Q. 64 An object moving in the xy-plane... [FREE SOLUTION]

91影视

Chapter 10: Q. 64 (page 259)

An object moving in the xy-plane is subjected to the force $\overset{鈫�}{F} = (2 x y \hat{谋} + 3 y \hat{确}) N$ , where x and y are in m.
a. The particle moves from the origin to the point with coordinates $(a, b)$ by moving first along the x-axis to $(a, 0)$ , then parallel to the y-axis. How much work does the force do?
b. The particle moves from the origin to the point with coordinates $(a, b)$ by moving first along the y-axis to $(0, b)$ , then parallel to the x-axis. How much work does the force do?
c. Is this a conservative force?

Short Answer

Expert verified

(a) The work done $= \frac{3 b^{2}}{2}$

(b) The work done $= \frac{3 b^{2}}{2} + a^{2} b$

Step by step solution

Given information (part a)

The force acting on the object is given as,

$\overset{鈫�}{F} = (2 x y \hat{i} + 3 y \hat{j}) N$

Explanation (part a)

The work done by the force is

$w_{1} = {鈭�}_{x_{i}}^{x_{f}} F d x w h e r e x_{i} a n d x_{f} i s i n i t i a l a n d f i n a l p o s i t i o n r e s p e c t i v e l y . T h e p a r t i c a l i s m o v e d a l o n g t h e x - a x i s f r o m (0, 0) t o (a, 0) x_{i} = 0, x_{f} = a \overset{鈫�}{F} = (2 x y \hat{i} + 3 y \hat{j}) N y = 0 for particle moving along x - axis w_{1} = {鈭�}_{x_{i}}^{x_{f}} (2 xy \hat{i} + 3 y \hat{j}) d x \hat{i} w_{1} = {鈭�}_{0}^{a} (0) d x \hat{i} = 0$

Now the particle is moving parallel to the y-axis.

$w_{2} = {鈭�}_{y_{i}}^{y_{f}} (2 x y \hat{i} + 3 y \hat{j}) d y S i n c e t h e p a r t i c l e i s m o v i n g f r o m (a, 0) t o (a, b) y_{i} = 0, y_{f} = b, x = a w_{2} = {鈭�}_{0}^{b} (2 a y \hat{i} + 3 y \hat{j}) d y \hat{j} = {鈭�}_{0}^{b} 3 y d y = {[\frac{3 y^{2}}{2}]}_{0}^{b} = (\frac{3 b^{2}}{2} - 0) = \frac{3 b^{2}}{2} T o t a l w o r k d o n e w = w_{1} + w_{2} w = 0 + \frac{3 b^{2}}{2} w = \frac{3 b^{2}}{2}$

Given information (part b)

The force acting on the object is given as,

$\overset{鈫�}{F} = (2 x y \hat{i} + 3 y \hat{j}) N$

Explanation (part b)

The work done by the force

$w = {鈭�}_{y_{i}}^{y_{f}} (2 x y \hat{i} + 3 y \hat{j}) d y y_{i} a n d y_{f} i s t h e i n i t i a l a n d f i n a l p o s i t i o n r e s p e c t i v e l y T h e p a r t i c l e i s m o v i n g a l o n g t h e y - a x i s f r o m (0, 0) t o (0, b) y_{i} = 0 a n d y_{i} = b w = {鈭�}_{0}^{b} (2 a y \hat{i} + 3 y \hat{j}) d y \hat{j} w = {鈭�}_{0}^{b} [3 y] d y w = {[\frac{3 y^{2}}{2}]}_{0}^{b} w = (\frac{3 b^{2}}{2} - 0) w = \frac{3 b^{2}}{2}$

Now the particle is moved parallel to the x-axis from $(0, b)$ to $(a, b)$ .

$w_{2} = {鈭�}_{x_{i}}^{x_{f}} (2 x y \hat{i} + 3 y \hat{j}) d x \hat{i} x_{i} = 0, x_{f} = a, y = b w_{2} = {鈭�}_{0}^{a} (2 x b \hat{i} + 3 y \hat{j}) d x \hat{i} w_{2} = {鈭�}_{0}^{a} (2 x b) d x w_{2} = {[x^{2} b]}_{0}^{a} w_{2} = a^{2} b T o t a l w o r k d o n e w = w_{1} + w_{2} w = \frac{3 b^{2}}{2} + a^{2} b$

Given information (part c)

The force acting on the object is given as,

$\overset{鈫�}{F} = (2 x y \hat{i} + 3 y \hat{j}) N$

Explanation (part c)

The force is non-conservative since the work done by the force is dependent on the path followed by the particle. Work done is different in both the paths followed while the particle reach (a, b)