Mechanical Power

Q 1 :

A body is moving unidirectionally under the influence of a constant power source. Its displacement in time $t$ is proportional to [2024]

$t$
$t^{2}$
$t^{2 / 3}$
$t^{3 / 2}$

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(4)

$\frac{d W}{d t} = P \Rightarrow dW = P d t$

$F \cdot d s = P d t$

$m d v \frac{d s}{d t} = P d t$

$\int_{0}^{v} v d v = \frac{P}{m} \int_{0}^{t} d t \Rightarrow \frac{v^{2}}{2} = \frac{P t}{m}$

Hence, $v \propto \sqrt{t} \Rightarrow \frac{d s}{d t} = C \sqrt{t},$ $C$ is a constant

$\int_{0}^{s} d s = C \int_{0}^{t} t^{1 / 2} d t$

$s = C \frac{2}{3} t^{3 / 2} \Rightarrow s \propto t^{3 / 2}$

$∴$ displacement of proportional to ${(t)}^{3 / 2}$

Q 2 :

A body of mass 2 kg begins to move under the action of a time dependent force given by $\vec{F} = (6 t \hat{i} + 6 t^{2} \hat{j}) N .$ The power developed by the force at the time $t$ is given by [2024]

$(6 t^{4} + 9 t^{5}) W$
$(3 t^{5} + 6 t^{5}) W$
$(9 t^{5} + 6 t^{3}) W$
$(9 t^{3} + 6 t^{5}) W$

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(4)

$\vec{F} = (6 t \hat{i} + 6 t^{2} \hat{j}) N$

$\vec{F} = m \vec{a} = (6 t \hat{i} + 6 t^{2} \hat{j})$

$\vec{a} = \frac{\vec{F}}{m} = (3 t \hat{i} + 3 t^{2} \hat{j})$

$\vec{v} = \int_{0}^{t} \vec{a} d t = \frac{3 t^{2}}{2} \hat{i} + t^{3} \hat{j}$

$P = \vec{F} \cdot \vec{v} = (9 t^{3} + 6 t^{5}) W$

Q 3 :

A body of mass 4 kg is placed on a plane at a point P having coordinate (3, 4) m. Under the action of force $\vec{F} = (2 \hat{i} + 3 \hat{j}) N$ , it moves to a new point Q having coordinates (6, 10) m in 4 sec. The average power and instantaneous power at the end of 4 sec are in the ratio of : [2025]

13 : 6
6 : 13
1 : 2
4 : 13

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(2)

$\vec{F} = (2 \hat{i} + 3 \hat{j}) N$ ; displacement $\vec{S} = (3 \hat{i} + 6 \hat{j}) m$

Acceleration, $\vec{a} = \frac{\vec{F}}{m} = (\frac{2 \hat{i} + 3 \hat{j}}{4}) m / s^{2}$

$\vec{S} = \vec{u} t + \frac{1}{2} \vec{a} t^{2}$

$\Rightarrow 3 \hat{i} + 6 \hat{j} = 4 \vec{u} + \frac{1}{2} (\frac{2 \hat{i} + 3 \hat{j}}{4}) \cdot 16 = 4 \vec{u} + 4 \hat{i} + 6 \hat{j}$

or $\vec{u} = \frac{1}{4} (3 \hat{i} + 6 \hat{j} - 4 \hat{i} - 6 \hat{j}) \Rightarrow \vec{u} = - \frac{1}{4} \hat{i}$

Average Power $< P > = \vec{F} \cdot \frac{\vec{S}}{t}$

$= \frac{(2 \hat{i} + 3 \hat{j}) \cdot (3 \hat{i} + 6 \hat{j})}{4} watt = 6 watt$

$\vec{v}$ $at t = 4 \sec = (\frac{1}{2} \hat{i} + \frac{3}{4} \hat{j}) \times 4 = (2 \hat{i} + 3 \hat{j})$

$P_{ins} = \vec{F} \cdot \vec{v} = (2 \hat{i} + 3 \hat{j}) (2 \hat{i} + 3 \hat{j}) = 13 W$

$\Rightarrow \frac{< P >}{P_{ins}} = \frac{6}{13}$

Q 4 :

