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

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

Q 5 :

A bob of mass m is suspended at a point O by a light string of length $l$ and left to perform vertical motion (circular) as shown in figure. Initially, by applying horizontal velocity $v_{0}$ at the point 'A'. The string becomes slack when, the bob reaches at the point 'D'. The ratio of the kinetic energy of the bob at the points B and C is [2025]

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(1)

Applying conservation of mechanical energy,

$\frac{1}{2} m v_{A}^{2} = \frac{1}{2} m v_{B}^{2} + m g h$

$\Rightarrow \frac{1}{2} m (5 g l) = \frac{1}{2} m v_{B}^{2} + m g \frac{l}{2}$

$\Rightarrow \frac{5 m g l}{2} - \frac{m g l}{2} = {KE}_{B}$

$\Rightarrow {KE}_{B} = 2 m g l$

$\frac{1}{2} m v_{C}^{2} = \frac{1}{2} m v_{D}^{2} + m g \frac{l}{2}$

$\Rightarrow \frac{1}{2} m g l + m g \frac{l}{2} = m g l$

${KE}_{C} = m g l$

$\Rightarrow \frac{{KE}_{B}}{KEc} = 2$

Q 6 :

A body of mass 100 g is moving in circular path of radius 2 m on vertical plane as shown in figure. The velocity of the body at point A is 10 m/s. The ratio of its kinetic energies at point B and C is: [2025]

(Take acceleration due to gravity as 10 $m / s^{2}$ )

$\frac{2 + \sqrt{3}}{3}$
$\frac{2 + \sqrt{2}}{3}$
$\frac{3 + \sqrt{3}}{2}$
$\frac{3 - \sqrt{2}}{2}$

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(3)

$\frac{1}{2} m \times 100 + 0 = \frac{1}{2} m V_{B}^{2} + m g (R - \frac{R \sqrt{3}}{2})$

$100 = V_{B}^{2} + 2 g R (1 - \frac{\sqrt{3}}{2})$

$V_{B}^{2} = 100 - 20 (2 - \sqrt{3})$

$\Rightarrow V_{B}^{2} = 60 + 20 \sqrt{3}$

$K . E ._{B} = \frac{1}{2} m V_{B}^{2} = \frac{m}{2} (60 + 20 \sqrt{3})$ ...(i)

$\frac{1}{2} m (100) = \frac{1}{2} m V_{C}^{2} + m g (\frac{3 R}{2})$

$100 = V_{C}^{2} + 60$

$V_{C}^{2} = 40$

$K . E ._{C} = \frac{1}{2} m V_{C}^{2} = \frac{1}{2} m (40)$

$\Rightarrow \frac{K_{B}}{K_{C}} = {(\frac{V_{B}}{V_{C}})}^{2} = \frac{3 + \sqrt{3}}{2}$

Q 7 :

A body of mass 'm' connected to a massless and unstretchable string goes in verticle circle of radius 'R' under gravity g. The other end of the string is fixed at the center of circle. If velocity at top of circular path is $n \sqrt{g R}$ , where $n \geq 1$ , then ratio of kinetic energy of the body at bottom to that at top of the circle is [2025]

$\frac{n}{n + 4}$
$\frac{n + 4}{n}$
$\frac{n^{2}}{n^{2} + 4}$
$\frac{n^{2} + 4}{n^{2}}$

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(4)

$v_{0} = \sqrt{v^{2} + 2 g (2 R)}$

$v_{0} = \sqrt{n^{2} g R + 4 g R}$

$∴ \frac{k_{bottom}}{k_{top}} = \frac{v_{0}^{2}}{v^{2}} = \frac{n^{2} + 4}{n^{2}}$

Q 8 :

In case of vertical circular motion of a particle by a thread of length $r$ , if the tension in the thread is zero at an angle $30 °$ shown in figure, the velocity at the bottom point (A) of the circular path is

(g = gravitational acceleration) [2026]

