Question

A pole of length 7.33 m, diameter 0.26 m, and mass 282 kg is floating in a pond, with one end tethered to the bottom, as shown. The depth of the pond is d = 8.95 m, and the length of the tether is h = 3.72 m. Find the angle θ between the pole and the surface

Answer 1

All you need to find theta is the amount of the pole that's underwater, and you can find that using the Archimedes principle - an object submersed or partially submersed is buoyed up by a force equal to the weight of the fluid displaced.

So,
\[m_{pole}g = F_w\]
\[m_{pole}g = m_{water} g\]
the density of water is
\[ \rho_w = 1000 kg/m^3\]
and the volume of the pole is that of a cylinder, so calling the submerged length of the pole D
\[ m_{pole} = \rho_w V\]
\[m_{pole}= \rho_w \frac{\pi d^2 D}{4}\]

and solve for D, then that and d-h gives you the definition of sin theta.

What do ya think? ^^

Answer 2

I think that's right, but it seems off somehow :/

Answer 3

@AllTehMaffs I don't believe that's right because when I solve the angle is 79 degrees

Answer 4

And that's not right? :/

Answer 5

That's what I got too (well, 80, but same diff)

Answer 6

Thought something was off with it, but I dunno what - Even if it's not right, I think that's a reasonable number though - almost 3/4 of the pole is submerged, so there would be quite a bit of force pushing up. And it's bout the density of wood, so it's not sinking by any means.

Answer 7

Ya it's incorrect but I understand your reasoning behind the equation

Answer 8

Hmph. Gotta think about it some more.... Do you have any theories (or know what the ballpark answer is?)

Answer 9

No ideas of what it might be i'm thinking between 50-65 degrees though.

Answer 10

I'm currently trying to solve though

Answer 11

k, good luck with the thinking - be back in a while (and as is often the case, sorry for the wrong answer and method! ^_^)

Answer 12

Posting so I get notifications from this thread — problem stumped me and I'm curious about the answer ^^.

Answer 13

@GavinxFiasco

This might be a dumb question, but are you saying that you think it's 50-65 degrees because of the picture?

Answer 14

@AllTehMaffs No not at all, I asked my TA and he said I would get an answer around that range obviously he couldn't give me the full answer.

Answer 15

But the way you solve the problem seemed correct to be so I'm very confused where you would have went wrong because I find your approached to be right

Answer 16

this is a very trick problem and one which i've never seen before.
i think that alltehmaffs has given you a good start but there's more to this than meets the eye.

in order for equilibrium to be maintained....
Force due to gravity of the entire pole + the upward force of bouyancy on the section of pole underwater + force of the rope holding the pole down = 0

I'll work out the formulas shortly

Answer 17

The approach was correct. The mass of the pole, 282kg, has to be offset by the same mass of water for it to float. That mass of water occupies 0.282m^3. Using the equation for the volume of a cylinder, it means that this skinny pole must be submerged 5.39m of its length.
The tether means that there is 8.95m-3.72m=5.23m of water above the tether. Using 5.39 sin(theta)= 5.23, I got theta=76o, so the pole is nearly vertical. Similar results may differ due to rounding?

Answer 18

Hi douglas. according to archimedes principle the bouyant force is caused by the mass of the water displaced, which is proportional to the volume of the pole under the water, not the total mass of the pole. I think your close

Answer 19

I think there may be troubles because the pole has volume - looking at the pole-water interface



That would definitely decrease the angle - just trying to figure out how to use this info...