Show that if the coefficient of friction were larger than a certain value, then the box would not start moving no matter how large a horizontal force is applied. What is this critical value for the friction coefficient? Please be sure your reasoning is clear.

u_critical = 1.33

Explanation:

Given:

- The complete question is:

[Box on Ramp] A 6.00-kg box sits on a ramp that is inclined at 37.0° above the horizontal. The coefficient of kinetic friction between the box and the ramp is th=0.300.

Find:

Show that if the coefficient of friction were larger than a certain value, then the box would not start moving no matter how large a horizontal force is applied. What is this critical value for the friction coefficient? Please be sure your reasoning is clear.

Solution:

- The equation of motion for the block on the ramp is given by:

                                Fhcos(θ) - mgsin(θ) - Ff = m*a

Where,    

m = mass of block , Ff = Frictional Force , Fh = Horizontal applied force, θ = Angle of the slope , a = acceleration of box

- When the block does not move then a = 0, we have:

                                Ff = Fhcos(θ) - mgsin(θ)

- The frictional force of the box Ff is given by:

                                Ff =< u*N

Where,

 N = Normal contact force, u = coefficient of static friction.

- The contact force N is given by the equilibrium equation in the direction normal to slope is:

                                Fhsin(θ) + mgcos(θ) = N

- The frictional force Ff is given by:

                                Ff =< [ Fhsin(θ) + mgcos(θ) ]*u

- Then substitute the two bold equations:

                               Fhcos(θ) - mgsin(θ) =< [ Fhsin(θ) + mgcos(θ) ]*u  

                               u >= [ Fhcos(θ) - mgsin(θ) ] / [ Fhsin(θ) + mgcos(θ) ]

- The critical value for u is given for limit Fh -> ∞ is:

             u>= Lim ( Fh -> ∞) {  [ Fhcos(θ) - mgsin(θ) ] / [ Fhsin(θ) + mgcos(θ) ]}

             u>= Lim ( Fh -> ∞) { [cos(θ) - mgsin(θ) / Fh ] / [sin(θ) + mgcos(θ)/Fh ]}

- Evaluate limit, we get:

            u >= cos(θ) / sin(θ)

            u >= cos(37) / sin(37)              

            u >= 1.33    

- The critical value for coefficient of friction is u_critical = 1.33.