Question

A variable force $F=5x$ $N$ acts on a body moving along x-axis. Find the work done by this force in displacing the body from $x=2$ m to $x=4$ m.
(K is constant)

• A. $\left(\frac{205}{2}K\right) J$
• B. $\left(\frac{105}{2}K\right) J$
• C. $52K J$
• D. $51K J$

Answer 1

Answer: (B)

$\left(\frac{105}{2}K\right) J$

Answer 2

When force remains constant and motion occurs along a straight path, work is calculated as the product of force and distance traveled. However, when force varies within the distance traveled, calculus aids in resolving the problem by breaking it down into infinitesimal steps where force can be considered constant.
Consequently, the work accomplished by a variable force is expressed as,
$W = ∫F dx$
Here, the force is described by $F = 5x$ N, and the displacement ranges from $x = 2$ m to $x = 4$ m.
Now the work performed,
$W=∫Fdx$
$W=∫5x dx$

$W= 5∫_2^4 x dx$

$W = 5[\frac {x^2}{2}]_2^4$

$W=\frac 52[4^2-2^2]$
$W= 5\times 6$
$W=30\ J$

So, the correct option is (B): $\left(\frac{105}{2}K\right) J$