Question

Explain whole rotational motion

Answer 1

Rotational motion involves kinematics, dynamics, key quantities, and rolling motion.

Answer 2

Rotational motion describes the movement of a rigid body where all its constituent particles move in circles centered on a common straight line called the axis of rotation. This axis can be fixed or moving.

Key Concepts in Rotational Motion:

1. Rotational Kinematics: Describes the motion without considering the forces causing it.

   • Angular Displacement ($\theta$): The angle swept by a particle around the axis of rotation. Measured in radians.
   • Angular Velocity ($\omega$): Rate of change of angular displacement. $\omega = \frac{d\theta}{dt}$ Measured in rad/s. Vector quantity, direction given by the right-hand rule along the axis of rotation.
   • Angular Acceleration ($\alpha$): Rate of change of angular velocity. $\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$ Measured in rad/s². Vector quantity.
   • Equations of Rotational Motion (for constant $\alpha$):
     • $\omega = \omega_0 + \alpha t$
     • $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$
     • $\omega^2 = \omega_0^2 + 2\alpha\theta$
   • Relationship between Linear and Angular Variables: For a particle at distance $r$ from the axis:
     • Linear velocity: $v = r\omega$
     • Tangential acceleration: $a_t = r\alpha$
     • Centripetal (radial) acceleration: $a_c = r\omega^2 = \frac{v^2}{r}$

2. Rotational Dynamics: Deals with the causes of rotational motion (forces and torques).

   • Moment of Inertia ($I$): The rotational analogue of mass. It measures a body's resistance to angular acceleration. For a system of particles: $I = \sum m_i r_i^2$ For a continuous body: $I = \int r^2 dm$ Measured in kg·m². Depends on mass distribution and axis of rotation.
     • Parallel Axis Theorem: $I = I_{CM} + Md^2$, where $I_{CM}$ is the moment of inertia about an axis through the center of mass, $M$ is the total mass, and $d$ is the perpendicular distance between the two parallel axes.
     • Perpendicular Axis Theorem (for planar bodies): $I_z = I_x + I_y$, where $I_x, I_y, I_z$ are moments of inertia about mutually perpendicular axes passing through the origin, and the body lies in the xy-plane.
   • Torque ($\tau$): The rotational analogue of force. It causes angular acceleration. $\tau = \vec{r} \times \vec{F}$ (vector cross product) Magnitude: $\tau = rF\sin\phi = F \times (\text{perpendicular distance from axis to force line})$ Measured in N·m. Newton's Second Law for Rotation: $\tau_{net} = I\alpha$
   • Angular Momentum ($\vec{L}$): The rotational analogue of linear momentum. For a particle: $\vec{L} = \vec{r} \times \vec{p} = \vec{r} \times (m\vec{v})$ For a rigid body rotating about a fixed axis: $L = I\omega$ Measured in kg·m²/s or J·s. Conservation of Angular Momentum: If the net external torque on a system is zero, its total angular momentum remains constant ($\vec{L}_{initial} = \vec{L}_{final}$).
   • Rotational Kinetic Energy ($KE_{rot}$): The kinetic energy due to rotation. $KE_{rot} = \frac{1}{2}I\omega^2$ Measured in Joules (J).

3. Rolling Motion: A combination of translational and rotational motion without slipping.

   • Total Kinetic Energy: $KE_{total} = KE_{translational} + KE_{rotational} = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}\omega^2$
   • For pure rolling, $v_{CM} = R\omega$, where $R$ is the radius of the rolling body.

Summary:

Rotational motion describes the motion of a rigid body around a fixed axis. It involves angular displacement, velocity, and acceleration (kinematics). The resistance to rotational motion is quantified by moment of inertia, and the cause of rotational motion is torque. Torque leads to angular acceleration, and angular momentum is conserved in the absence of external torques. Rolling motion is a special case combining translation and rotation.