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physics.fisikastudycenter.com, physics olympiads series problem 20. Inclined plane, force, statics friction and newton's laws.

Problem 20
A mass m = 10 kg is held at rest on an inclined planed, whose slope is θ = 53°, by means of a horizontal force F (see Figure).


The coefficient of statics friction is μ = 0.2, find the maximum F allowed before the body starts to move up the plane, use the acceleration due to gravity g = 10 m ⋅ s⁻² and tan 53°= 4/3

Solution
We take the x - direction up the slope and y-direction normal to it.



At the moment when the mass begins to move, F = F_max and the resultant forces are :

Σ F_x = F_max cos θ − mg sin θ − μ_s N
0 = F_max (0.6) − (10)(10)(0.8) − (0.2) N
0.6 F_max = 80 + 0.2 N
Equation (1)

Σ F_y = N − F_max sin θ − mg cos θ

Both forces must vanish, so we can use the second equation to write

N = F_max sin θ + mg cos θ
N = F_max (0.8) + (10)(10)(0.6)
N = 0.8 F_max + 60
Equation (2)

and then from combining (1) and (2) we get:



Literatures:
-Physics with Answers, A.R King and O. Regev
-Mekanika I, Yohanes Surya