Answer :
a) c = 1.85
b) c = 0.8
c) c = 2.33
Explanation:
a)
The displacement of the particle is given by
[tex]d=2.2i+cj[/tex]
While the force applied on the particle is
[tex]F=3.2i-3.8 j[/tex]
So we have a problem in 2-dimensions.
The work done on the particle is given by the scalar product between force and displacement:
[tex]W=F\cdot d[/tex] (1)
Here the work done on the particle is zero, so
W = 0
Therefore from eq(1) we find:
[tex]0=(3.2i-3.8j)\cdot (2.2i+cj)=7.04-3.8c\\3.8c=7.04\\c=\frac{7.04}{3.8}=1.85[/tex]
b)
In this problem, the work done on the particle is
[tex]W=4.0 J[/tex]
The force and displacement are still
[tex]d=2.2i+cj[/tex] (displacement)
[tex]F=3.2i-3.8 j[/tex] (force)
Therefore, by calculting the scalar product between force and displacement and equating it to the work done (4.0 J), we find:
[tex]W=F\cdot d[/tex]
[tex]4.0 =(3.2i-3.8j)\cdot (2.2i+cj)=7.04-3.8c\\3.8c=3.04\\c=\frac{3.04}{3.8}=0.8[/tex]
c)
In this problem instead, the work done on the particle is negative:
[tex]W=-1.8 J[/tex]
As before, the force and displacement are
[tex]d=2.2i+cj[/tex] (displacement)
[tex]F=3.2i-3.8 j[/tex] (force)
And so again, we calculate the scalar product between force and displacement and we equate it to the work done on the particle, -1.8 J.
Doing so, we find:
[tex]W=F\cdot d[/tex]
[tex]-1.8=(3.2i-3.8j)\cdot (2.2i+c)=7.04-3.8c\\3.8c=8.84\\c=\frac{8.84}{3.8}=2.33[/tex]