A sand dropper drops sand of mass m(t) on a conveyer belt at a rate proportional to the square root of speed (v) of the belt, i.e., $\frac{d m}{d t} \propto \sqrt{v}$ . If P is the power delivered to run the belt at constant speed then which of the following relationship is true? [2025]

$P^{2} \propto v^{3}$
$P \propto \sqrt{v}$
$P \propto v$
$P^{2} \propto v^{5}$

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(4)

Power = $\vec{F} \cdot \vec{v} and F = \frac{d p}{d t} = Rate of change of linear momentum$

$F = v \cdot \frac{d m}{d t} = K v^{3 / 2}, K is constant$

Power $(P) = {(K v)}^{3 / 2} \cdot (v) = K v^{5 / 2}$

So, $P^{2} \propto v^{5}$

Q 5 :

An object of mass 1000 g experiences a time dependent force $\vec{F} = (2 t \hat{i} + 3 t^{2} \hat{j}) N$ . The power generated by the force at time t is: [2025]

$(2 t^{3} + 3 t^{3}) W$
$(2 t^{2} + 18 t^{3}) W$
$(3 t^{3} + 5 t^{5}) W$
$(2 t^{3} + 3 t^{5}) W$

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(4)

m = 1000 gram = 1 kg

$\vec{F} = (2 t \hat{i} + 3 t^{2} \hat{j})$

$\frac{d v}{d t} = \frac{F}{m} = (2 t \hat{i} + 3 t^{2} \hat{j})$

$\Rightarrow \int_{0}^{v} d v = \int_{0}^{t} (2 t \hat{i} + 3 t^{2} \hat{j}) d t$

$\Rightarrow \vec{v} = t^{2} \hat{i} + t^{3} \hat{j}$

Power, $P = \vec{F} \cdot \vec{v}$

$P = (2 t \hat{i} + 3 t^{2} \hat{j}) \cdot (t^{2} \hat{i} + t^{3} \hat{j})$

$P = (2 t^{3} + 3 t^{5}) W$

Q 6 :

The ratio of powers of two motors is $\frac{3 \sqrt{x}}{\sqrt{x} + 1}$ , that are capable of raising 300 kg water in 5 minutes and 50 kg water in 2 minutes respectively from a well of 100 m deep. The value of $x$ will be [2023]

2
4
2.4
16

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(4)

$Average Power = \frac{total work done}{total time} = \frac{m g h}{t}$

$\frac{P_{1}}{P_{2}} = \frac{\frac{m_{1} g h}{t_{1}}}{\frac{m_{2} g h}{t_{2}}} = \frac{m_{1}}{t_{1}} \frac{t_{2}}{m_{2}}$

$\frac{P_{1}}{P_{2}} = \frac{300 \times 2}{5 \times 50} = \frac{12}{5} = \frac{3 \sqrt{x}}{\sqrt{x} + 1}$

$\Rightarrow x = 16$

Q 7 :

A body of mass $1 kg$ begins to move under the action of a time dependent force $\vec{F} = (t \hat{i} + 3 t^{2} \hat{j}) N.$ where $\hat{i}$ and $\hat{j}$ are the unit vectors along $x$ and $y$ axis. The power developed by the above force, at the time $t = 2 s$ will be _______ W. [2023]

(100)

$\vec{F} = t \hat{i} + 3 t^{2} \hat{j}$

$\frac{m d \vec{v}}{d t} = t \hat{i} + 3 t^{2} \hat{j}$

$m = 1 kg$ , $\int_{0}^{v} d v = \int_{0}^{t} t d t \hat{i} + \int_{0}^{t} 3 t^{2} d t \hat{j}$

$\vec{v} = \frac{t^{2}}{2} \hat{i} + t^{3} \hat{j}$

$Power = \vec{F} \cdot \vec{v} = \frac{t^{3}}{2} + 3 t^{5}$

$At t = 2, Power = \frac{8}{2} + 3 \times 32 = 100$

Q 8 :

A body of mass 2 kg is initially at rest. It starts moving unidirectionally under the influence of a source of constant power P. Its displacement in 4 sec is $\frac{1}{3} α^{2} \sqrt{P} m .$ The value of $α$ will be ________ . [2023]

