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Uniform circular motion - (Concept)

  • Introduction - 

Circular motion is one of the examples of motion in two dimensions. In the case of circular motion, the particle moves in a circular path on the circumference of a circle. The velocity of a particle moving on a circular path is along the tangent at that point.


                                                          

 

  • Terms related to circular motion- 

  • Radius vector

  • Vector joining the centre of the circular path to the position on the circular path is called radius vector

Angular position

  • Angle made by the radius vector with reference line (arbitrarily chosen diameter) is called angular position.
  • The direction of angular position can be clockwise or anticlockwise depending upon the choice of frame of reference.
  • The angular position of the particle at position "P" is denoted by angle \theta in the diagram above.

Angular displacement

  • The change in angular position is called angular displacement.
  • It is the angle through which the radius vector rotates during the given circular motion.
  • The angular displacement between positions 'P' and 'Q' is denoted by \Delta \theta in the diagram above.
  • S.I unit of angular position and angular displacement is Radian.                                                        

  1. Angular velocity

  • -Denoted by $\omega$ (omega)
    - $\omega$-Rate of change of angular displacement.
    - Average angular velocity-

    $
    \omega_{a v g}=\frac{\Delta \theta}{\Delta t}
    $

    - Instantaneous angular velocity-

    $
    \omega=\frac{d \theta}{d t}
    $

    - S.I. units- Radian per second (rad per sec )
    - $\omega$ is a vector quantity
    - The direction of $\omega$ is given by the Right-hand rule.
    - According to the right-hand rule, if you hold the axis with your right hand and rotate the fingers in the direction of motion of the rotating body then the thumb will point the direction of the angular velocity.
    - Relation between angular velocity and linear velocity-
    - $\vec{v}=\vec{\omega} \times \vec{r}$
    3. Angular Acceleration
    - The rate of change of angular velocity with time is said to be Angular Acceleration.
    - $\alpha=\frac{\Delta \omega}{\Delta t}$
    - SI units- $\operatorname{rad.}(\mathrm{sec})^{-2}$

  • Angular  Acceleration is a vector quantity.

  • The direction of Angular  Acceleration   

       a) If angular velocity is increasing then the direction of Angular  Acceleration

           is in the direction of angular velocity.

       b) If angular velocity is decreasing then the direction of Angular  Acceleration

           is in the direction which is opposite to the direction angular velocity.

4. Time period-

  • Time is taken to complete one rotation

  • Formula-

                 

$
T=\frac{2 \pi}{\omega}
$


Where $\omega=$ angular velocity
If $\mathrm{N}=$ no. of revolutions and $\mathrm{t}=$ total time then

$
T=\frac{t}{N}_{\text {or }} \quad\left(\omega=\frac{2 \pi N}{t}\right)
$

- S.I unit seconds (s)
5. Frequency-
- The total number of rotations in one second.
- Formula-

$
\nu=\frac{1}{T}
$

- S.I. unit = Hertz
- We can write relation between angular frequency and frequency as

$
w=2 \pi \nu
$

6. Centripetal acceleration and Tangential acceleration -
a. Centripetal acceleration-
- When a body is moving in a uniform circular motion, a force is responsible to change the direction of its velocity.This force acts towards the centre of the circle and is called centripetal force. Acceleration produced by this force is centripetal acceleration.
- Formula-

$
a_c=\frac{V^2}{r}
$
 

                      Where a_c=Centripetal acceleration,

                      V= linear velocity

                      r = radius

                                                   

                                                                      Figure Shows Centripetal acceleration 

      b. Tangential acceleration -

  •   During circular motion, if the speed is not constant, then along with centripetal acceleration there is also a tangential         

            acceleration, Which is equal to the rate of change of magnitude of linear velocity.

          

$
a_t=\frac{\mathrm{d} v}{\mathrm{~d} t}
$

- Relation between angular acceleration and tangential acceleration-

$
\overrightarrow{a_t}=\vec{\alpha} \times \vec{r}
$


Where $\overrightarrow{a_t}=$ tangential acceleration
$\vec{r}=$ radius vector
$\alpha=$ angular acceleration

       c. Total acceleration- 

  • The vector sum of Centripetal acceleration and tangential acceleration is called Total acceleration.

  • Formula- 

                 

$
a_n=\sqrt{a_c^2+a_t^2}
$

d. Angle between Net acceleration and tangential acceleration ( $\theta$ )
- From the above diagram-

$
\tan \theta=\frac{a_c}{a_t}
$
 

Exam Chapter
JEE MAIN Kinematics
Physics Part I Textbook for Class XI
Page No. : 79
Line : 77

Concepts of Physics Part-1
Page No. : 102
Line : 29

Understanding Physics (Volume-1)
Page No. : 432
Line : 9

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