Applications of Newton's Laws of Motion in Different Frames

Q 1 :

A light stretchable string passing over a smooth light pulley connects two blocks of masses $m_{1}$ and $m_{2}$ . If the acceleration of the system is $\frac{g}{8}$ , then the ratio of the masses $\frac{m_{2}}{m_{1}}$ is: [2024]

9 : 7
4 : 3
5 : 3
8 : 1

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(1)

$a_{c} = \frac{(m_{2} - m_{1}) g}{(m_{1} + m_{2})} \Rightarrow \frac{g}{8} = (\frac{m_{2} - m_{1}}{m_{1} + m_{2}}) g$

$\frac{1}{8} = (\frac{\frac{m_{2}}{m_{1}} - 1}{\frac{m_{2}}{m_{1}} + 1}) \Rightarrow \frac{m_{2}}{m_{1}} + 1 = 8 \frac{m_{2}}{m_{1}} - 8 \Rightarrow \frac{m_{2}}{m_{1}} = \frac{9}{7}$

Q 2 :

A wooden block of mass 5 kg rests on a soft horizontal floor. When an iron cylinder of mass 25 kg is placed on top of the block, the floor yields, and the block and the cylinder together go down with an acceleration of 0.1 ${ms}^{- 2}$ .

The action force of the system on the floor is equal to [2024].

294 N
196 N
297 N
291 N

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(4)

$30 g - N = 30 a$

Taking $g = 9.8 {m/s}^{2}$

$30 \times 9.8 - N = 30 \times 0.1$

$N = 291 N$

Q 3 :

A light string passing over a smooth light pulley connects two blocks of masses $m_{1}$ and $m_{2}$ (where $m_{2} > m_{1}$ ). If the acceleration of the system is $\frac{g}{\sqrt{2}}$ , then the ratio of the masses $\frac{m_{1}}{m_{2}}$ is: [2024]

$\frac{\sqrt{3} + 1}{\sqrt{2} - 1}$
$\frac{1 + \sqrt{5}}{\sqrt{2} - 1}$
$\frac{1 + \sqrt{5}}{\sqrt{5} - 1}$
$\frac{\sqrt{2} - 1}{\sqrt{2} + 1}$

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(4)

$a = (\frac{m_{2} - m_{1}}{m_{1} + m_{2}}) g \Rightarrow \frac{g}{\sqrt{2}} = (\frac{m_{2} - m_{1}}{m_{1} + m_{2}}) g$

$(m_{1} + m_{2}) = \sqrt{2} m_{2} - \sqrt{2} m_{1}$

$\Rightarrow \frac{m_{1}}{m_{2}} = (\frac{\sqrt{2} - 1}{\sqrt{2} + 1})$

Q 4 :

A body of weight 200 N is suspended from a tree branch through a chain of mass 10 kg. The branch pulls the chain by a force equal to (if g = 10 $m / s^{2}$ ) [2024]

300 N
150 N
100 N
200 N

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(1)

Chain block system is in equilibrium, so T = 200 + 100 = 300 N.

Q 5 :

All surfaces shown in the figure are assumed to be frictionless and the pulleys and the string are light.

The acceleration of the block of mass 2 kg is [2024].

$g$
$\frac{g}{3}$
$\frac{g}{2}$
$\frac{g}{4}$

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(2)

By constrained motion

$- a_{1} + a_{2} + a_{2} = 0$

$a_{1} - 2 a_{2} = 0 or a_{2} = \frac{a_{1}}{2}$

$T - 2 g \sin 30 ° = 2 a_{1}$

$\Rightarrow T - g = 2 a_{1}$ ...(1)

$4 g - 2 T = 4 a_{2}$

$\Rightarrow 4 g - 2 T = 2 a_{1}$

$\Rightarrow 2 g - T = a_{1}$ ...(2)

Adding equation (1) and (2), $g = 3 a_{1} \Rightarrow a_{1} = \frac{g}{3}$

Q 6 :

Three blocks A, B, and C are pulled on a horizontal smooth surface by a force of 80 N as shown in the figure.