(4)

$\frac{1}{2} m v^{2} = P t$

$v = \sqrt{\frac{2 P t}{m}}$

$\frac{d x}{d t} = \sqrt{\frac{2 P t}{m}}$

$x = \sqrt{\frac{2 P}{m}} \frac{2}{3} {[t^{3 / 2}]}_{0}^{4}$

$x = \frac{16 \sqrt{P}}{3}$ $= \frac{1}{3} \times 16 \sqrt{P}$

$\Rightarrow α = 4$

Q 9 :

If the maximum load carried by an elevator is 1400 kg (600 kg – passenger + 800 kg – elevator), which is moving up with a uniform speed of $3 {ms}^{- 1}$ and the frictional force acting on it is 2000 N, then the maximum power used by the motor is _______ kW $(g = 10 {m/s}^{2})$ [2023]

(48)

$P_{\max} = F_{\max} \times v$

$F_{\max} = 1400 g + friction$

$= 14000 + 2000 = 16000$

$P_{\max} = 16000 \times 3 = 48000 W = 48 kW$

Q 10 :

A block of mass 5 kg starting from rest pulled up on a smooth inclined plane making an angle of $30 °$ with the horizontal with an effective acceleration of $1 {ms}^{- 2}$ . The power delivered by the pulling force at $t = 10 s$ from the start is ________ W. [Use $g = 10 {ms}^{- 2}$ ] (calculate the nearest integer value) [2023]

(300)

$F - 5 \sin 30 ° = 5 a$

$\Rightarrow F = 5 + 25 = 30 N$

$v_{10} = u + a t$

$\Rightarrow v_{10} = 0 + 1 (10) = 10 m/s$

$P_{10} = F v = 300 W$

Q 11 :

A body of mass 2 kg begins to move under the action of a time dependent force given by $\vec{F} = (6 t \hat{i} + 6 t^{2} \hat{j})$ $N$ . The power developed by the force at the time $t$ is given by:

$(6 t^{4} + 9 t^{5}) W$
$(3 t^{3} + 6 t^{5}) W$
$(9 t^{5} + 6 t^{3}) W$
$(9 t^{3} + 6 t^{5}) W$

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(4)

$\vec{F} = (6 t \hat{i} + 6 t^{2} \hat{j}) N$

$\vec{F} = m \vec{a} = (6 t \hat{i} + 6 t^{2} \hat{j})$

$\vec{a} = \frac{\vec{F}}{m} = (3 t \hat{i} + 3 t^{2} \hat{j})$

$\vec{v} = \int_{0}^{t} \vec{a} d t = \frac{3 t^{2}}{2} \hat{i} + t^{3} \hat{j}$

$P = \vec{F} \cdot \vec{v} = (9 t^{3} + 6 t^{5}) W$

Q 12 :

A particle (bob) of mass 1 kg is performing vertical circular motion. Then:

Power delivered by tension at point (4) is negative.
Power delivered by tension at point (2) is positive.
Power delivered by gravitational force at point (3) is zero.
Power delivered by gravitational force at point (4) is negative.

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(4)

$P = \vec{F} \cdot \vec{v},$ $Tension is always perpendicular to velocity$

Q 13 :

A body of mass 2 kg begins to move under the influence of time dependent force $\vec{F} = (2 t \hat{i} + 6 t^{2} \hat{j}) N$ , where $\hat{i}$ and $\hat{j}$ are unit vectors along $x$ and $y$ -axis respectively. The power produced by the force at $t = 2 s$ is _______ $W$ . [2026]

(200)

$Given: mass of body = 2 kg$

$\vec{F} = (2 t \hat{i} + 6 t^{2} \hat{j}) N$

$acceleration \vec{a} = \frac{\vec{F}}{m} = \frac{2 t \hat{i} + 6 t^{2} \hat{j}}{2} = (t \hat{i} + 3 t^{2} \hat{j}) {ms}^{- 2}$

$\vec{v} = \int_{0}^{2} \vec{a} \cdot d t = {[\frac{t^{2}}{2} \hat{i} + \frac{3 t^{3}}{3} \hat{j}]}_{0}^{2} = (2 \hat{i} + 8 \hat{j}) m / s$

