Uniform rectangular plate equilibrium problem

[kudoushinichi88]

Homework Statement

A uniform rectangular plate of width d, height h, and weight W is supported with its top and bottom edges horizontal. At the lower left corner there is a inge, and the upper right corner there is a cable. For what angle $\theta$ with the vertical will the tension in the cable be the least, and what is the tension?

Homework Equations

$\tau=Fd$

The Attempt at a Solution

for the angle, it's easy,
$tan \theta = d/h$
$\theta=\arctan{d/h}$

but I'm having trouble with the tension of the cable. I managed to derive
$\frac{Wd}{2}=Td\cos{\theta}+Th\sin{\theta}$

which gives T as

$T=\frac{Wd}{2\left(d\cos{\theta}+h\sin{\theta})}$

the answer given is

$T=(Wd/2)\sqrt{h^2+d^2}$

I seem to fail to see the connection
$\sqrt{h^2+d^2}=\frac{1}{d\cos{\theta}+h\sin{\theta}}$

can anyone show me why is this so?

 

[kuruman]

Homework Statement

A uniform rectangular plate of width d, height h, and weight W is supported with its top and bottom edges horizontal. At the lower left corner there is a inge, and the upper right corner there is a cable. For what angle $\theta$ with the vertical will the tension in the cable be the least, and what is the tension?

You did only part of the problem. You found the tension at some angle θ. Now you need to find at what angle the tension has the least value, then find what the tension is.

 

[tomcue]

It seems that there may be a typo in the given answer. The correct equation for the tension T should be:

T = (Wd/2) / cos\theta + sin\theta

This can be derived from the equation you already have:

Wd/2 = Td cos\theta + Th sin\theta

Rearranging and solving for T gives us:

T = (Wd/2) / (d cos\theta + h sin\theta)

To simplify this further, we can use the trigonometric identity:

cos\theta = h/sqrt(h^2 + d^2)
sin\theta = d/sqrt(h^2 + d^2)

Substituting these into the equation for T, we get:

T = (Wd/2) / (d(h/sqrt(h^2 + d^2)) + h(d/sqrt(h^2 + d^2)))
= (Wd/2) / (d^2 + h^2)
= (Wd/2) / sqrt(h^2 + d^2)

Which gives us the correct answer:

T = (Wd/2) * sqrt(h^2 + d^2)

This shows that the tension in the cable is directly proportional to the weight of the plate and the length of the plate (d), and is also affected by the angle \theta. It also shows that the minimum tension in the cable occurs when \theta = 0, meaning the cable is vertical and the plate is in perfect equilibrium.