Motion in a Vertical Circle

Q 1 :
A body of m kg slides from rest along the curve of a vertical circle from point A to B in a frictionless path. The velocity of the body at B is ______.

$(given, R = 14 m, g = 10 {m/s}^{2} and \sqrt{2} = 1.4)$ [2024]

21.9 m/s
10.6 m/s
19.8 m/s
16.7 m/s

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4

(1)

By conservation of mechanical energy, decrease in P.E. = increase in K.E.

$m g (R + \frac{R}{\sqrt{2}}) = \frac{1}{2} m v_{B}^{2} - 0 \Rightarrow 2 g R (1 + \frac{1}{\sqrt{2}}) = v_{B}^{2}$

$\Rightarrow v_{B}^{2} = 2 \times 10 \times 14 \times (1 + \frac{1}{1.4})$

$v_{B} = \sqrt{20 \times 24} = 4 \sqrt{30} \approx 21.9 m/s$

Q 2 :
A ball suspended by a thread swings in a vertical plane so that its magnitude of acceleration in the extreme position and lowest position are equal. The angle (θ) of thread deflection in the extreme position will be ______. [2024]

$\tan^{- 1} (\sqrt{2})$
$2 \tan^{- 1} (\frac{1}{2})$
$\tan^{- 1} (\frac{1}{2})$
$2 \tan^{- 1} (\frac{1}{\sqrt{5}})$

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(2)

Loss in kinetic energy = Gain in potential energy

$\Rightarrow \frac{1}{2} m v^{2} = m g ℓ (1 - \cos θ)$

$\Rightarrow \frac{v^{2}}{ℓ} = 2 g (1 - \cos θ)$

Acceleration at lowest point $= \frac{v^{2}}{ℓ}$

Acceleration at extreme point $= g \sin θ$

$∴$ $\frac{v^{2}}{ℓ} = g \sin θ$

$∴$ $\sin θ = 2 (1 - \cos θ)$

$\Rightarrow \tan \frac{θ}{2} = \frac{1}{2} \Rightarrow θ = 2 \tan^{- 1} (\frac{1}{2})$

Q 3 :
A stone of mass 900 g is tied to a string and moved in a vertical circle of radius 1 m, making 10 rpm. The tension in the string when the stone is at the lowest point is (if $π^{2}$ = 9.8 and g = 9.8 $m / s^{2}$ ) [2024]

17.8 N
8.82 N
97 N
9.8 N

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(4)

Given that, $m = 900 gm = \frac{900}{1000} kg = \frac{9}{10} kg$

$r = 1 m$

$ω = \frac{2 π N}{60} = \frac{2 π (10)}{60} = \frac{π}{3} rad/sec$

$T - m g = m r ω^{2} \Rightarrow T = m g + m r ω^{2}$

$\Rightarrow T = \frac{9}{10} \times 9.8 + \frac{9}{10} \times 1 {(\frac{π}{3})}^{2}$

$= 8.82 + \frac{9}{10} \times \frac{π^{2}}{9} = 9.80 N$

Q 4 :
A bob of mass 'm' is suspended by a light string of length 'L'. It is imparted a minimum horizontal velocity at the lowest point A such that it just completes a half-circle, reaching the topmost position B. The ratio of kinetic energies $\frac{{(K . E)}_{A}}{{(K . E)}_{B}}$ is ______. [2024]

3 : 2
5 : 1
2 : 5
1 : 5

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(2)

Apply energy conservation

$\frac{1}{2} m V_{L}^{2} = \frac{1}{2} m V_{H}^{2} + m g (2 L)$

$∵$ $V_{L} = \sqrt{5 g L}$

So, $V_{H} = \sqrt{g L}$

$\frac{{(K . E)}_{A}}{{(K . E)}_{B}} = \frac{\frac{1}{2} m {(\sqrt{5 g L})}^{2}}{\frac{1}{2} m {(\sqrt{g L})}^{2}} = \frac{5}{1}$

Topic Question Set

Q 1 :
A body of m kg slides from rest along the curve of a vertical circle from point A to B in a frictionless path. The velocity of the body at B is ______.

$(given, R = 14 m, g = 10 {m/s}^{2} and \sqrt{2} = 1.4)$ [2024]

Q 2 :
A ball suspended by a thread swings in a vertical plane so that its magnitude of acceleration in the extreme position and lowest position are equal. The angle (θ) of thread deflection in the extreme position will be ______. [2024]

Q 3 :
A stone of mass 900 g is tied to a string and moved in a vertical circle of radius 1 m, making 10 rpm. The tension in the string when the stone is at the lowest point is (if $π^{2}$ = 9.8 and g = 9.8 $m / s^{2}$ ) [2024]

Q 4 :
A bob of mass 'm' is suspended by a light string of length 'L'. It is imparted a minimum horizontal velocity at the lowest point A such that it just completes a half-circle, reaching the topmost position B. The ratio of kinetic energies $\frac{{(K . E)}_{A}}{{(K . E)}_{B}}$ is ______. [2024]

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Topic Question Set

Q 1 : A body of m kg slides from rest along the curve of a vertical circle from point A to B in a frictionless path. The velocity of the body at B is ______. (given, R=14 m, g=10 m/s2 and 2=1.4 ) [2024]

Q 2 : A ball suspended by a thread swings in a vertical plane so that its magnitude of acceleration in the extreme position and lowest position are equal. The angle (θ) of thread deflection in the extreme position will be ______. [2024]

Q 3 : A stone of mass 900 g is tied to a string and moved in a vertical circle of radius 1 m, making 10 rpm. The tension in the string when the stone is at the lowest point is (if π2 = 9.8 and g = 9.8 m/s2) [2024]

Q 4 : A bob of mass 'm' is suspended by a light string of length 'L'. It is imparted a minimum horizontal velocity at the lowest point A such that it just completes a half-circle, reaching the topmost position B. The ratio of kinetic energies (K.E)A(K.E)B is ______. [2024]

Want Us to Email you About Special Offers & Updates?

Q 1 :
A body of m kg slides from rest along the curve of a vertical circle from point A to B in a frictionless path. The velocity of the body at B is ______.

$(given, R = 14 m, g = 10 {m/s}^{2} and \sqrt{2} = 1.4)$ [2024]

Q 2 :
A ball suspended by a thread swings in a vertical plane so that its magnitude of acceleration in the extreme position and lowest position are equal. The angle (θ) of thread deflection in the extreme position will be ______. [2024]

Q 3 :
A stone of mass 900 g is tied to a string and moved in a vertical circle of radius 1 m, making 10 rpm. The tension in the string when the stone is at the lowest point is (if $π^{2}$ = 9.8 and g = 9.8 $m / s^{2}$ ) [2024]

Q 4 :
A bob of mass 'm' is suspended by a light string of length 'L'. It is imparted a minimum horizontal velocity at the lowest point A such that it just completes a half-circle, reaching the topmost position B. The ratio of kinetic energies $\frac{{(K . E)}_{A}}{{(K . E)}_{B}}$ is ______. [2024]