Engineering Mechanics

(^142) „„„„„ A Textbook of Engineering Mechanics
Solution. Given: Load (W ) = 300 N; Force (P 1 ) = 60 N and angle at which force is
inclined (θ) = 30°,
Let α = Angle of inclination of the plane.
First of all, consider the load lying on a smooth plane inclined at an angle (α) with the
horizontal and subjected to a force of 60 N acting at an angle 30° with the plane as shown in Fig.
8.18 (a).
Fig. 8.18.
We know that in this case, because of the smooth plane μ = 0 or φ = 0. We also know that
the force required, when the load is at the point of sliding upwards (P),
sin ( ) sin
60 300
cos ( – ) cos 30
W
α+φ α
=× = ×
θ φ °
sin
300 346.4 sin
0.866
α
=× = α
...(Q φ = 0)
or
60
sin 0.1732
346.4
α= = or α = 10°
Now consider the load lying on the rough plane inclined at an angle of 10° with the horizontal
as shown in Fig. 8.18. (b). We know that in this case, μ = 0.3 = tan φ or φ = 16.7°.
We also know that force required to move the load up the plane,
sin ( ) sin (10 16.7 )
300 N
cos cos 16.7
PW
α+φ °+ °
=× = ×
φ°
sin 26.7 0.4493
300 300 140.7 N
cos 16.7 0.9578
°
=× =× =
°
Ans.
Alternative method
Ist case
Given: In this case load (P) = 60 N; Angle (θ) = 30° and force of friction F = 0 (because of
smooth plane). Resolving the forces along the inclined plane,
60 cos 30° = 300 sin α
∴
60 cos 30 60 0.866
sin 0.1732
300 300
°×
α= = = or α = 10°
2nd case
Given: In this case, coefficient of friction (μ) = 0.3 = tan φ or φ = 16.7°
Let P = Force required to move the load up the plane,
R = Normal reaction, and
F = Force of friction equal to 0.3 R.
Resolving the forces along the plane,
P = F + 300 sin 10° = 0.3 R + (300 × 0.1732) = 0.3 R + 51.96 N ...(i)