Physics, find the coefficent of static fristion for the furniture/floor surfaces

Question

Supposed the minimum magnitude of the force F needed to push the furniture at this same angle in order to get it to start sliding is 657N. What is the coefficient of static friction for the furniture/floor surfaces?

There is a picutre with this question, it is a box on the floor with the force at an able with the arrow pointign to the upper right corner. The angle is 30 degrees and the mass of the furnature is 100kg.

I can't figure out how to solve for the coefficient of static friction.

Answer 1

Draw a sketch:

Realizing that the force can be broken down into two components, the force in the y-direction, F_y (downwards) and the force in the x-direction, F_x (sideways) which form a right triangle, we can use a trig ratio to solve for F_x:

cos(θ) = adjacent/hypotenuse

cos(30°) = F_x/657

657cos(30°) = F_x

F_x = 568.98 N

A free-body diagram looks like:

W = mass•g = (100)(9.8) = 980 N

We can sum the forces in the y-direction and, since the object is not accelerating in the y-direction (in fact, it's not moving at all in that up/down direction), we can set the sum of the forces in the y-direction equal to zero:

∑F_y = F_N - W = 0

F_N - 980 = 0

F_N = 980 N

We can now use the equation for frictional forces that:

F_F = μF_N or

μ = F_F/F_N

Since the force needed to start the furniture moving is 657 N at the angle of 30º and since the x-direction component of that is 568.98 N, then we can say F_F = 568.98 N therefore since μ = F_F/F_N:

μ = 568.98/980

μ = 0.58