Engineering Mechanics

(^132) „„„„„ A Textbook of Engineering Mechanics

1. Minimum force (P 1 ) which will keep the body in equilibrium, when it is at the point of sliding
   downwards.

Fig. 8.8.

In this case, the force of friction (F 1 = μ.R 1 ) will act upwards, as the body is at the point of

sliding downwards as shown in Fig. 8.8 (a). Now resolving the forces along the plane,

P 1 = W sin α – μ.R 1 ...(i)

and now resolving the forces perpendicular to the plane.

R 1 = W cos α ...(ii)

Substituting the value of R 1 in equation (i),

P 1 = W sin α – μ W cos α = W (sin α – μ cos α)

and now substituting the value of μ = tan φ in the above equation,

P 1 = W (sin α – tan φ cos α)

Multiplying both sides of this equation by cos φ,

P 1 cos φ = W (sin α cos φ – sin φ cos α) = W sin (α – φ)

∴ 1

sin ( – )

cos

PW

α φ

=×

φ

2. Maximum force (P 2 ) which will keep the body in equilibrium, when it is at the point of sliding
   upwards.
   In this case, the force of friction (F 2 = μ.R 2 ) will act downwards as the body is at the point of
   sliding upwards as shown in Fig. 8.8 (b). Now resolving the forces along the plane,
   P 2 = W sin α + μ.R 2 ...(i)
   and now resolving the forces perpendicular to the plane,
   R 2 = W cos α ...(ii)
   Substituting the value of R 2 in equation (i),
   P 2 = W sin α + μ W cos α = W (sin α + μ cos α)
   and now substituting the value of μ = tan φ in the above equation,
   P 2 = W (sin α + tan φ cos α)
   Multiplying both sides of this equation by cos φ,
   P 2 cos φ = W (sin α cos φ + sin φ cos α) = W sin (α + φ)

∴ 2

sin ( )

cos

PW

α+φ

=×

φ