Tag: circular motion and gravitation

Questions Related to circular motion and gravitation

A block of mass $m$ at the end of a string is whirled round in a vertical circle of radius $R$. The critical speed of the block at the top of its swing below which the string would slacken before the block reaches the top is:

examples of circular motion uniform circular motion circular motion and gravitation physics

  1. $Rg$

  2. ${(Rg)}^{2}$

  3. $R/g$

  4. $\sqrt {Rg}$

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Correct Option: D
Explanation:

At the top if the block is moving with critical speed the tension in the string would be zero and therefore force of gravity would provide the necessary centripetal force:

$mg=\cfrac { m{ v }^{ 2 } }{ R } \ \Rightarrow v=\sqrt { Rg } $

A bob of mass $100\ g$ tied at the end of a string of length $50\ cm$ is revolved in a vertical circle with constant speed of $1\ ms^{-1}$. When the tension in the string is $0.7N$, the angle made by the vertical is $(g=10\ ms^{-2})$

examples of circular motion uniform circular motion circular motion and gravitation physics

  1. $0^{o}$

  2. $90^{o}$

  3. $180^{o}$

  4. $60^{o}$

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Correct Option: C

A weightless thread can withstand tension upto 30 N.A. stone of mass 0.5kg is tied to it and is revolved in a circular path of radius 2m i n a vertical plane. If $g=10m/s^2$, then the maximum angular velocity of the stone can be;-

examples of circular motion uniform circular motion circular motion and gravitation physics

  1. $5\, rad / s$

  2. $\sqrt{30} \ rad/s$

  3. $\sqrt{60} \ rad/s$

  4. $10\, rad / s$

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Correct Option: A

An inclined plane ends into vertical loop of radius $R$. A particle is released from height $3R$. Can it loop the loop?

examples of circular motion uniform circular motion circular motion and gravitation physics

  1. yes

  2. no

  3. cannot say

  4. yes if fraction is present

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Correct Option: A
Explanation:

The minimum speed required at the bottom of the circle(r) to complete the circular motion is $\sqrt {5gR}$
For this speed, the minimum height required is
$0.5mv^2=mgh$
$\Rightarrow 0.5(5gR)=gh$
$\Rightarrow h=\dfrac {5}{2}R$
So, the minimum height from which the body has to released is 2.5R. We have a height of 3R, so the particle completes the vertical loop.
Option A.

A block of mass $m$ at the end of a string is whirled round in a vertical circle of radius $r$. The critical speed of the block at the top of its swing below which the string would slacken before block reaches the top is

examples of circular motion uniform circular motion circular motion and gravitation physics

  1. $\sqrt{2rg}$

  2. $\sqrt{3rg}$

  3. $\sqrt{rg}$

  4. $\sqrt{5rg}$

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Correct Option: C
Explanation:

the string will not slack if the centrifugal force on the particle is greater then or equal to the weight of mass i.e.
$\dfrac{mv^2}{r} \ge mg \ \Rightarrow v \ge \sqrt{rg} \ \Rightarrow v _c=\sqrt{rg} $

The maximum tension that an inextensible ring of radius 1 m and mass density 0.1 kg ${ m }^{ -1 }$ can bear is 40 N . The maximum angular velocity with which it can be rotated in a  circular path is 

examples of circular motion uniform circular motion circular motion and gravitation physics

  1. 20 rad/s

  2. 18 rad/s

  3. 16 rad/s

  4. 15 rad/s

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Correct Option: A

A body is allowed to slide on a frictionless track from rest under-gravity.The track ends in a circular of diameter D. What should be the mini-mum height of the body in terms of D.          So that it may successfully complete the loop?

examples of circular motion uniform circular motion circular motion and gravitation physics

  1. $\dfrac{9}{4}D $

  2. $ \frac {5}{4} D $

  3. D

  4. 2 D

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Correct Option: A
Explanation:

Velocity at top of the circle is $\sqrt{5gr}$ 

Energy lost to reach height D from height H  $ = mg(H-D)$

Energy gained in the form of $K.E= \dfrac12 mv^2= \dfrac12 m(5gr) = \dfrac{5mgD}{4}$ 

Equating K.E. and P.E. ,

$\dfrac{5mgD}{4} = mg(H-D)$

$H=\dfrac{9D}{4}$

A body is a allowed to slide down a frictionless track from rest position at its top under gravity. The track ends in a circular loop of diameter $D$. Then, the minimum height of the inclined track  (in terms of $D$ ) so that it may complete successfully the loop is:

examples of circular motion uniform circular motion circular motion and gravitation physics

  1. $\dfrac{7D}{4}$

  2. $\dfrac{9D}{4}$

  3. $\dfrac{5D}{4}$

  4. $\dfrac{3D}{4}$

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Correct Option: C
Explanation:

$V _{0}$ required for a complete vertical circle: $V _{0} =\sqrt{\dfrac{5}{2}gD}$
WET between top and bottom:
              $mgH  = \dfrac{1}{2}m V _{0}^{2}$


                 $gH  = \dfrac{5}{4}gD$

                   $H  = \dfrac{5}{4}D$

The length of simple pendulum is $1m$ and mass of its bob is $50 gram$. The bob is given  sufficient velocity so that the bob describes vertical circle whose radius equal to length of pendulum, the tension in the string at lowest extreme position is:

examples of circular motion uniform circular motion circular motion and gravitation physics

  1. $2.5 N$

  2. $1N$

  3. $1.5 N$

  4. $2N$

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Correct Option: A
Explanation:

Using conservation of energy, 

$mg \times 2R = mv^2/2$
$v=2 \sqrt{Rg}$ .........(1)

At lower most point, from free body digram,
$T - mg = m v^2/R$ ...........(2)
From (1) and (2), 
$T = 5mg$
$T = 2.5N$

A weightless thread can bear tension up to $3.7kg-wt$. A stone of mass $500g$ is tied to it and revolved in a circular path of radius $4$m in a verticle plane. If $g=10m/s^2$, then the maximum angular velocity of the stone will be:

examples of circular motion uniform circular motion circular motion and gravitation physics

  1. $2 rad/s$

  2. $4 rad/s$

  3. $16 rad/s$

  4. $\sqrt{21} rad/s$

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Correct Option: B
Explanation:

Tension is maximum at lowest point of a verticle circle $T _{max}=mr\omega^2+mg$
$3.7g=0.5\times 4\times \omega^2+0.5g$
$2\omega^2=3.2g$
$\omega=\sqrt{1.6g}=\sqrt{1.6\times 10}$
$\omega=4rad/s$