Uniform Circular Motion MCQ - Practice Questions with Answers

• Introduction - 

Circular motion is one of the examples of motion in two dimensions. To cause a body to have a circular motion, it must be given some initial velocity and there is a force must act on the body which is always directed at right angles to instantaneous velocity. The body moves in circular path have direction of velocity in the tangential direction of the circular motion always.

                                                           

 

• Terms related to circular motion- 

1. Angular  Displacement

• For a circular motion  angular displacement is defined as the angle in radians (degrees, revolutions) through which a point  has been rotated about a point or specified axis.

• Denoted by  (theta)

• S.I unit is Radian

                                                                     

2. Angular velocity

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

$
\omega=\frac{\Theta}{t}_{\text {or }} \omega=\frac{d \Theta}{d t}
$

• S.I. units- Radian per second (rad per sec)
• $\omega$ is a vector quantity
• Direction of $\omega$ is given by 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 thumb will point the direction of the angular velocity.

                                                                     

                                                                           Fig. Shows angular velocity

 

3. Angular  Acceleration

• The rate of change of angular velocity with time is said to be Angular  Acceleration.

$
\alpha=\frac{\Delta \omega}{\Delta t}
$

• Sl units- $\operatorname{rad} .(s e c)^{-2}$

• Angular  Acceleration is a vector quantity.

• 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 of angular velocity.

 

4. Time period-

• Time taken to complete one rotation

• Formula-

                   

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

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

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

• S.I unit seconds (s)

    5.  Frequency-

•    Total number of rotations in one second.

•    Formula-

                   $\nu=\frac{1}{T}$

• S.I. unit = Hertz

• We can write the 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 for changing the direction of its velocity. This force acts towards the center of the circle and is called centripetal force. The acceleration produced by this force is centripetal acceleration.

• Formula-

                 

$
a_c=\frac{V^2}{r}
$

Where $a_{c=\text { Centripetal acceleration, }}$

                      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. 

• Formula-

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

Where $\overrightarrow{a_t}=$ tangential acceleration

$$
\begin{aligned}
& r=\text { radius } \\
& \alpha=\text { angular acceleration }
\end{aligned}
$$
 

       c. Total acceleration- 

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

• Formula- 

                       

$
a=\sqrt{a_t^2+\left(\frac{v^2}{r}\right)^2}
$

Where $a_t=$ tangential acceleration

$
\frac{v^2}{r}={ }_{\text {centripetal acceleration }}
$

     

d. Angle between Total acceleration and centripetal acceleration $\left({ }^\phi\right)$
- Formula-

$
\tan \phi=\frac{a_t}{a_c}=\frac{r^2 \alpha}{V^2}
$

Where $\alpha=$ angular acceleration

$
\begin{aligned}
V & =\text { velocity } \\
r & =\text { radius of circle }
\end{aligned}
$