Concepts of Rotational Motion

Rotational motion refers to the movement of an object around a fixed axis. It is a complex concept that requires an understanding of several related concepts. Some of the important concepts related to rotational motion include angular displacement, angular velocity, angular acceleration, torque, the moment of inertia, centripetal force, kinetic energy, angular momentum, and the conservation of angular momentum.

A system of particles is a collection of several individual particles that are interacting with each other. In physics, the behaviour of such a system is studied to understand the interactions between the individual particles and the resulting motion of the entire system.

What is Rotational Motion?

Rotational motion is the motion of an object around a fixed axis of rotation. It is a type of motion that occurs when an object is rotating about a point, rather than moving in a straight line.

In rotational motion, an object is characterized by its angular position, angular velocity, and angular acceleration. The angular position of an object is its orientation with respect to a reference point and is measured in radians. The angular velocity of an object is its rate of change of angular position and is measured in radians per second. The angular acceleration of an object is its rate of change of angular velocity and is measured in radians per second squared.

Important Concepts Related to Rotational Motion

An important concept in rotational motion is torque, which is a force that causes rotation. Torque is equal to the product of the force applied and the perpendicular distance from the axis of rotation to the line of action of the force. The net torque acting on an object is proportional to its angular acceleration.

Another important concept in rotational motion is the moment of inertia, which is a measure of an object’s resistance to rotational motion. The moment of inertia depends on the distribution of mass within the object and the distance of each particle from the axis of rotation. The net torque acting on an object is proportional to its angular acceleration, and the angular acceleration is inversely proportional to the moment of inertia.

In rotational motion, the kinetic energy of rotation is equal to one-half the product of the moment of inertia and angular velocity squared. This energy can be used to describe the rotational motion of an object, as an object with a large amount of kinetic energy of rotation will have a high angular velocity and a large moment of inertia.

Rotational motion is important in many areas of physics, including mechanics, astronomy, and engineering. It is used to describe the motion of objects such as gears, wheels, planets, and satellites, and plays a crucial role in the study of rotating systems and the dynamics of rotating objects.
Rotational motion refers to the motion of an object as it rotates around a fixed axis. The study of rotational motion includes the analysis of rotational kinematics, rotational dynamics, and rotational energy.

Angular Displacement

Angular displacement is the change in an object’s orientation with regard to its initial position, measured in radians. It is a vector quantity that describes the size and direction of the orientation change.

Angular Velocity

Angular velocity is the rate at which an object rotates, expressed in radians per second. It is a vector quantity describing the amplitude and direction of rotational motion.

The image below represents the tangential velocity and angular velocity(ω).

Angular Acceleration

Angular acceleration is the rate at which angular velocity changes, expressed in radians per second squared. It is a scalar quantity that describes the rate at which rotational velocity changes.

Torque is a force that induces rotation and is equal to the product of the applied force and the perpendicular distance from the axis of rotation to the force’s line of action. Torque can be used to explain an object’s rotational motion since the net torque exerted on it is proportional to its angular acceleration.

Moment of Inertia

An object’s moment of inertia is a measure of its resistance to rotational motion that depends on its mass distribution and the distance of each particle from the axis of rotation. The moment of inertia can be used to characterize an object’s rotational motion since the net torque acting on an item is proportional to its angular acceleration, which is inversely proportional to the moment of inertia.

Kinetic Energy of Rotation

The kinetic energy of rotation is the energy that an object has as a result of its rotational motion, which is equal to one-half the product of its moment of inertia and angular velocity squared. This energy can be used to describe an object’s rotational motion, as an object with a high angular velocity and a big moment of inertia will have a high kinetic energy of rotation.

Examples,

• Earth rotates around its own axis.
• A spinning top.
• A wheel rotating on an axle.
• A figure skater spinning on the ice.

Formulas used in Rotational Motion

Several formulas are used in the study of systems of particles and rotational motion. Some of the most important formulas are,

Center of Mass

The formula for the centre of mass of a system of particles is given by,

R_cm = (m₁r₁ + m₂r₂ + … + m_nr_n) / (m₁ + m₂ + … + m_n)

where,
R_cm is the position vector of the centre of mass, 
m₁, m₂, …, m_n are the masses of the particles,
r₁, r₂, …, r_n are the position vectors of the particles.

Linear Momentum

The formula for the linear momentum of a system of particles is given,

P = m₁v₁ + m₂v₂ + … + m_nv_n 

where, 
P is the linear momentum of the system, 
m₁, m₂, …, m_n are the masses of the particles,
v₁, v₂, …, v_n are the velocity vectors of the particles.

Linear Kinetic Energy

The formula for the linear kinetic energy of a system of particles is given,

T = (1/2)m₁v₁² + (1/2)m₂v₂² + … + (1/2)m_nv_n²

where
T is the linear kinetic energy of the system, 
m₁, m₂, …, m_n are the masses of the particles,
v₁, v₂, …, v_n are the velocity vectors of the particles.

Torque

The formula for the torque on a particle is given,

τ = r x F

where 
τ is the torque, 
r is the position vector of the particle with respect to the axis of rotation,
F is the force applied to the particle.

