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Introduction-
Rotational Motion is a fundamental concept in physics that describes the motion of the object as Rotation or Spin around a axis. When an object rotates around a fixed axis, it is undergoing rotational motion. This motion is characterized by the object spinning or revolving around a central point, axis, or line. Examples of rotational motion include a spinning top, the Earth rotating on its axis, or a wheel turning around its axle.
Kinematics of rotational motion-
Angular Displacement (Ꝋ)-
The change in angular position of an object as it undergoes rotational motion. It is generally measured in degree or radian. It is the difference in the initial and final angular positions of the object. Angular Displacement is a vector quantity.
Angular Velocity (ὠ)-
Change of angular displacement per unit time is called angular velocity. SI unit of angular velocity is radian per second (rad s-1). Angular velocity is also a vector quantity
Formula for Angular velocity: $$ \omega = \frac{\theta}{t} $$
Angular Acceleration (𝛼)-
Rate of change of angular velocity with respect to time. SI unit of angular acceleration is (rad s-2).
Angular acceleration formula: $$ \alpha = \frac{\omega_2 - \omega_1}{t} $$
Key Equations in Rotational Motion
$$ \omega_2 = {\omega_1 + \alpha t} $$
$$ \theta = \omega_1 t + \frac{1}{2} \alpha t^2 $$
$$ \omega_2^2 = \omega_1^2 + 2 \alpha \theta^2 $$
Dynamics of rotational motion-
Topics-
- Torque
- Inertia
- Angular momentum
- Newton’s Second Law for Rotation
- Rotational Kinetic Energy
Torque-
Torque is a measure of the rotational force applied to an object. It is the product of the force applied and the distance from the axis of rotation (lever arm). Torque is a vector quantity. It's direction is given by right hand curl rule.
$$ \tau = r \times F $$
In magnitude form:- $$ \tau = r F \sin(\theta) $$
Moment of Inertia-
Inertia is also known as a resistance. The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution of the object relative to the axis of rotation.
$$ I = \sum m_i r_i^2 $$
Newton's second law for rotation-
Just as Newton's second law for linear motion relates force and linear acceleration, Newton's second law for rotational motion relates torque and angular acceleration.
$$ \tau = I \alpha $$
Rotational Kinetic Energy-
When an object rotates it possesses some amount of Kinetic energy
$$ K = \frac{1}{2} I \omega^2 $$
Angular Momentum-
Angular momentum is a measure of the quantity of rotation of an object and is conserved in a closed system with no external torques.
$$ L = I \omega $$
Conservation of Angular Momentum
Introduction-
The conservation of angular momentum is a fundamental principle in physics that states that the angular momentum of a closed system remains constant if no external torques act on it.
Conservation Law
The principle of conservation of angular momentum states that if no external torques act on a system, the total angular momentum of the system remains constant. Mathematically, this is expressed as: L Initial=L Final
or
I initial ὠ initial = I Final ὠ Final
Applications of Rotational Motion
- Rotational Engines and Motors
- Turbines
- Gyroscopes
- Satellites
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