Explore the fundamental concepts of rotational motion in classical mechanics through interactive simulations. Learn about angular velocity, torque, moment of inertia, angular momentum, and rotational kinetic energy with real-world examples and visualizations.
Rotational motion refers to the movement of an object around a fixed axis. Unlike translational motion where all points on an object move in the same direction, in rotational motion, different points on the object follow circular paths around the axis of rotation.
Rotational motion is governed by its own set of kinematic and dynamic equations, analogous to those of linear motion but with rotational quantities such as angular displacement, angular velocity, and angular acceleration.
The rate of change of angular displacement with respect to time
Angular velocity (ω) is the rate at which an object rotates around a fixed axis. It is measured in radians per second (rad/s).
ω = Δθ/Δt
Where:
For uniform circular motion, the relationship between linear velocity (v) and angular velocity (ω) is:
v = ωr
Where r is the radius of the circular path
The rotational equivalent of force that causes angular acceleration
Torque (τ) is the measure of the force that can cause an object to rotate about an axis. It is the rotational equivalent of linear force.
τ = r × F = rFsin(θ)
Where:
Torque is related to angular acceleration (α) through the moment of inertia (I):
τ = Iα
The resistance of an object to changes in its rotational motion
Moment of inertia (I) is a measure of an object's resistance to changes in its rotational motion about a particular axis. It depends on both the mass of the object and its distribution relative to the axis of rotation.
I = Σmr²
For continuous objects:
I = ∫r² dm
I = (2/5)MR²
I = (2/3)MR²
I = (1/2)MR²
The rotational equivalent of linear momentum
Angular momentum (L) is the rotational equivalent of linear momentum. It is a conserved quantity in isolated systems, similar to linear momentum.
L = Iω
Where:
For a point particle:
L = r × p = rmv sin(θ)
The energy associated with rotational motion
Rotational kinetic energy is the energy associated with the rotation of an object. It is analogous to translational kinetic energy but uses rotational quantities.
KE_rot = (1/2)Iω²
Where:
For objects undergoing both translational and rotational motion:
KE_total = (1/2)mv² + (1/2)Iω²
Explore rotational motion concepts through hands-on experiments
Experience how moment of inertia affects rotational motion with disks of different mass distributions.
Investigate gyroscopic precession and the conservation of angular momentum.
Study the rotational motion of physical pendulums with different shapes and pivot points.
Compare rolling motion on different surfaces and analyze energy transformations.