${\vec{F}}_{t = 2 s} = (4 \hat{i} + 24 \hat{j}) N$

$P = \vec{F} \cdot \vec{v} = 8 + 192 = 200 W$

Topic Question Set

Q 1 :

A body is moving unidirectionally under the influence of a constant power source. Its displacement in time $t$ is proportional to [2024]

Q 2 :

A body of mass 2 kg begins to move under the action of a time dependent force given by $\vec{F} = (6 t \hat{i} + 6 t^{2} \hat{j}) N .$ The power developed by the force at the time $t$ is given by [2024]

Q 4 :

A sand dropper drops sand of mass m(t) on a conveyer belt at a rate proportional to the square root of speed (v) of the belt, i.e., $\frac{d m}{d t} \propto \sqrt{v}$ . If P is the power delivered to run the belt at constant speed then which of the following relationship is true? [2025]

Q 5 :

An object of mass 1000 g experiences a time dependent force $\vec{F} = (2 t \hat{i} + 3 t^{2} \hat{j}) N$ . The power generated by the force at time t is: [2025]

Q 6 :

The ratio of powers of two motors is $\frac{3 \sqrt{x}}{\sqrt{x} + 1}$ , that are capable of raising 300 kg water in 5 minutes and 50 kg water in 2 minutes respectively from a well of 100 m deep. The value of $x$ will be [2023]

Q 7 :

A body of mass $1 kg$ begins to move under the action of a time dependent force $\vec{F} = (t \hat{i} + 3 t^{2} \hat{j}) N.$ where $\hat{i}$ and $\hat{j}$ are the unit vectors along $x$ and $y$ axis. The power developed by the above force, at the time $t = 2 s$ will be _______ W. [2023]

Q 8 :

A body of mass 2 kg is initially at rest. It starts moving unidirectionally under the influence of a source of constant power P. Its displacement in 4 sec is $\frac{1}{3} α^{2} \sqrt{P} m .$ The value of $α$ will be ________ . [2023]

Q 9 :

If the maximum load carried by an elevator is 1400 kg (600 kg – passenger + 800 kg – elevator), which is moving up with a uniform speed of $3 {ms}^{- 1}$ and the frictional force acting on it is 2000 N, then the maximum power used by the motor is _______ kW $(g = 10 {m/s}^{2})$ [2023]

Q 11 :

A body of mass 2 kg begins to move under the action of a time dependent force given by $\vec{F} = (6 t \hat{i} + 6 t^{2} \hat{j})$ $N$ . The power developed by the force at the time $t$ is given by:

Q 12 :

A particle (bob) of mass 1 kg is performing vertical circular motion. Then:

Q 13 :

A body of mass 2 kg begins to move under the influence of time dependent force $\vec{F} = (2 t \hat{i} + 6 t^{2} \hat{j}) N$ , where $\hat{i}$ and $\hat{j}$ are unit vectors along $x$ and $y$ -axis respectively. The power produced by the force at $t = 2 s$ is _______ $W$ . [2026]

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Topic Question Set

Q 1 : A body is moving unidirectionally under the influence of a constant power source. Its displacement in time t is proportional to [2024]

Q 2 : A body of mass 2 kg begins to move under the action of a time dependent force given by F→=(6ti^+6t2j^)N. The power developed by the force at the time t is given by [2024]

Q 3 : A body of mass 4 kg is placed on a plane at a point P having coordinate (3, 4) m. Under the action of force F→=(2i^+3j^)N, it moves to a new point Q having coordinates (6, 10) m in 4 sec. The average power and instantaneous power at the end of 4 sec are in the ratio of : [2025]

Q 4 : A sand dropper drops sand of mass m(t) on a conveyer belt at a rate proportional to the square root of speed (v) of the belt, i.e., dmdt∝v. If P is the power delivered to run the belt at constant speed then which of the following relationship is true? [2025]

Q 5 : An object of mass 1000 g experiences a time dependent force F→=(2ti^+3t2j^)N. The power generated by the force at time t is: [2025]

