Optimizing Friction for Moving Sand: What Angle Should the Cable Be?

[king_naeem]

Question: 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 1100 N. 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?

b)What is the weight of the sand and box in that situation?

My solution that gives me the 'incorrect' answer:

Eqn 1--> Tcos(theta) - u(static coefficient)F(normal)=0

Equation 2--> Force Normal + tsin(theta) - mg=0

I can solve for the force normal to be equal to = mg/1.35

For there i assumed that the angle should be 45 degrees (i know..wrong assumption) but have no clue what to do next...

any help would be appreciated

 

[Doc Al]

Eqn 1--> Tcos(theta) - u(static coefficient)F(normal)=0

Equation 2--> Force Normal + tsin(theta) - mg=0

These two equations are fine. Rearrange them to show the weight (mg) as a function of the angle. Find the angle that maximizes the weight.

 

[king_naeem]

how would i rearrange the first equation to get it interms of mg?

b/c the i have

Tcos(theta) - u(static coefficient)F(normal)=0

force normal does not equal the force of gravity because the force applied on the string is at an angle so i can't just say 'force of gravity=force normal' can i? unless I'm wrong yet again...ahh!

 

[king_naeem]

okay...is this right so far

i get by combining and rearranging the following new equation:

Cos (theta) + Sin(theta)= 0.003121363m

where m is the mass of the object...what should i do next?

 

[dextercioby]

how would i rearrange the first equation to get it interms of mg?

b/c the i have

Tcos(theta) - u(static coefficient)F(normal)=0

force normal does not equal the force of gravity because the force applied on the string is at an angle so i can't just say 'force of gravity=force normal' can i? unless I'm wrong yet again...ahh!

1.I should like to congratulate DocAl for his 3000 posts.
2.From the first equation express N (the normal force of pressing) in terms of µ,T and theta.Plug it in the second equation and obtain the function m=m(µ,theta,T,g).Since g is constant,for the max of m you must have max tension T (1100 N) and a special angle which is found by the condition of maximum imposed to m seen as a function of theta (considering T as a parameter).My guess is that angle should be arctan (µ).Then insert all numerical data and get the maximized mass.

I believe it's all clear now.

 

[Doc Al]

1.I should like to congratulate DocAl for his 3000 posts.

Thanks, Daniel!

2... My guess is that angle should be arctan (µ).

Good answer (and good "guess").