The tensions $T_{1}$ and $T_{2}$ in the string are respectively: [2024]

40 N, 64 N
60 N, 80 N
88 N, 96 N
80 N, 100 N

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(1)

$a = \frac{80}{10} = 8 {m/s}^{2}$

F.B.D of 5 kg

$T_{1} = 5 (a)$

$T_{1} = 5 (8) = 40 N$

$80 - T_{2} = 2 (8)$

$T_{2} = 80 - 16 = 64 N$

Q 7 :

In the given arrangement of a doubly inclined plane, two blocks of masses M and m are placed. The blocks are connected by a light string passing over an ideal pulley as shown. The coefficient of friction between the surface of the plane and the blocks is 0.25. The value of m, for which M = 10 kg will move down with an acceleration of 2 $m / s^{2}$ , is (take g = 10 $m / s^{2}$ and tan 37° = 3/4) [2024]

6.5 kg
4.5 kg
2.25 kg
9 kg

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(2)

For M block

$10 g \sin 53 ° - μ (10 g) \cos 53 ° - T = 10 \times 2$

$T = 80 - 15 - 20$

$T = 45 N$

For m block

$T - m g \sin 37 ° - μ m g \cos 37 ° = m \times 2$

$45 = 10 m$

$m = 4.5 kg$

Q 8 :

A light string passing over a smooth light fixed pulley connects two blocks of masses $m_{1}$ and $m_{2}$ .

If the acceleration of the system is g/8, then the ratio of masses is [2024].

$\frac{9}{7}$
$\frac{8}{1}$
$\frac{4}{3}$
$\frac{5}{3}$

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(1)

$a = \frac{(m_{1} - m_{2}) g}{(m_{1} + m_{2})} = \frac{g}{8}$

$8 m_{1} - 8 m_{2} = m_{1} + m_{2}$

$7 m_{1} = 9 m_{2}$

$\frac{m_{1}}{m_{2}} = \frac{9}{7}$

Q 9 :

Three blocks $M_{1}$ , $M_{2}$ , $M_{3}$ having masses 4 kg, 6 kg, and 10 kg respectively are hanging from a smooth pulley using rope 1, 2, and 3 as shown in the figure. The tension in the rope 1, $T_{1}$ , when they are moving upward with an acceleration of 2 ${ms}^{- 2}$ , is ________ N (if g = 10 $m / s^{2}$ ). [2024]

(240)

FBD of $M_{1}$

$T_{1} - 200 = (4 + 6 + 10) \times 2$

$∴$ $T_{1} = 240 N$

Q 10 :

A balloon and its content having mass M is moving up with an acceleration 'a'. The mass that must be released from the content so that the balloon starts moving up with an acceleration '3a' will be: (Take 'g' as acceleration due to gravity) [2025]

$\frac{3 M a}{2 a - g}$
$\frac{3 M a}{2 a + g}$
$\frac{2 M a}{3 a + g}$
$\frac{2 M a}{3 a - g}$

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(3)

$F - M g = M a \Rightarrow F = M a + M g$ ... (i)

$F - (M - x) g = (M - x) 3 a$ ... (ii)

From (i) and (ii)

$M a + M g - M g + x g = 3 M a - 3 x a \Rightarrow x = \frac{2 M a}{g + 3 a}$

Q 11 :

Given below are two statements:

Statement I: An elevator can go up or down with uniform speed when its weight is balanced with the tension of its cable.

Statement II: Force exerted by the floor of an elevator on the foot of a person standing on it is more than his/her weight when the elevator goes down with increasing speed.

In the light of the above statements, choose the correct answer from the options given below: [2023]

Statement I is false but Statement II is true
Statement I is true but Statement II is false
Both Statement I and Statement II are false
Both Statement I and Statement II are true

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(2)

$Statement-1$

When elevator is moving with uniform speed, $T = F_{g}$

$Statement-2$

When elevator is going down with increasing speed, its acceleration is downward.

