A particle moves along a straight path through a displacement d = 1.5i + cj while a force F = -8.5i + -6.5j acts on it. The displacement is measured in meters and the force is measured in Newtons. (Other forces also act on the particle.)
1. What is the value of c if the work done by F on the particle is (a) zero, (b) 7.8 J, and (c) -5.6 J

(a) c = - 1.96

(b) c = - 3.16

(c) c = - 1.1

The work done (W) on a particle is the dot product of its force vector (F) and its displacement vector (d).

i.e  

W = F . d          (i)

Given in the question;

F = -8.5i + -6.5j

d = 1.5i + cj

Calculate the dot product of F and d as follows;

F . d =  (-8.5i + -6.5j) . (1.5i + cj)

F . d =  (-8.5 x 1.5) i² + (-6.5 x c) j²    Note:  [i² = j² = 1]

F . d =  -12.75 - 6.5c

Substitute F . d =  -12.75 - 6.5c  into equation (i) as follows;

W = -12.75 - 6.5c    (ii)

(a) when W = 0

Substitute W = 0 into equation (ii) as follows;

0 = -12.75 - 6.5c

6.5c = - 12.75  

Solve for c;

c = - 12.75 / 6.5

c = - 1.96

(b) when W = 7.8

Substitute W = 7.8 into equation (ii) as follows;

7.8 = -12.75 - 6.5c

6.5c = - 12.75 - 7.8

6.5c = - 20.55

Solve for c;

c = - 20.55 / 6.5

c = - 3.16

(c) when W = -5.6

Substitute W = -5.6 into equation (ii) as follows;

-5.6 = -12.75 - 6.5c

6.5c = - 12.75 + 5.6

6.5c = - 7.15

Solve for c;

c = - 7.15 / 6.5

c = - 1.1

if they have a finite resultant