## System of Particles and Rotational Motion

# 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 = mr
^{2}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. I
_{1}w_{1}= I_{1}w_{2} - I
_{1}n_{1}= I_{2}n_{2}→ I_{1}/ T_{1}= I_{2}/ T_{2} - (I
_{1}w_{1}+ I_{2}w_{2}) = (I_{1}+ I_{2}) w. - (I
_{1}n_{1}+ I_{2}n_{2}) = (I_{1}+ I_{2}) 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

Disclaimer: Compete.etutor.co may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. Compete.etutor.co makes no representations whatsoever concerning the content of these sites and the fact that compete.etutor.co has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by compete.etutor.co with respect to such sites, its services, the products displayed, its owners, or its providers.

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}