Physics Centre of Mass and Rotational Motion

Study Notes

Centre of Mass

The centre of mass (COM) of a body is a point representing the average position of all the mass in the body.
It's a theoretical point where the total mass of the body can be considered concentrated.
The location of the COM depends on the mass distribution within the body.
For a system of discrete point masses, the x-coordinate of the COM is calculated by: x_COM = (Σ mᵢxᵢ) / Σ mᵢ where:
- mᵢ is the mass of the i-th particle.
- xᵢ is the x-coordinate of the i-th particle.
For a continuous object, the COM's position is calculated by integrating over the object's mass distribution.
The COM is crucial for determining an object's motion under external forces and torques.

Rotational Motion

Rotational motion describes a body's motion about an axis of rotation.
Key concepts include:
- Angular displacement (θ): The angle a body rotates through, measured in radians (rad).
- Angular velocity (ω): The rate of change of angular displacement, measured in radians per second (rad/s).
- Angular acceleration (α): The rate of change of angular velocity, measured in radians per second squared (rad/s²).
Relationships between linear and rotational quantities:
- Linear velocity (v) = ωr
- Linear acceleration (a) = αr
- Linear displacement (s) = θr
- Where r is the distance to the axis of rotation.
Newton's second law for rotational motion: τ = Iα, where:
- τ is the torque.
- I is the moment of inertia. Moment of inertia depends on the mass distribution and the axis of rotation. Different shapes have unique moment of inertia formulas.
Types of rotation:
- Constant angular acceleration
- Uniform circular motion
- Angular momentum (L) = Iω

Work, Power, and Energy

Work: Work is done when a force causes a displacement.
W = Fd cos θ where:
- W is work.
- F is force.
- d is displacement.
- θ is the angle between force and displacement.
Power: Power is the rate of doing work.
P = W/t or P = Fv cos θ where:
- P is power.
- W is work.
- t is time.
- F is force.
- v is velocity.
Energy: Energy is the ability to do work.
- Forms of energy include kinetic, potential, thermal, chemical, and nuclear.
Kinetic Energy: Energy of motion.
KE = (1/2)mv² where:
- m is mass.
- v is velocity.
Potential Energy: Energy due to position (gravitational or elastic).
Gravitational potential energy (PE) = mgh where:
- m is mass.
- g is acceleration due to gravity.
- h is height.
Work-Energy Theorem: Work done on an object equals the change in its kinetic energy. W = ΔKE.
Conservation of Energy: Energy cannot be created or destroyed, only transformed. The total energy of an isolated system remains constant.

Description

This quiz covers essential concepts related to the centre of mass (COM) and rotational motion in physics. Learn about how the COM represents mass distribution in a body and delve into the principles of rotational motion, including angular displacement and velocity. Test your understanding through engaging questions.