Week 4: Rotational Motion

Static Equilibrium

A condition where the sum of all forces and the sum of all torques acting on a structure are both zero.

Torque

A measure of the rotational force on an object, calculated about a specific axis.

Right Hand Rule

A method used to determine the direction of forces based on the orientation of the fingers of the right hand for the cross product of two vectors.

Vector Cross Product

A mathematical operation used to calculate torque, defined as $aXb=ab\sin\theta$ .The result is a vector that is perpendicular to the plane formed by the two vectors being multiplied.

Maximum Torque

Occurs when the force is applied perpendicular to the lever arm.

Torque in 3D

Torque is a vector that is perpendicular to the plane defined by the cross product of the position vector and force vector.

Positive Torque

Torque directed along the positive z-axis.

Negative Torque

Torque directed along the negative z-axis.

Parallel Axis Theorem

Allows the calculation of the moment of inertia of an object about any axis parallel to one through its center of mass.

Perpendicular Axis Theorem

States that for a plane object, the moment of inertia about an axis perpendicular to its plane is the sum of the moments of inertia about two axes in its plane.

Angular Velocity (ω)

The rate of change of angular displacement of an object, typically measured in radians per second.

Continuous Mass Distribution

When mass is spread out over a volume rather than concentrated at discrete points.

Compound Object

An object made up of two or more different bodies, each with its own moment of inertia.

Distribution of Mass

The arrangement of mass within an object, affecting its moment of inertia with respect to a chosen axis.

Center of Mass (CoM)

The point in a body or system of bodies where the mass can be considered to be concentrated for the purpose of analyzing translational motion.

Conservation of Angular Momentum

If external forces produce no net torque on a system, the angular momentum remains constant.

Angular Momentum

A quantity defined as the product of an object's moment of inertia and its angular velocity.

Gyroscope

A device consisting of a spinning disk which is free to assume any orientation, used to demonstrate angular momentum conservation.

Precession

The phenomenon where the axis of a spinning object moves in response to an external torque, changing the direction of its angular momentum. This results in a gradual shift of the rotation axis, typically observed in gyroscopes or spinning tops.

Torque (τ)

A measure of the force that produces or tends to produce rotation or torsion.

External Torque

A torque that results from forces applied from outside the system.

Invariant Magnitude of Angular Momentum

During precession, the magnitude of angular momentum remains constant while its direction changes.

Rate of Precession (Ω)

The rate at which the axis of a spinning body precesses around the vertical axis.

Angular Momentum

A measure of the rotational motion of a particle or rigid body, defined as the cross product of the position vector and momentum vector.

Rigid Body

An object with a fixed shape that does not deform under the application of forces, maintaining the distance between any two points.

Angular Momentum of a Rigid Body

For a rigid body rotating about an axis, angular momentum is calculated as $L=I{ω}$ , where $I$ is the moment of inertia and ${ω}$ is the angular velocity.

Newton's Second Law for Rotation

The change in angular momentum of a system is equal to the net external torque applied to it, expressed as $\frac{dL}{dt} = \tau$.

Rotational Kinetic Energy

The kinetic energy of an object due to its rotation, expressed as $K_{r}={}\frac12I\omega^2$ .It is the energy possessed by a body due to its rotational motion about an axis.

Total Kinetic Energy

The sum of rotational and translational kinetic energies, expressed as $K = K_t + K_r$.

Angular Velocity ($\theta$)

The rate of change of the angle with respect to time, typically denoted by \frac{d\theta}{\differentialD t} .

Linear Velocity or Tangential velocity ($v$)

The speed of a point in a rotating object, given by the product of the radius and angular speed: $v_{i}=r_{i}\omega$ .It represents how fast the object is moving along a circular path, where $r$ is the radius and $\omega$ is the angular velocity.

Angular Displacement

The change in the angle (in radians) during rotation, represented as ∆𝜃 = 𝜃(B) − 𝜃(A).

Kinematic Equations for Constant Angular Acceleration

Equations that relate angular displacement, angular acceleration, and angular velocities in rotational motion.

Relationship Between Angular and Translational Quantities

Particles in a rotating rigid body share the same angular displacement, velocity, and acceleration, but translational quantities depend on their radial distance from the axis.

Angular Acceleration

The rate of change of angular velocity of a rigid body due to the sum of external torques.

Work-Energy Theorem

A principle that relates the work done on an object to its change in kinetic energy, used to analyze rotation.

Moment of Inertia (I)

A quantity expressing a body's tendency to resist angular acceleration, dependent on mass distribution.

Power (P) for Rotational Motion

The rate at which work is done in rotational motion, given by P=\tau\frac{d\theta}{\differentialD t}=\tau\omega.

Translational Motion vs. Rotational Motion

Translational motion refers to movement along a path (linear), while rotational motion refers to movement around an axis.

Centre of Mass for multiple particles

Given particles of mass $m_1$ and $m_2$, the CoM can be calculated using the formula $x_C = \frac{m_1x_1 + m_2x_2}{m_1 + m_2}$.

Centre of Mass for continuous objects

For an extended body like a wire, the CoM is calculated as $x_c = \frac{\sum x_i \Delta m}{\sum \Delta m}$ where $\Delta m$ is an infinitesimally small mass.

Locating Centre of Mass

The process of finding the CoM involves locating the CoM of each object and treating them as point masses.

Newton’s 2nd Law for a System of Particles

The motion of the centre of mass depends only on the vector sum of all external forces acting on the system.