How to find the angle that will pull the most weight

[Erratus]

Homework Statement

Straight out of the book: "An initially stationary box of sand is to be pulled across a floor by means of a cable in which the tension should not exceed 1100N. The coefficient of static friction between the box and the floor is 0.35. (a) What should be the angle between the cable and the horizontal in order to pull the greatest possible amount of sand and (b) what is the weight of the sand and box in that situation?"

Homework Equations

F = ma
F_s = μ_sF_N

The Attempt at a Solution

So, I've decided that the x component, that is, the 1100N (I call this F_c) times cosθ, has to be equal to static friction. If it wasn't, then more sand could be added until it was (is this an incorrect assumption?). From there I tried to solve it like my professor has been doing by analzying the sum of the forces in the x direction and y direction separately to try to get two equations with two variables. The idea then would be that it would be an algebra problem, but I keep getting three unknowns (F_N, F_g, and θ). I'll go ahead and show my work, I am probably going something wrong or going about it the wrong way (if I wasn't, this problem would have been done long ago).

Ʃ F_x = F_cCosθ - F_s = 0

Or

F_cCosθ = μ_sF_N


Ʃ F_y = F_cSinθ + F_N - F_g= 0

or

F_cSinθ + F_N = F_g

So, I don't know θ, F_N, or F_g and I don't see any third equation that I can use to eliminate a variable. Do I have to set the y component of F_c to something like I did the x component? Or was that my mistake from the beginning? Also, I will probably be spamming this forum with physics problems, sorry and if there is a limit do say. But I am tired of struggling with these homework problems, and the professor's hours are kind of awkward for me.

 

[nasu]

I think you are doing OK so far.
Use the first equation to replace F_N in the second. You'll have Fg, or the weight of the box, as a function of the angle (Fc is a constant).
Find the angle which maximizes that function.

 

[azizlwl]

FCosθ=Nμ ...(1)
N=mg-FSinθ ...(2)

Sub.(2) in (1).
FCosθ=(mg-FSinθ)μ
FCosθ=mgμ-FSinθμ

mgμ=F(Cosθ+Sinθμ)

Maximum value of (Cosθ+Sinθμ) is when Cosθμ-Sinθ=0
Edit: Errors . Thanks Nasu

 

[nasu]

You forgot to multiply N by Fsinθ when you multiplied the parenthesis (fourth equation).

 

[Nickie]

I would approach this problem by first identifying the relevant equations and variables involved. In this case, the equations of Newton's laws of motion (F = ma) and the equation for static friction (Fs = μsFN) are applicable. The variables in this problem are the tension in the cable (Fc), the coefficient of static friction (μs), the normal force (FN), the weight of the box and sand (Fg), and the angle between the cable and the horizontal (θ).

To find the angle that will pull the most weight, we need to consider the maximum possible tension in the cable (Fc) and the maximum possible static friction force (Fs). Using the equation Fs = μsFN, we can see that the maximum static friction force is equal to μsFN. We also know that the maximum tension in the cable is 1100N. Therefore, to pull the most weight, we need to find the angle that maximizes the product of these two values.

To do this, we can use the equations for the x and y components of the forces (Ʃ Fx = FcCosθ - Fs = 0 and Ʃ Fy = FcSinθ + FN - Fg = 0) and solve for the normal force (FN) and the weight (Fg). Once we have these values, we can plug them into the equation for the maximum static friction force (Fs = μsFN) and solve for the angle θ.

In summary, to find the angle that will pull the most weight, we need to use the equations of Newton's laws and static friction, solve for the normal force and weight, and then use these values to determine the angle that maximizes the product of the tension and static friction force.