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 = 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}