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.                                              

report!

explanation:

physics means: "the branch of science concerned with the nature and properties of matter and energy. the subject matter of physics, distinguished from that of chemistry and biology, includes mechanics, heat, light and other radiation, sound, electricity, magnetism, and the structure of atoms."

total distance/total time

(85 + 82)/(5+3)

167/8

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