Kinematics of Circular Motion questions | Kinematics of Circular Motion jee questions

Q 1 :

A man carrying a monkey on his shoulder does cycling smoothly on a circular track of radius 9 m and completes 120 revolutions in 3 minutes. The magnitude of centripetal acceleration of monkey is (in $m / s^{2}$ ) [2024]

$4 π^{2} m s^{- 2}$
$57600 π^{2} m s^{- 2}$
$16 π^{2} m s^{- 2}$
Zero

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(3)

Given : R = 9m and 120 revolutions in 3 min

Angular velocity, $ω = \frac{120 R e v .}{3 m i n .} = \frac{120 \times 2 π r a d}{3 \times 60 s e c} = \frac{4 π}{3} r a d / s$

$a_{c e n t r i p e t a l} = ω^{2} R = {(\frac{4 π}{3})}^{2} \times 9 = 16 π^{2} m / s^{2}$

Q 2 :

A clock has 75 cm, 60 cm long second hand and minute hand respectively. In 30 minutes, duration the tip of second hand will travel $x$ distance more than the tip of minute hand. The value of $x$ in meter is nearly (Take $π$ = 3.14) [2024]

118.9
220.0
139.4
140.5

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(3)

$x_{m i n} = π \times r_{m i n} = π \times \frac{60}{100} m .$

$x_{s e c o n d} = 30 \times 2 π \times r_{s e c o n d} = 30 \times 2 π \times \frac{75}{100}$

$x = x_{s e c o n d} - x_{m i n} = 139.4 m$

Q 3 :

A particle is moving in a circle of radius 50 cm in such a way that at any instant the normal and tangential components of its acceleration are equal. If its speed at t = 0 is 4 m/s, the time taken to complete the first revolution will be $\frac{1}{α} [1 - e^{- 2 π}] s$ , where $α =$ _____________ . [2024]

0%

(8) $| {\vec{a}}_{c} | = | {\vec{a}}_{t} |$

$\frac{v^{2}}{r} = \frac{d v}{d t} \Rightarrow \int_{4}^{v} \frac{d v}{v^{2}} = \int_{0}^{t} \frac{d t}{r}$

$\Rightarrow {[\frac{- 1}{v}]}_{4}^{v} = \frac{t}{r} \Rightarrow \frac{- 1}{v} + \frac{1}{4} = 2 t$

$\Rightarrow v = \frac{4}{1 - 8 t} = \frac{d s}{d t}$

$4 \int_{0}^{t} \frac{d t}{1 - 8 t} = \int_{0}^{s} d s$

$(r = 0.5 m s = 2 π r = π)$

$4 \times \frac{{[l n (1 - 8 t)]}_{0}^{t}}{- 8} = π$

$l \ln (1 - 8 t) = - 2 π$

$1 - 8 t = e^{- 2 π} \Rightarrow t = \frac{1}{8} (1 - e^{- 2 π})$

$S o, α = 8$

Q 4 :

A particle is moving with constant speed in a circular path. When the particle turns by an angle 90°, the ratio of instantaneous velocity to its average velocity is $π : x \sqrt{2}$ . The value of $x$ will be [2023]

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7
5

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(2)

$A B = R \sqrt{2}$

Let instantaneous velocity be $v$ , time,

$t = \frac{Arc length}{v} = \frac{2 π \frac{R}{4}}{v} = \frac{π R}{2 v}$

Average velocity, $< v > = \frac{A B}{t} = \frac{R \sqrt{2} (2 v)}{π R} = \frac{2 \sqrt{2} v}{π}$

$\Rightarrow \frac{v}{⟨ v ⟩} = \frac{π}{2 \sqrt{2}}$

Q 5 :

As shown in the figure, a particle is moving with constant speed $π$ m/s. Considering its motion from A to B, the magnitude of the average velocity is [2023]

$π$ m/s
$\sqrt{3}$ m/s
$2 \sqrt{3}$ m/s
$1.5 \sqrt{3}$ m/s

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(4)

$| < \vec{v} > |$ $= \frac{| {\vec{r}}_{f} - {\vec{r}}_{i} |}{Δ t}$

$= \frac{2 R \cos [\frac{π - θ}{2}]}{\frac{2 π R}{3 v}} = 3 \cos 30 ° = 1.5 \sqrt{3}$ $m/s$

Correct option is (4)

Q 6 :

For particle P revolving round the centre $O$ with radius of circular path $r$ and angular velocity $ω$ , as shown in below figure, the projection of OP on the $x$ -axis at time $t$ is [2023]

$x (t) = r \cos (ω t + \frac{π}{6})$
$x (t) = r \sin (ω t + \frac{π}{6})$
$x (t) = r \cos (ω t)$
$x (t) = r \cos (ω t - \frac{π}{6} ω)$

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(1)

$x (t) = r \cos (ω t + 30 °)$

$x (t) = r \cos (ω t + \frac{π}{6})$

Topic Question Set

Q 1 :

A man carrying a monkey on his shoulder does cycling smoothly on a circular track of radius 9 m and completes 120 revolutions in 3 minutes. The magnitude of centripetal acceleration of monkey is (in $m / s^{2}$ ) [2024]

Q 2 :

A clock has 75 cm, 60 cm long second hand and minute hand respectively. In 30 minutes, duration the tip of second hand will travel $x$ distance more than the tip of minute hand. The value of $x$ in meter is nearly (Take $π$ = 3.14) [2024]

0%

Q 4 :

A particle is moving with constant speed in a circular path. When the particle turns by an angle 90°, the ratio of instantaneous velocity to its average velocity is $π : x \sqrt{2}$ . The value of $x$ will be [2023]

Q 5 :

As shown in the figure, a particle is moving with constant speed $π$ m/s. Considering its motion from A to B, the magnitude of the average velocity is [2023]

Q 6 :

For particle P revolving round the centre $O$ with radius of circular path $r$ and angular velocity $ω$ , as shown in below figure, the projection of OP on the $x$ -axis at time $t$ is [2023]

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