NCERT Class 11 Physics Chapter 7 notes System of Particles and Rotational Motion - Download PDF

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NCERT Class 11 Physics Chapter 7 Notes

Rotational Motion

Rotational motion describes the movement of an object as it spins or revolves around an axis. Consider a spinning top, a rotating wheel, or the Earth turning on its axis; these are all examples of rotational motion.

In this type of motion, each point in the object follows a circular path around a fixed line known as the axis. The motion is a rotation rather than a linear translation.

Centre of Mass

The centre of mass is an imaginary point in a system where all of the system's mass is concentrated. When two bodies are joined, their centre of mass is along the line that connects them.

(Here m₁ & m₂ are two bodies in which m₁ is at a distance x₁, & m₂ at a distance x₂ from the origin O).

• The coordinates of a system's centre of mass (X, Y, Z) are generally given by

• Considering a thin rod of length l, the origin is at the geometric centre of the rod. and the X-axis is along the length of the rod, and for every element dm of the rod at x, there is an element of the same mass dm located at –x.

• The total contribution of each such integral and the integral x dm is zero. Therefore, the COM coincides with its geometric centre.

• The centre of mass coincides with the geometric centre and with the same geometry is applicable to homogenous rings, discs, spheres, or even thick rods of circular or rectangular cross-sections;

Motion of COM

The centre of mass of a system of particles moves in a manner such that the entire mass of the system is considered to be concentrated at the centre, and all external forces are treated as if they act solely on that point.

Differentiating w.r.t. time, we get,

Again, differentiating w.r.t. time, we get,

Ma = Fext

• Instead of considering extended bodies as single particles, we can consider them as systems of particles.

The translational component of their motion could be attained by taking the mass of the whole system and all the external forces on the system to be acting at the centre of mass.

Linear Momentum of System of Particles

The linear momentum of a system of particles is the product of the system's total mass and the velocity of its centre of mass. This relationship is expressed as follows:

Linear Momentum=Total Mass×Velocity of Center of Mass

P = p₁ + p₂ + …. + p_n

= m₁v₁ + m₂v₂ + …. + m_nv_n

P = MV;

Where P = Momentum particles and V = Velocity of COM

Newton’s Second Law in reference to the system of particles is represented by the following expression:

dP/dt = Fext

• In case when the total external force acting on a system of particles is zero (Fext = 0), then, in that case, the total linear momentum of the system is constant and can be represented as:

(dP/dt = 0; P = constant).

• The Centre of mass having velocity also remains constant (Since P = mv = Constant).

• If the total external force on a body is zero, then the internal forces cause complex trajectories of individual particles in spite of the COM moving with a constant velocity.

Angular Velocity & Linear Velocity

Each particle of a rotating body is considered to move in a circle. We define the angular displacement of a given particle about its centre in unit time as angular velocity.

• Average angular velocity =∆θ/∆t
• Instantaneous angular velocity, ω=dθ/dt
• In a pure rotation motion, all the parts of the moving body having the same angular velocity.
• It is a vector quantity
• v = ω x r; where v is the linear velocity of a particle and r is the radius of the circle
• Angular acceleration is outlined as the rate of amendment of angular speed with relevance time.α = dω / dt

Torque & Angular Momentum

The rotational analogue of force is a moment of force which is also known as Torque).

Torque is a vector quantity.

Equilibrium of Rigid Body

• The translational state of the motion of the rigid body is due to the changes in force, i.e. its total linear momentum changes.

• The rotational state of motion of the rigid body is due to the changes in torque i.e. the total angular momentum of the body changes.

Unless stated otherwise, only external forces and torques should be dealt.

A rigid body is said to be in mechanical equilibrium when both its linear momentum and angular momentum are constant with time. This means

Total force zero = Translational Equilibrium

Total torque zero = Rotational Equilibrium

A couple or torque can be defined as a pair of equal and opposite forces with different lines of action is termed as couple or torque.

Without translation, the couple can produce the rotation.

For example, when you open the lid of a jar, you put a couple on it.

Centre of Gravity

The centre of gravity of a body is the point on the body where the total gravitational torque on the body is zero.

In uniform gravity or gravity-free space, the centre of gravity of the body coincides with the centre of mass.

The condition for the centre of gravity and centre of mass will not coincide is that if the value of g will vary from part to part of the body.

Moment of Inertia

Moment of inertia (I) is the analogue of the mass in rotational motion.

A change in rotational motion is resisted by the moment of inertia about a given axis of rotation. It can be regarded as a measure of rotational inertia of the body.

It is a measure of how the different parts of the body are distributed at different distances from the axis.

The moment of inertia of a rigid body depends on the mass of the body, its form and size, and therefore the distribution of mass concerning the axis of rotation.

The radius of gyration of a body concerning association in axis is outlined as the distance from the axis of a mass whose mass is up to the mass of the total body and whose moment of inertia is up to the instant of inertia of the body concerning the axis.

I = M k², where k is the radius of gyration.

The Theorem of Perpendicular Axis

Perpendicular Axis Theorem: The moment of inertia of a two-dimensional body (lamina) regarding an axis perpendicular to its plane is the sum of its moments of inertia regarding two perpendicular axes synchronous with perpendicular axis and lying within the plane of the body.

It is applicable only to planar bodies.

The Theorem of Parallel Axis

Parallel Axis Theorem: The moment of inertia of a body about an axis parallel to an axis passing through the center of mass is equal to the sum of the moment of inertia of body about an axis passing through center of mass and product of mass & square of the distance between two axes.This theorem is applicable to a body of any form.

Kinematics of Rotational Motion for a Fixed Axis

We can derive equations of motion similar to translational motion.

Translational motion

v=v₀+at

x=x₀+v₀t+0.5at²

v²=v₀²+2as

Rotational Motion

ω=ω₀+αt

θ=θ₀+ω₀t+0.5αt²

ω²=ω₀²+2αθ

Dynamics of Rotational Motion for a Fixed Axis

The components of torques are considered along the direction of the fixed axis and this is so because the component of the torque is perpendicular to the axis of rotation and this will tend to turn the axis from its position.

The forces that lie in planes perpendicular to the axis are only considered. Forces parallel to the axis give torques perpendicular to the axis.

Also, the components of the position vectors which are perpendicular to the axis are only considered. Components of position vectors on the axis can lead to torques perpendicular to the axis.

Rolling Motion

Rolling motion may be defined as a combination of rotation and translation motion.

Particles of a rolling body have two velocities.

1. Transnational, which is the velocity of COM.

2. The linear velocity of rotational motion.

In the figure, we observe that every point has two velocities.

One Velocity is found in the direction COM and the other velocity is found to be perpendicular to the line joining centre and the point.

At Point P_o have velocities and both are in opposite direction to each other.

For no-slipping condition: Velocity is found has zero value, so

V_com = ω R

There is an instantaneous axis of rotation that can be seen through P_O and parallel to ‘w’ .

At Point, P₁ Velocities are added up.

Kinetic Energy of Rolling Motion

K.E.rolling = K.E.translation + K.E.rotation

Kinetic energy of a rolling body

where k is radius of gyration of body

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