JEE Advanced PYQ Work, Energy and Power | Chapterwise Previous Year Questions

Q 1 :

The work done on a particle of mass $m$ by a force, $K [\frac{x}{{(x^{2} + y^{2})}^{3 / 2}} \hat{i} + \frac{y}{{(x^{2} + y^{2})}^{3 / 2}} \hat{j}]$ ( $K$ being a constant of appropriate dimensions), when the particle is taken from the point $(a, 0)$ to the point $(0, a)$ along a circular path of radius $a$ about the origin in the $x - y$ plane is [2013]

$\frac{2 K π}{a}$
$\frac{K π}{a}$
$\frac{K π}{2 a}$
$0$

1

2

3

4

(4)

$Radius of circular path = a$

$The equation of circle is x^{2} + y^{2} = a^{2}$

$Given : force$

$\vec{F} = K [\frac{x \hat{i}}{{(x^{2} + y^{2})}^{3 / 2}} + \frac{y \hat{j}}{{(x^{2} + y^{2})}^{3 / 2}}]$

$\vec{F} = K [\frac{x \hat{i}}{{(a^{2})}^{3 / 2}} + \frac{y \hat{j}}{{(a^{2})}^{3 / 2}}]$

[IMAGE 106]

$\vec{F} = \frac{K}{a^{3}} [x \hat{i} + y \hat{j}]$

The force acts radially outwards as shown in the figure and the displacement is tangential to the circular path.

Here the angle between the force which acts radially outwards and displacement which is tangential to the circular path is $90 °$

$∴$ $Work done, W = F S \cos θ = 0$

Q 2 :

If $W_{1}$ , $W_{2}$ and $W_{3}$ represent the work done in moving a particle from $A$ to $B$ along three different paths 1, 2 and 3 respectively (as shown) in the gravitational field of a point mass $m$ , find the correct relation between $W_{1}$ , $W_{2}$ and $W_{3}$ . [2003]

[IMAGE 107]

$W_{1} > W_{2} > W_{3}$
$W_{1} = W_{2} = W_{3}$
$W_{1} < W_{2} < W_{3}$
$W_{2} > W_{1} > W_{3}$

1

2

3

4

(2)

In a conservative field work done does not depend on the path i.e., path independent. The gravitational field is a conservative field.

$∴$ $W_{1} = W_{2} = W_{3}$

Q 3 :

A light inextensible string that goes over a smooth fixed pulley as shown in the figure connects two blocks of masses 0.36 kg and 0.72 kg. Taking $g = 10 {m/s}^{2}$ , find the work done (in joules) by the string on the block of mass 0.36 kg during the first second after the system is released from rest. [2009]

[IMAGE 108]

(8)

[IMAGE 109]

When the system is released,

$T - m g = m a ...(i)$

$M g - T = M a ...(ii)$

$From eq. (i) & (ii)$

$a = \frac{(M - m) g}{M + m} = \frac{g}{3}$

$and T = \frac{4 m g}{3}$

$For block m = 0.36 kg$

$u = 0, a = \frac{g}{3}, t = 1, s = ?$

$s = u t + \frac{1}{2} a t^{2} = 0 + \frac{1}{2} \times \frac{g}{3} \times 1^{2} = \frac{g}{6} (m = 0.72 kg)$

$∴$ $Work done by the string on m$

$W = T s \cos 0 ° = \frac{4 m g}{3} \times \frac{g}{6} \times 1 = \frac{4 \times 0.36 \times 10 \times 10}{3 \times 6} = 8 J$

Q 4 :

A particle is moved along a path $AB - BC - CD - DE - EF - FA$ , as shown in figure, in presence of a force $\vec{F} = (α y \hat{i} + 2 α x \hat{j}) N,$ where $x$ and $y$ are in meter and $α = - 1 {N m}^{- 1}$ . The work done on the particle by this force $\vec{F}$ will be _____ Joule. [2019]

[IMAGE 110]

(0.75)

Given : Force, $\vec{F} = (α y \hat{i} + 2 α x \hat{j})$

$and α = - 1 {N m}^{- 1}$

$We know that d W = \vec{F} \cdot d \vec{r} = (α y \hat{i} + 2 α x \hat{j}) \cdot (d x \hat{i} + d y \hat{j})$

$∴$ $d W = α y d x + 2 α x d y$

$Work done$

$From A \to B, d y = 0, as y = 1$

$∴$ $W_{1} = \int_{0}^{1} α y d x = α \int_{0}^{1} d x = α$

$From B \to C, d x = 0, as x = 1$

$∴$ $W_{2} = \int_{1}^{0.5} 2 α x d y = \int_{1}^{0.5} 2 α d y = 2 α (0.5 - 1) = - α$

$From C \to D, d y = 0, as y = 0.5$

$∴$ $W_{3} = \int_{1}^{0.5} α \times 0.5 d x = \frac{α}{2} (0.5 - 1) = - \frac{α}{4}$

$From D \to E, d x = 0, as x = 0.5$

$∴$ $W_{4} = \int_{0.5}^{0} 2 α \times 0.5 d y = α (0 - 0.5) = - \frac{α}{2}$

$From E \to F, d y = 0, as y = 0$

$∴$ $W_{5} = \int α \times 0 d x = 0$

$From F \to A, d x = 0, as x = 0$

$∴$ $W_{6} = \int 2 α \times 0 d y = 0$

$∴$ $Total work done W = W_{1} + W_{2} + W_{3} + W_{4} + W_{5} + W_{6}$

$W = α - α - \frac{α}{4} - \frac{α}{2} = - \frac{3 α}{4} = - \frac{3 (- 1)}{4} = 0.75 J$

Topic Question Set

Q 2 :

If $W_{1}$ , $W_{2}$ and $W_{3}$ represent the work done in moving a particle from $A$ to $B$ along three different paths 1, 2 and 3 respectively (as shown) in the gravitational field of a point mass $m$ , find the correct relation between $W_{1}$ , $W_{2}$ and $W_{3}$ . [2003]

[IMAGE 107]

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Topic Question Set

Q 1 : The work done on a particle of mass m by a force, K[x(x2+y2)3/2i^+y(x2+y2)3/2j^] (K being a constant of appropriate dimensions), when the particle is taken from the point (a,0) to the point (0,a) along a circular path of radius a about the origin in the x–y plane is [2013]

Q 2 : If W1, W2 and W3 represent the work done in moving a particle from A to B along three different paths 1, 2 and 3 respectively (as shown) in the gravitational field of a point mass m, find the correct relation between W1, W2 and W3. [2003] [IMAGE 107]

Q 4 : A particle is moved along a path AB-BC-CD-DE-EF-FA, as shown in figure, in presence of a force F→=(αyi^+2αxj^) N, where x and y are in meter and α=-1 N m-1. The work done on the particle by this force F→ will be _____ Joule. [2019] [IMAGE 110]

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Q 2 :

If $W_{1}$ , $W_{2}$ and $W_{3}$ represent the work done in moving a particle from $A$ to $B$ along three different paths 1, 2 and 3 respectively (as shown) in the gravitational field of a point mass $m$ , find the correct relation between $W_{1}$ , $W_{2}$ and $W_{3}$ . [2003]

[IMAGE 107]