• Mar 8th 2007, 02:38 PM
the_sensai
Hey people,
im really stuck on this and wondered if you could help please.

Q: A heavy crate of mass m is pulled along a rough horizontal surface at a constant speed by a rope. The coefficient of friction between the crate and surface is u (substituted for mew as i dont know how to insert that)

http://www.mathhelpforum.com/math-he...20Pictures%5Ct

A) Show that T = umg/(cos(theta)+usin(theta).

B) By finding dT/d(theta) and letting dT/d(theta) = 0, show that for a minimum value of T , tan(theta) = u. ( you may assume that, for a maximum value of T, (theta) = 0).

any help would be really greatful guys as im really stuck.
thanks.
• Mar 9th 2007, 12:25 AM
ticbol
Quote:

Originally Posted by the_sensai
Hey people,
im really stuck on this and wondered if you could help please.

Q: A heavy crate of mass m is pulled along a rough horizontal surface at a constant speed by a rope. The coefficient of friction between the crate and surface is u (substituted for mew as i dont know how to insert that)

http://www.mathhelpforum.com/math-he...20Pictures%5Ct

A) Show that T = umg/(cos(theta)+usin(theta).

B) By finding dT/d(theta) and letting dT/d(theta) = 0, show that for a minimum value of T , tan(theta) = u. ( you may assume that, for a maximum value of T, (theta) = 0).

any help would be really greatful guys as im really stuck.
thanks.

So where is the diagram?
• Mar 9th 2007, 08:02 AM
topsquark
Quote:

Originally Posted by the_sensai
Hey people,
im really stuck on this and wondered if you could help please.

Q: A heavy crate of mass m is pulled along a rough horizontal surface at a constant speed by a rope. The coefficient of friction between the crate and surface is u (substituted for mew as i dont know how to insert that)

http://www.mathhelpforum.com/math-he...20Pictures%5Ct

A) Show that T = umg/(cos(theta)+usin(theta).

B) By finding dT/d(theta) and letting dT/d(theta) = 0, show that for a minimum value of T , tan(theta) = u. ( you may assume that, for a maximum value of T, (theta) = 0).

any help would be really greatful guys as im really stuck.
thanks.

Still no diagram, but I think I know what's going on here.

First of all, the Greek letter "mew" (shudders) is spelled "mu." It's not a cat!

My diagram has a crate being dragged to the right by a rope. The rope makes an angle t (in place of theta) with the horizontal.

My Free-Body Diagram on the crate has a weight (w) acting directly downward, a normal force (N) acting directly upward, a kinetic friction force (f) acting to the left, and a tension (T) acting upward and to the right at an angle t with the horizontal. I have a +x axis to the right and a +y axis directly upward. The coefficient of kinetic friction is u.

The speed is constant, so the acceleration is 0 m/s^2. Thus Newton's 2nd in the x and y directions say:
SumFx = -f + Tcos(t) = 0
SumFy = N - w + Tsin(t) = 0

Thus
N = w - Tsin(t) = mg - Tsin(t)

Thus
f = uN = umg - uTsin(t)

So
-(umg - uTsin(t)) + Tcos(t) = 0

-umg + uTsin(t) + Tcos(t) = 0

T(usin(t) + cos(t)) = umg

T = umg/(usin(t) + cos(t))

For the second part, find dT/dt:
dT/dt = -umg/(usin(t) + cos(t))^2 * (ucos(t) - sin(t))

Setting dT/dt = 0 gives:
-umg(ucos(t) - sin(t))/(usin(t) + cos(t))^2 = 0

So the numerator must be 0 (as long as the denominator isn't 0 for this angle). Thus
ucos(t) - sin(t) = 0

u = tan(t)