Angular Momentum

The formula for the angular momentum of a system of particles is given,

L = Iω

where,
L is the angular momentum of the system, 
I is the moment of inertia of the system,
ω is the angular velocity of the system.

Angular Kinetic Energy

The formula for the angular kinetic energy of a system of particles is given,

K = (1/2)Iω² 

where 
K is the angular kinetic energy of the system, 
I is the moment of inertia of the system,
ω is the angular velocity of the system.

These formulas provide a foundation for the study of systems of particles and rotational motion and are used to analyze the motion of rotating objects and to predict the behaviour of rotating systems.

Read More,

• Rotation
• Rigid Body
• Dynamics of Rotational Motion

Solved Examples

Example 1: A system of three particles with masses m₁ = 3 kg, m₂ = 4 kg, and m₃ = 5 kg are located at positions r₁ = (2, 3, 0), r₂ = (4, 0, 0), and r₃ = (0, 4, 0) respectively. Find the position of the centre of mass.

Solution:

Position of the centre of mass can be found using the formula,

R_cm = (m₁r₁ + m₂r₂ + m₃r₃) / (m₁ + m₂ + m₃)

R_cm = (3(2, 3, 0) + 4(4, 0, 0) + 5(0, 4, 0)) / (3 + 4 + 5)

R_cm = (6, 9, 0) / 12

R_cm = (1/2, 3/4, 0)

So the centre of mass of the system is located at (1/2, 3/4, 0)

Example 2: A system of two particles with masses m₁ = 2 kg and m₂ = 3 kg are moving with velocities v₁ = (4, 5, 0)m /s and v₂ = (3, 2, 0) m/s respectively. Find the linear momentum of the system.

Solution:

Linear momentum of the system can be found using the formula,

P = m₁v₁ + m₂v₂

P = 2(4, 5, 0) + 3(3, 2, 0)

P = (8, 10, 0) + (9, 6, 0)

P = (17, 16, 0) kgms^-1

So, linear momentum of the system is (17, 16, 0) kgms^-1

Example 3: A force of 10 N is applied to a particle located at position r = (3, 4, 0) with respect to an axis of rotation. Find the torque on the particle.

Solution:

Torque on the particle can be found using the formula,

τ = r x F

τ = (3, 4, 0) x (0, 0, 10)

τ = (0, 0, -30) N-m

So the torque on the particle is (0, 0, -30) N-m

Example 4: A system of two particles with masses m₁ = 2 kg and m₂ = 3 kg is rotating with angular velocities ω₁ = 2 rad/s and ω₂ = 3 rad/s respectively. The moments of inertia of the particles are I₁ = 5 kg m² and I₂ = 7 kg m². Find the angular momentum of the system.

Solution:

Angular Momentum of the system can be found using the formula:

L = I₁ω₁ + I₂ω₂

L = 5(2) + 7(3)

L = 14

So the Angular Momentum of the system is 14 kg m²/s

Example 5: A system of two particles with masses m₁ = 2 kg and m₂ = 3 kg is rotating with angular velocities ω₁ = 2 rad/s and ω₂ = 3 rad/s respectively. The moments of inertia of the particles are I₁ = 5 kg m² and I₂ = 7 kg m². Find the angular kinetic energy of the system.

Solution:

Angular Kinetic Energy of the system can be found using the formula,

K = (1/2)I₁ω₁² + (1/2)I₂ω₂²

K = (1/2)5(2)² + (1/2)7(3)²

K = (1/2)20 + (1/2)63

K = 43/2

So the Angular Kinetic Energy of the system is 43/2 J

FAQs on Rotational Motion

Q1: What is Rotational Motion?

Answer:

Rotational motion is a type of motion in which an object rotates around a fixed axis. The rotation can be described in terms of angular velocity, angular displacement, and angular acceleration.

Q2: What is the Moment of Inertia?

Answer:

The moment of inertia is a measure of an object’s resistance to rotational motion. It is calculated as the sum of the products of the mass of each particle in the object and the square of its distance from the axis of rotation.

Q3: What is Torque?

Answer:

Torque is a measure of the twisting force that causes rotational motion. It is calculated as the cross product of the position vector of a particle and the force acting on it.

Q4: What is Angular Momentum?

Answer:

Angular momentum is a measure of an object’s rotational motion. It is calculated as the product of the moment of inertia and the angular velocity.

Q5: What is the Centre of Mass?

Answer:

The center of mass is the point at which the total mass of an object can be considered to be concentrated. The position of the center of mass can be found by taking the weighted average of the positions of all the particles in the object.

Q6: What is the Difference between Linear and Angular Momentum?

Answer:

Linear momentum is a measure of an object’s motion in a straight line, while angular momentum is a measure of an object’s rotational motion. The difference lies in the direction and nature of the motion being considered.

Q7: How is Angular Velocity related to Linear Velocity?

Answer:

Angular velocity is related to linear velocity through the radius of rotation. The linear velocity of a particle can be found by multiplying the angular velocity by the radius of the particle’s path.