using the right-hand rule. Curl the fingers of the right hand in the direction of rotation; the thumb points in the direction of angular momentum.

For rotation about a fixed axis, angular momentum remains parallel to the axis at all times.

6. Relation Between Torque and Angular Momentum

The net external torque acting on a rigid body equals the time rate of change of its angular momentum:

\[ \vec{\tau} = \frac{d\vec{L}}{dt} \]

For rotation about a fixed axis, moment of inertia \( I \) remains constant. Substituting \( \vec{L} = I\vec{\omega} \):

\[ \begin{aligned} \vec{\tau} &= \frac{d}{dt}(I\vec{\omega}) \\ &= I\frac{d\vec{\omega}}{dt} \\ &= I\vec{\alpha} \end{aligned} \]

This equation represents the rotational analogue of Newton’s second law.

7. Conservation of Angular Momentum

If no external torque acts on the rotating body:

\[ \vec{\tau} = 0 \Rightarrow \frac{d\vec{L}}{dt} = 0 \]

\[ L = I\omega = \text{constant} \]

This means that any change in moment of inertia is accompanied by a corresponding change in angular velocity.

8. Important Aspects and Observations

• Angular momentum depends on both angular velocity and moment of inertia.
• Moment of inertia depends on mass distribution and axis of rotation.
• Only external torque can change angular momentum.
• For a fixed axis, angular momentum always remains along the axis.

9. Physical Significance

Angular momentum about a fixed axis helps explain the motion of wheels, flywheels, spinning tops, turbines, and rotating machinery. It also provides deep insight into why rotating bodies respond differently when their mass distribution changes.

Exam Tip: For rotation about a fixed axis, always start with the relation \( L = I\omega \) and clearly explain the role of moment of inertia.

The law of conservation of angular momentum is one of the most powerful principles in rotational motion. It explains why rotating systems behave in a predictable way even when their shape or speed changes, provided no external turning effect acts on them.

1. Definition of Conservation of Angular Momentum

The angular momentum of a system remains constant if the net external torque acting on the system is zero.

In the absence of external torque, angular momentum is conserved.

2. Mathematical Statement

The relationship between torque and angular momentum is given by:

\[ \vec{\tau}_{\text{external}} = \frac{d\vec{L}}{dt} \]

If the net external torque is zero:

\[ \vec{\tau}_{\text{external}} = 0 \]

then,

\[ \frac{d\vec{L}}{dt} = 0 \]

\[ \vec{L} = \text{constant} \]

3. Proof of Conservation of Angular Momentum

Starting from the fundamental equation of rotational motion:

\[ \vec{\tau} = \frac{d\vec{L}}{dt} \]

For an isolated system, no external torque acts:

\[ \vec{\tau} = 0 \]

Hence,

\[ \begin{aligned} \frac{d\vec{L}}{dt} &= 0 \\ \vec{L} &= \text{constant} \end{aligned} \]

This proves that angular momentum remains unchanged in magnitude and direction when no external torque acts on the system.

4. Conservation in Case of Rotation About a Fixed Axis

For a rigid body rotating about a fixed axis, angular momentum is given by:

\[ L = I\omega \]

If no external torque acts on the system:

\[ I\omega = \text{constant} \]

This implies:

\[ \begin{aligned} I_1 \omega_1 &= I_2 \omega_2 \end{aligned} \]

A decrease in moment of inertia results in an increase in angular velocity, and vice versa.

5. Important Aspects and Observations

• Angular momentum is conserved only when external torque is zero.
• Internal forces cannot change the total angular momentum of a system.
• The law is valid for particles, rigid bodies, and systems of particles.
• Angular momentum conservation holds even if shape and speed change.

6. Physical Significance

The conservation of angular momentum explains why rotating systems resist changes in their rotational state. It allows prediction of motion without knowing detailed forces acting inside the system.

7. Real-Life Examples

• A spinning skater pulls in their arms and rotates faster.
• Divers tuck their bodies to increase rotation speed.
• Planetary motion around the Sun.
• Stability of spinning tops and gyroscopes.

8. Relation with Newton’s Laws

The law of conservation of angular momentum is the rotational counterpart of the conservation of linear momentum. Both arise due to the absence of external influences on the system.

Exam Tip: Always begin angular momentum conservation problems by checking whether the net external torque on the system is zero.