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, Ch-4 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, Ch-4 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, Ch-7, 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, Ch-7, 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