$Hence W - N = \frac{W}{g} \times a$

$N = W (1 - \frac{a}{g}), i.e., less than weight.$

Q 12 :

A small block of mass m slides down from the top of a frictionless inclined surface, while the inclined plane is moving towards left with constant acceleration $a_{0}$ . The angle between the inclined plane and ground is $θ$ and its base length is L. Assuming that initially the small block is at the top of the inclined plane, the time it takes to reach the lowest point of the inclined plane is ________. [2026]

$\sqrt{\frac{2 L}{g \sin θ - a_{0} \cos θ}}$
$\sqrt{\frac{4 L}{g \sin 2 θ - a_{0} (1 + \cos 2 θ)}}$
$\sqrt{\frac{4 L}{g \cos^{2} θ - a_{0} \sin θ \cos θ}}$
$\sqrt{\frac{2 L}{g \sin 2 θ - a_{0} (1 + \cos 2 θ)}}$

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(2)

$m g s i n θ - m a_{0} \cos θ = m a$

$a = g \sin θ - a_{0} \cos θ$

$Now using,$

$S = u t + \frac{1}{2} a_{down} t^{2}$

$\frac{L}{\cos θ} = \frac{1}{2} (g \sin θ - a_{0} \cos θ) t^{2}$

$t = \sqrt{\frac{2 L}{g \sin θ \cos θ - a_{0} \cos^{2} θ}}$

$t = \sqrt{\frac{4 L}{g \sin 2 θ - a_{0} (1 + \cos 2 θ)}}$

Q 13 :

A block is sliding down on an inclined plane of slope $θ$ and at an instant $t = 0$ this block is given an upward momentum so that it starts moving up on the inclined surface with velocity u. The distance (S) travelled by the block before its velocity becomes zero is ______.

(g = gravitational acceleration) [2026]

$\frac{u^{2}}{4 g \sin θ}$
$\frac{2 u^{2}}{g \cos θ}$
$\frac{u^{2}}{\sqrt{2} g \cos θ}$
None of these

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(4)

$a = - g \sin θ$

$V^{2} = U^{2} + 2 a s$

$0 = u^{2} - 2 g \sin θ \cdot s$

$s = \frac{u^{2}}{2 g \sin θ}$

Q 14 :

A particle of mass $m$ falls from rest through a resistive medium having resistive force, $F = - k v$ , where $v$ is the velocity of the particle and $k$ is a constant. Which of the following graphs represents velocity (v) versus time (t)? [2026]

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(2)

$m \cdot \frac{d v}{d t} = m g - k v$

$\int_{0}^{v} \frac{d v}{m g - k v} = \int_{0}^{t} \frac{d t}{m}$

$- \frac{1}{k} l n (\frac{m g - k v}{m g}) = \frac{t}{m}$

$v = \frac{m g}{k} (1 - e^{- k t / m})$

Topic Question Set

Q 1 :

A light stretchable string passing over a smooth light pulley connects two blocks of masses $m_{1}$ and $m_{2}$ . If the acceleration of the system is $\frac{g}{8}$ , then the ratio of the masses $\frac{m_{2}}{m_{1}}$ is: [2024]

Q 3 :

A light string passing over a smooth light pulley connects two blocks of masses $m_{1}$ and $m_{2}$ (where $m_{2} > m_{1}$ ). If the acceleration of the system is $\frac{g}{\sqrt{2}}$ , then the ratio of the masses $\frac{m_{1}}{m_{2}}$ is: [2024]

Q 4 :

A body of weight 200 N is suspended from a tree branch through a chain of mass 10 kg. The branch pulls the chain by a force equal to (if g = 10 $m / s^{2}$ ) [2024]

Q 5 :

All surfaces shown in the figure are assumed to be frictionless and the pulleys and the string are light.

The acceleration of the block of mass 2 kg is [2024].

Q 6 :

Three blocks A, B, and C are pulled on a horizontal smooth surface by a force of 80 N as shown in the figure.

The tensions $T_{1}$ and $T_{2}$ in the string are respectively: [2024]

Q 8 :

A light string passing over a smooth light fixed pulley connects two blocks of masses $m_{1}$ and $m_{2}$ .

If the acceleration of the system is g/8, then the ratio of masses is [2024].

