 # Angular Velocity, Torque and Angular Momentum

• The rate of angular displacement with time is called angular velocity.
• Angular velocity ω = dθ/ dt
• Unit of angular velocity is rad/sec
• Angular velocity is a axial vector.
• Relation between angular velocity and linear velocity is V = rw. \tt \left\{\overline{v}=\overline{w}\times\overline{r}\right\}.
• Angular velocity of seconds hand of water is = 2π/60
• Angular velocity of minutes hand of water is = 2π/3600
• Angular velocity of Hours hand of water is = 2π/43200
• Direction of angular velocity is perpendicular to the plane of rotation.
• Torque is the turning effect of force.
• Magnitude of torque is the product of force and perpendicular distance between the line of action of force and perpendicular to the plane of rotation.
• Torque \tt \tau=\overline{r}\times\overline{F}=r\ F\sin\theta
• SI unit of torque is Nm.
• Magnitude of torque is maximum when \tt \overline{r}\ and\ \overline{F} are perpendicular to each other.
• Two equal and opposite non collinear forces simultaneously acting on a body constitute couple.
• Couple always produces turning effect.
• When a rigid body rotates with uniform angular velocity all particles under go same angular displacement but different lineal displacements.
• All particles rotate with same angular velocity but with different linear velocities.
• Particles on the axis of rotation remain at rest.
• The moment of linear momentum is called angular momentum.
• Angular momentum L = nvr
• Angular momentum \tt L=\overline{r}\times \overline{p}
• Angular momentum is a axial vector.
• Angular momentum & direction is perpendicular to the plane of rotation.
• For a particle in circular motion L = mr2 ω.
• For a particle in circular torque T’ = Iα.
• Torque T’ = I (dw/dt) (I = moment of inertia). Torque T’ = (dL/dt)
• Law of conservation of angular momentum states there when no external torque acts the angular momentum of the system remains constant.
• When no external torque is acting on a system L = Iω = constant. I1 w1 = I1 w2
• I1 n1 = I2 n2 → I1 / T1 = I2 / T2
• (I1 w1 + I2 w2) = (I1 + I2) w.
• (I1 n1 + I2 n2) = (I1 + I2) n.
• Rotational KE = \tt \frac{1}{2}\ \frac{I_{1}I_{2}}{I_{1}+I_{2}}\left(\omega_{1}-\omega_{2}\right)^{2}

### View the Topic in this video From 09:33 To 52:26

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1. Angular velocity : \omega = \frac{d\theta}{dt}

2. If the two particles are moving in the same direction then \tt T_{relative} = \frac{2 \pi}{\omega_{B} - \omega_{A}} = \frac{T_{A}T_{B}}{T_{A} - T_{B}}

3. Relationship between linear velocity and angular velocity \vec{v} = \vec{\omega} \times \vec{r}

4. Torque or moment of a force about the axis of rotation τ = r × F = rF sin θ \hat{n}

5. In rotational motion, torque, τ = I α

6. Angular acceleration : \vec{\alpha} = \frac{d \ \vec{\omega}}{dt}