Notes from Toppers
Vector Products, Angular Velocity, and Angular Acceleration
1. Vector Products:
Concepts
 (a) Cross Product: Scalar Triple Product (Ref. NCERT, Class 12, Chapter – Vectors, Exemplar Problem 20 )
 (b) Vector Product of two vectors A and B: (\overrightarrow A \times \overrightarrow B)
(Ref: NCERT, Class 12, Ch4 Vectors, Page No. 94)
 (c) Cross Product in determinant form:
$$\overrightarrow A \times \overrightarrow B = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_1 & A_2 & A_3 \\ B_1 & B_2 & B_3 \end{vmatrix}$$
(Ref: NCERT, Class 12, Ch4 Vectors, Page No. 96)

(d) Properties:

$$\overrightarrow A \times \overrightarrow B=AB \text{ sin }\theta$$

(\overrightarrow A \times \overrightarrow B= \overrightarrow B \times \overrightarrow A)

(\overrightarrow A \times (\overrightarrow B + \overrightarrow C) = \overrightarrow A \times \overrightarrow B + \overrightarrow A\times\overrightarrow C)

(e) Applications:

Work Done by a Torque:
$$\overrightarrow \tau = \overrightarrow r\times \overrightarrow F$$
 Torque produced by a Force:
$$\overrightarrow N=q(\overrightarrow v\times \overrightarrow B)$$
 Angular Momentum:
$$\overrightarrow L=m\overrightarrow r \times \overrightarrow v$$
2. Angular Velocity:
Concepts

(a) Concept: Angular velocity is a vector quantity that specifies the rate of change of angular displacement of an object rotating about an axis.

(b) Units: Radian per second (rad/s).

(c) Relation with linear velocity: $$v=r\omega$$

(d) Angular velocity in Rotational Motion: (\omega = \frac{d\theta}{dt})
(Ref: NCERT, Class 11, Ch7, System of Particles and Rotational Motion, Page No. 161)
3. Angular Acceleration:
Concepts

(a) Concept: It is a vector quantity that describes the rate of change of angular velocity.

(b) Units: Radian per second square (rad/s2).

(c) Calculation:
$$\alpha = \frac{d\omega}{dt}=\frac{d^2\theta}{dt^2}$$ (Ref: NCERT, Class 11, Ch7, System of Particles and Rotational Motion, Page No. 161)
4. Interconnection:

Angular acceleration and angular velocity are interrelated through the equation: $$\alpha = \frac{d\omega}{dt}$$

Torque acting on a rigid body leads to angular acceleration:
$$\overrightarrow \tau=I\overrightarrow \alpha$$
 Moment of Inertia (I): Resistance of an object to angular acceleration.
5. Applications:
 Solving Rotational Dynamics Problems
 Rigid Body Dynamics
 Gyroscope and Motion