Q 10 :

A balloon and its content having mass M is moving up with an acceleration 'a'. The mass that must be released from the content so that the balloon starts moving up with an acceleration '3a' will be: (Take 'g' as acceleration due to gravity) [2025]

Q 14 :

A particle of mass $m$ falls from rest through a resistive medium having resistive force, $F = - k v$ , where $v$ is the velocity of the particle and $k$ is a constant. Which of the following graphs represents velocity (v) versus time (t)? [2026]

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Topic Question Set

Q 1 : A light stretchable string passing over a smooth light pulley connects two blocks of masses m1 and m2. If the acceleration of the system is g8, then the ratio of the masses m2m1 is: [2024]

Q 2 : A wooden block of mass 5 kg rests on a soft horizontal floor. When an iron cylinder of mass 25 kg is placed on top of the block, the floor yields, and the block and the cylinder together go down with an acceleration of 0.1 ms-2. The action force of the system on the floor is equal to [2024].

Q 3 : A light string passing over a smooth light pulley connects two blocks of masses m1 and m2 (where m2>m1). If the acceleration of the system is g2, then the ratio of the masses m1m2 is: [2024]

Q 4 : A body of weight 200 N is suspended from a tree branch through a chain of mass 10 kg. The branch pulls the chain by a force equal to (if g = 10 m/s2) [2024]

Q 5 : All surfaces shown in the figure are assumed to be frictionless and the pulleys and the string are light. The acceleration of the block of mass 2 kg is [2024].

Q 6 : Three blocks A, B, and C are pulled on a horizontal smooth surface by a force of 80 N as shown in the figure. The tensions T1 and T2 in the string are respectively: [2024]

Q 8 : A light string passing over a smooth light fixed pulley connects two blocks of masses m1 and m2. If the acceleration of the system is g/8, then the ratio of masses is [2024].

Q 9 : Three blocks M1, M2, M3 having masses 4 kg, 6 kg, and 10 kg respectively are hanging from a smooth pulley using rope 1, 2, and 3 as shown in the figure. The tension in the rope 1, T1, when they are moving upward with an acceleration of 2 ms-2, is ________ N (if g = 10 m/s2). [2024]

Q 10 : A balloon and its content having mass M is moving up with an acceleration 'a'. The mass that must be released from the content so that the balloon starts moving up with an acceleration '3a' will be: (Take 'g' as acceleration due to gravity) [2025]

Q 14 : A particle of mass m falls from rest through a resistive medium having resistive force, F=−kv, where v is the velocity of the particle and k is a constant. Which of the following graphs represents velocity (v) versus time (t)? [2026]

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Q 1 :

A light stretchable string passing over a smooth light pulley connects two blocks of masses $m_{1}$ and $m_{2}$ . If the acceleration of the system is $\frac{g}{8}$ , then the ratio of the masses $\frac{m_{2}}{m_{1}}$ is: [2024]

Q 3 :

A light string passing over a smooth light pulley connects two blocks of masses $m_{1}$ and $m_{2}$ (where $m_{2} > m_{1}$ ). If the acceleration of the system is $\frac{g}{\sqrt{2}}$ , then the ratio of the masses $\frac{m_{1}}{m_{2}}$ is: [2024]

Q 4 :

A body of weight 200 N is suspended from a tree branch through a chain of mass 10 kg. The branch pulls the chain by a force equal to (if g = 10 $m / s^{2}$ ) [2024]

Q 5 :

All surfaces shown in the figure are assumed to be frictionless and the pulleys and the string are light.

The acceleration of the block of mass 2 kg is [2024].

Q 6 :

Three blocks A, B, and C are pulled on a horizontal smooth surface by a force of 80 N as shown in the figure.

The tensions $T_{1}$ and $T_{2}$ in the string are respectively: [2024]

Q 8 :

A light string passing over a smooth light fixed pulley connects two blocks of masses $m_{1}$ and $m_{2}$ .

If the acceleration of the system is g/8, then the ratio of masses is [2024].

Q 10 :

A balloon and its content having mass M is moving up with an acceleration 'a'. The mass that must be released from the content so that the balloon starts moving up with an acceleration '3a' will be: (Take 'g' as acceleration due to gravity) [2025]

Q 14 :

A particle of mass $m$ falls from rest through a resistive medium having resistive force, $F = - k v$ , where $v$ is the velocity of the particle and $k$ is a constant. Which of the following graphs represents velocity (v) versus time (t)? [2026]