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Work Done When Force Is Varying

Question

The work done by the force $\to F = A (y^{2}^i + 2 x^{2}^j)$ , where $A$ is a constant and $x$ & $y$ are in meters around the path shown is :

$A d$
$z e r o$
$A d^{2}$
$A d^{3}$

A

A d^{2}

B

A d^{3}

C

z e r o

D

A d

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Solution

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Variable force is apllied across the path $O A$ to $A B$ to $B C$ to $C O$

$W = \int ¯ F . d ¯ r$

$= \int A (y^{2} ¯ i + 2 x^{2} ¯ j) . (d x ¯ i + d y . ¯ j)$

$= \int A (y^{2} d x + 2 x^{2} d y)$

Now following the path of displacement along $O A$ the $y = 0$

$W_{O A} = \int_{x = 0}^{x = d} A (0 + 2 x^{2} .0) = 0 + 0 = 0$

Similerly, for $W_{A B}, x = d$

$W_{A B} = A [0 + 2 d^{2} d] = 2 A d^{3}$

Similarly, for $W_{B C}, x = d, y = d$

$W_{B C} = \int_{x = d}^{x = 0} A (d^{2} d x + 0)$

$W_{B C} = A [d^{2} (- d) + 0] = - A d^{3}$

$W_{C O} = A [0 + 0]$

$W = 0 + 2 A d^{3} - A d^{3} + 0 = A d^{3}$

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Similar Questions

Q1

The work done by the force

\to F = A (y^{2}^i + 2 x^{2}^j)

, where

A

is a constant and

x

&

y

are in meters around the path shown is :

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Q2

The work done by a force

\to F = x^{2}^i + y^{2}^j

around the path shown in the figure is:

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Q3

If

K_{1}, K_{2} a n d K_{3}

are equilibrium constant for formation of

A D, A D_{2} a n d A D_{3}

respectively as follow

A + D ⇌ A D, A D + D ⇌ A D_{2}, A D_{2} + D ⇌ A D_{3}

then equilibrium constant 'K" for

A + 3 D ⇌ A D_{3}

is related as

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Q4

If

K_{1}, K_{2} a n d K_{3}

are equilibrium constant for formation of

A D, A D_{2} a n d A D_{3}

respectively as follow

A + D ⇌ A D, A D + D ⇌ A D_{2}, A D_{2} + D ⇌ A D_{3}

then equilibrium constant 'K" for

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is related as

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Q5

If

K_{1}, K_{2}, K_{3}

are equilibrium constant for formation of

A D_{1}, A D_{2}, A D_{3}

, respectively as follows

A + D ⇌ A D_{1}

,

A D_{1} + D ⇌ A D_{2}, A D_{2} + D ⇌ A D_{3}

. Then equilibrium constant 'K' for

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