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

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.