Question

The force acting on a particle varies as in the figure below. (The x-axis is marked in increments of 5.00 m.)A coordinate plane has a horizontal axis labeled x (m) and a vertical axis labeled F_x (N). There are five points, connected by straight lines in order from A to E.

• Point A is at (0,0). Point B is at (20,6). Point C is at (40,0). Point D is at (50,−3). Point E is at (60,0).

Find the work done by the force as the particle moves across the following distances.

(a) from x = 0 m

 to x = 40.0 m

__ J

(b) from x = 40.0 m

 to x = 60.0 m

__ J

(c) from x = 0 m

 to x = 60.0 m

 __J

Answer 1

First, we have to calculate the force along the displacement:

We proceed to find the following equations:

$F_{AB}=\frac{3x}{10} \\ F_{BC}=12-\frac{3x}{10} \\ F_{CD}=12-\frac{3x}{10} \\ F_{DE}=\frac{3x}{10}-18$

With this information we can find the work done:

$W_{AC}=\int^C_A F\,{dx}=\int^B_A[F_{AB}]\,{dx}+\int^C_B[F_{BC}]\,{dx} \\ \text{ } \\W_{AC} =\int^{20}_0[3x/10]\,{dx}+\int^{40}_{20}[12-3x/10]\,{dx} \\ \text{ } \\W_{AC} =[3x^2/20]^{20}_0+[12x-3x^2/20]^{40}_{20} \\ \text{ } \\W_{AC} =[3(20)^2/20-0]+[12(40-20)-(3/20)(40^2-20^2)]J \\ \text{ } \\W_{AC} =[3(20)+12(20)-(3)(60)]J=(60+240-180)J=120\,J$

For the second part:

$W_{CE}=\int^E_C F\,{dx}=\int^D_C[F_{CD}]\,{dx}+\int^E_D[F_{DE}]\,{dx} \\ \text{ } \\W_{CE} =\int^{50}_{40}[12-3x/10]\,{dx}+\int^{60}_{50}[-18+3x/10]\,{dx} \\ \text{ } \\W_{CE} =[12x-3x^2/20]^{50}_{40}+[-18x+3x^2/20]^{60}_{50} \\ \text{ } \\W_{CE} =([12(50-40)-(3/20)(50^2-40^2)]+[-18(60-50)+(3/20)(60^2-50^2)])J \\ \text{ } \\W_{CE} =(12(10)-(3/2)(90)-18(10)+(3/2)(110))J=(120-135-180+165)J \\ W_{CE}=-30\,J$

The total work for the whole process will be the sum:

$W_T=W_{AC}+W_{CE}=(120-30)J=90\,J$

In conclusion, the work done by the force as the particle moves across the following distances is:
(a) from x = 0 m  to x = 40.0 m W=120 J(b) from x = 40.0 m  to x = 60.0 m W= -30 J(c) from x = 0 m to x = 60.0 m W= 90 J

Reference:

• Sears, F. W., & Zemansky, M. W. (1973). University physics.