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A particle starts from the point (0, 8m) and moves with uniform velocity of m/s. After 2 seconds, the angular speed of the particle about the origin will be
A) $\eqalign{
  & \dfrac{3}{{10}}rad/s \cr
  & \cr} $
B) $\eqalign{
  & \dfrac{6}{{25}}rad/s \cr
  & \cr} $
C) $\dfrac{{12}}{{25}}rad/s$
D) $\dfrac{3}{{50}}rad/s$

Last updated date: 19th Jun 2024
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Hint: Apply distance formula for this question, we have to calculate the distance of the particle after traveling 2 seconds. With this value we create a new position of the particle, for this position we have to apply a distance formula to find the distance of this new position to the origin. After this we calculate angular velocity.

Complete Step by step solution:
To calculate the distance of the particle after traveling for 2 seconds. We consider the particle starts from (0, 8) m from rest with a uniform velocity of 3m/s in the x-direction so after 2 seconds the distance a particle will traverse along the x-axis is $3 \times 2 = 6m$.
So we get the new position of the particle is (6m, 8m).

Hence, the distance of this new position to the origin is =$\sqrt {{{\left( {x - 0} \right)}^2} + {{\left( {y - 0} \right)}^2}} $
  & \Rightarrow \sqrt {{{\left( {6 - 0} \right)}^2} + {{\left( {8 - 0} \right)}^2}} \cr
  & \Rightarrow \sqrt {{6^2} + {8^2}} \cr
  & \Rightarrow \sqrt {36 + 64} \cr
  & \Rightarrow \sqrt {100} \cr
  & \therefore 10 \cr} $

Calculate, Angular velocity about origin is given by
$\omega = \dfrac{v}{r}$
V is the velocity of the particle
r is the new distance of particle
$\omega $ is the angular velocity.
Substitute given values we get

$\therefore \omega = \dfrac{3}{{10}}$ rad/sec.

Additional information:
We know that Speed is measured as the ratio between the distance and time and its SI unit of speed is m/s, speed is considered as a scalar quantity.
The student knows the distance formula can be derived from the Pythagorean Theorem so, practice this derivation. We usually used this in geometry.
Distance is equal to the square root of the square of the sum of the quantity as shown in the formula.

In the question clearly mentioned to take the formula of distance from the point of origin. Angular velocity is defined as the rate of change of angular position of a rotating body. So, it is measured in the unit of rad/sec.