Question 231: Work, Energy and Power & Circular Motion

Question

A force $\mathrm{f}=\mathrm{x}^2 \mathrm{y} \hat{\mathrm{i}}+\mathrm{y}^2 \hat{\mathrm{j}}$ acts on a particle in a plane $\mathrm{x}+\mathrm{y}=10$. The work done by this force during a displacement from $(0,0)$ to $(4 \mathrm{~m}, 2 \mathrm{~m})$ is _________ Joule (round off to the nearest integer)

Accepted Answer

To calculate the work done by the force $\mathrm{f} = \mathrm{x}^2 \mathrm{y} \hat{\mathrm{i}} + \mathrm{y}^2 \hat{\mathrm{j}}$ during the displacement from $(0,0)$ to $(4 	ext{ m}, 2 	ext{ m})$, we proceed as follows:
Given the constraint that the particle moves in the plane $\mathrm{x} + \mathrm{y} = 10$, we can express $\mathrm{y}$ in terms of $\mathrm{x}$:
$  y = 10 - x  $
The work done, $W$, is the integral of the force along the path of the displacement:
$  W = \int_{0}^{4} x^2 (10-x) \, dx + \int_{0}^{2} y^2 \, dy  $
Calculate the integral of the $\hat{\mathrm{i}}$ component:
$ \begin{aligned} & \int_{0}^{4} x^2 (10 - x) \, dx = \int_{0}^{4} (10x^2 - x^3) \, dx \ & = \left[ \frac{10x^3}{3} - \frac{x^4}{4} ight]_{0}^{4} \ & = \left( \frac{10(4)^3}{3} - \frac{(4)^4}{4} ight) - \left( \frac{10(0)^3}{3} - \frac{(0)^4}{4} ight) \ & = \frac{640}{3} - \frac{256}{4} \ \end{aligned} $
Calculate the integral of the $\hat{\mathrm{j}}$ component:
$ \begin{aligned} & \int_{0}^{2} y^2 \, dy = \left[ \frac{y^3}{3} ight]_{0}^{2} \ & = \frac{(2)^3}{3} - \frac{(0)^3}{3}\ & = \frac{8}{3} \ \end{aligned} $
Combine the results:
$ W = \left( \frac{640}{3} - 64 ight) + \frac{8}{3} $
Simplify:
$ \begin{aligned} W & = \frac{640}{3} - 64 + \frac{8}{3} \ & = \frac{640 + 8}{3} - 64 \ & = \frac{648}{3} - 64 \ & = 216 - 64 \ & = 152 	ext{ J} \end{aligned} $
Thus, the work done by the force is 152 Joules.

Work, Energy and Power & Circular Motion

A force \mathrm{f}=\mathrm{x}^2 \mathrm{y} \hat{\mathrm{i}}+\mathrm{y}^2 \hat{\mathrm{j}} acts on a particle in a plane \mathrm{x}+\mathrm{y}=10. The work done by this force during a displacement from (0,0) to (4 \mathrm{~m}, 2 \mathrm{~m}) is _________ Joule (round off to the nearest integer)

Integer / Numeric Answer Type