Pearson | PTE
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Uniform circular motion is considered one of the most asked concept.
26 Questions around this concept.
For a particle in a uniform circular motion, the acceleration $\bar{a}$ at a point $P(R, \Theta)$ on the circle of radius $R$ is (Here $\Theta$ is measured by $x$-axis)
Two cars of masses m1 and m2 are moving in circles of radii r1 and r2 respectively. Their speeds are such that they make complete circles in the same time t. The ratio of their centripetal acceleration is:
Circular motion is an example of
Tangent to the circular path of the body gives -
A particle is moving in a uniform circular motion, the acceleration at a point P(R,$\theta$) on the circle of radius R is (Here $\theta$ is measured from the X-axis):
The angular velocity (in radian/sec) of a particle rotating in a circular orbit 100 times per minute is:
A particle moves in a circle of radius 5 cm with constant speed and time period 0.2s. The acceleration of the particle is:
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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-
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
Angular velocity
$
\omega=\frac{\Theta}{t}_{\text {or }} \omega=\frac{d \Theta}{d t}
$
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}
$
Angular Acceleration is a vector quantity.
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}
$
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