Q 6 : The ratio of powers of two motors is 3xx+1, that are capable of raising 300 kg water in 5 minutes and 50 kg water in 2 minutes respectively from a well of 100 m deep. The value of x will be [2023]

Q 7 : A body of mass 1 kg begins to move under the action of a time dependent force F→=(ti^+3t2j^)N. where i^ and j^ are the unit vectors along x and y axis. The power developed by the above force, at the time t=2 s will be _______ W. [2023]

Q 8 : A body of mass 2 kg is initially at rest. It starts moving unidirectionally under the influence of a source of constant power P. Its displacement in 4 sec is 13α2P m. The value of α will be ________ . [2023]

Q 9 : If the maximum load carried by an elevator is 1400 kg (600 kg – passenger + 800 kg – elevator), which is moving up with a uniform speed of 3 ms-1 and the frictional force acting on it is 2000 N, then the maximum power used by the motor is _______ kW (g=10 m/s2) [2023]

Q 11 : A body of mass 2 kg begins to move under the action of a time dependent force given by F→=(6ti^+6t2j^)N. The power developed by the force at the time t is given by:

Q 12 : A particle (bob) of mass 1 kg is performing vertical circular motion. Then:

Q 13 : A body of mass 2 kg begins to move under the influence of time dependent force F→=(2ti^+6t2j^) N, where i^ and j^ are unit vectors along x and y-axis respectively. The power produced by the force at t=2 s is _______ W. [2026]

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Q 1 :

A body is moving unidirectionally under the influence of a constant power source. Its displacement in time $t$ is proportional to [2024]

Q 2 :

A body of mass 2 kg begins to move under the action of a time dependent force given by $\vec{F} = (6 t \hat{i} + 6 t^{2} \hat{j}) N .$ The power developed by the force at the time $t$ is given by [2024]

Q 4 :

A sand dropper drops sand of mass m(t) on a conveyer belt at a rate proportional to the square root of speed (v) of the belt, i.e., $\frac{d m}{d t} \propto \sqrt{v}$ . If P is the power delivered to run the belt at constant speed then which of the following relationship is true? [2025]

Q 5 :

An object of mass 1000 g experiences a time dependent force $\vec{F} = (2 t \hat{i} + 3 t^{2} \hat{j}) N$ . The power generated by the force at time t is: [2025]

Q 6 :

The ratio of powers of two motors is $\frac{3 \sqrt{x}}{\sqrt{x} + 1}$ , that are capable of raising 300 kg water in 5 minutes and 50 kg water in 2 minutes respectively from a well of 100 m deep. The value of $x$ will be [2023]

Q 7 :

A body of mass $1 kg$ begins to move under the action of a time dependent force $\vec{F} = (t \hat{i} + 3 t^{2} \hat{j}) N.$ where $\hat{i}$ and $\hat{j}$ are the unit vectors along $x$ and $y$ axis. The power developed by the above force, at the time $t = 2 s$ will be _______ W. [2023]

Q 8 :

A body of mass 2 kg is initially at rest. It starts moving unidirectionally under the influence of a source of constant power P. Its displacement in 4 sec is $\frac{1}{3} α^{2} \sqrt{P} m .$ The value of $α$ will be ________ . [2023]

Q 9 :

If the maximum load carried by an elevator is 1400 kg (600 kg – passenger + 800 kg – elevator), which is moving up with a uniform speed of $3 {ms}^{- 1}$ and the frictional force acting on it is 2000 N, then the maximum power used by the motor is _______ kW $(g = 10 {m/s}^{2})$ [2023]

Q 11 :

A body of mass 2 kg begins to move under the action of a time dependent force given by $\vec{F} = (6 t \hat{i} + 6 t^{2} \hat{j})$ $N$ . The power developed by the force at the time $t$ is given by:

Q 12 :

A particle (bob) of mass 1 kg is performing vertical circular motion. Then:

Q 13 :

A body of mass 2 kg begins to move under the influence of time dependent force $\vec{F} = (2 t \hat{i} + 6 t^{2} \hat{j}) N$ , where $\hat{i}$ and $\hat{j}$ are unit vectors along $x$ and $y$ -axis respectively. The power produced by the force at $t = 2 s$ is _______ $W$ . [2026]