$\sqrt{4 g r}$
$\sqrt{5 g r}$
$\sqrt{\frac{7}{2} g r}$
$\sqrt{\frac{5}{2} g r}$

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(3)

$T + m g \cos 60 ° = \frac{m V^{2}}{ℓ}$

$T = 0$

$V^{2} = \frac{g ℓ}{2} where V is the speed at point A$

$M.E.C.$

$\frac{1}{2} m u^{2} = m g (ℓ + ℓ \cos 60 °) + \frac{1}{2} m V^{2}$

$u^{2} = 3 g ℓ + \frac{g ℓ}{2}$

$u = \sqrt{\frac{7 g ℓ}{2}}$

Q 9 :

A pendulum of length $ℓ = 1$ m having a bob of mass $m = 1$ kg is hanging from a rigid support. If the bob is projected horizontally with a velocity $v_{0} = \sqrt{35}$ m/s. The tension in the string is 6k Newton when angle made by the string is $60 °$ from vertical as shown. Find the value of $k$ .

(5)

$\frac{1}{2} m v_{0}^{2} = \frac{1}{2} m v^{2} + m g l (1 - \cos θ), T - m g \cos θ = \frac{m v^{2}}{R}$

Q 10 :

A smooth inclined plane ends in a vertical circular loop, as shown in the figure. A small body is released from height $h$ as shown. If the body exerts a force of three times its weight on the plane at the highest point of circle then the height $h = α R$ . The value of $α$ is [2026]

[IMAGE 19]

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(2)

[IMAGE 20]

$At highest point$

$m g + N = \frac{m v_{top}^{2}}{R}$

$m g + 3 m g = \frac{m v_{top}^{2}}{R} \Rightarrow v_{top} = \sqrt{4 g R}$

$Now C.O.M.E:$

$m g (h - 2 R) = \frac{1}{2} m \cdot 4 g R \Rightarrow h = 4 R$

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]

Q 6 :

A body of mass 100 g is moving in circular path of radius 2 m on vertical plane as shown in figure. The velocity of the body at point A is 10 m/s. The ratio of its kinetic energies at point B and C is: [2025]

(Take acceleration due to gravity as 10 $m / s^{2}$ )

Q 8 :

In case of vertical circular motion of a particle by a thread of length $r$ , if the tension in the thread is zero at an angle $30 °$ shown in figure, the velocity at the bottom point (A) of the circular path is

(g = gravitational acceleration) [2026]

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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]

Q 6 : A body of mass 100 g is moving in circular path of radius 2 m on vertical plane as shown in figure. The velocity of the body at point A is 10 m/s. The ratio of its kinetic energies at point B and C is: [2025] (Take acceleration due to gravity as 10 m/s2)

Q 8 : In case of vertical circular motion of a particle by a thread of length r, if the tension in the thread is zero at an angle 30° shown in figure, the velocity at the bottom point (A) of the circular path is (g = gravitational acceleration) [2026]

Q 9 : A pendulum of length ℓ=1 m having a bob of mass m=1 kg is hanging from a rigid support. If the bob is projected horizontally with a velocity v0=35 m/s. The tension in the string is 6k Newton when angle made by the string is 60° from vertical as shown. Find the value of k.

Q 10 : A smooth inclined plane ends in a vertical circular loop, as shown in the figure. A small body is released from height h as shown. If the body exerts a force of three times its weight on the plane at the highest point of circle then the height h=αR. The value of α is [2026] [IMAGE 19]

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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]

Q 6 :

A body of mass 100 g is moving in circular path of radius 2 m on vertical plane as shown in figure. The velocity of the body at point A is 10 m/s. The ratio of its kinetic energies at point B and C is: [2025]

(Take acceleration due to gravity as 10 $m / s^{2}$ )

Q 8 :

In case of vertical circular motion of a particle by a thread of length $r$ , if the tension in the thread is zero at an angle $30 °$ shown in figure, the velocity at the bottom point (A) of the circular path is

(g = gravitational acceleration) [2026]