Amug rests on an inclined surface, as shown in (figure 1) , θ=17∘. what is the magnitude of the frictional force exerted on the mug? what is the minimum coefficient of static friction required to keep the mug from sliding?

Refer to the figure shown below.

g =  9.8 m/s², the acceleration due to gravity.
W = mg, the weight of the mug.
θ = 17°, the angle of the ramp.

Let μ = the coefficient of static friction.

The force acting down the ramp is
F = W sin θ = W sin(17°) = 0.2924W N
The normal reaction is
N = W cosθ = W cos(17°) = 0.9563W N
The resistive force due to friction is
R = μN = 0.9563μW N

For static equilibrium,
μN = F
0.9563μW =0.2924W
μ = 0.3058

The frictional force is F = μN = 0.2924W
The minimum value of μ required to prevent the mug from sliding satisfies
 the condition
R > F
0.9563μW > 0.2924W
μ > 002924/.9563 = 0.306

The frictional force is 0.2924mg, where m = the mass of the mug.
The minimum coefficient of static friction is 0.306