Problem 1 : A block of 2 kg is brought up from the bottom to the top along a rough incline of length 10 m and height 5 m by applying an external force parallel to the surface. If the coefficient of kinetic friction between surfaces is 0.1, find work done by the friction during the motion. (consider, g = 10
m
/
s
2
m /
s
2
).
Solution : We see here that there are four forces on the block : (i) weight (ii) normal force (iii) friction and (iv) force, "F" parallel to incline. The magnitude of external force is not given. We are, however, required to find work by friction. Thus, we need to know the magnitude of friction and its direction. As the block moves up, kinetic friction acts downward. Here, displacement is equal to the length of incline, which is 10 m.
From the figure, it is clear that friction force is given as :
F
k
=
μ
k
N
=
μ
k
m
g
cos
θ
Here,
sin
θ
=
5
10
and
cos
θ
=
(
1

sin
2
θ
)
⇒
F
=
0.1
x
2
x
10
x
(
1

5
2
10
2
)
=
2
x
0.866
=
1.732
N
F
k
=
μ
k
N
=
μ
k
m
g
cos
θ
Here,
sin
θ
=
5
10
and
cos
θ
=
(
1

sin
2
θ
)
⇒
F
=
0.1
x
2
x
10
x
(
1

5
2
10
2
)
=
2
x
0.866
=
1.732
N
To evaluate work in terms of "Frcosφ", we need to know the angle between force and displacement. In this case, this angle is 180° as shown in the figure below.
We denote "φ" instead of "θ" as angle between force and displacement to distinguish this angle from the angle of incline.
W
=
F
r
cos
φ
W
=
F
r
cos
φ
⇒
W
F
=
F
r
cos
φ
=
1.732
x
10
x
cos
180
0
⇒
W
F
=
1.732
x
10
x
(

1
)
=

17.32
J
⇒
W
F
=
F
r
cos
φ
=
1.732
x
10
x
cos
180
0
⇒
W
F
=
1.732
x
10
x
(

1
)
=

17.32
J
This example brings out the concept of work by named force (friction). The important point to note here is that we could calculate work by friction even though we were not knowing the magnitude of force of external force, "F". Yet another point to note here is that computation of work by friction is actually independent of  whether block is accelerated or not? In addition, this example illustrates how the evaluation of the cosine of angle between force and displacement determines the sign of work.