Find the Upward Force of a Boat with 6 People Aboard

[gjh]

Homework Statement

This was posed as a challenge problem by Professor Richard Wolfson as part of a physics course he recorded for TGC company. Problem: A flatboat has dimensions 3m long, 1 m wide and .24 m deep and has mass of 120 kg and is on a lake (fresh water). What is the upward force on the boat and how many 85 kg people can get on the boat before it is swamped?

Relevant Equations

density = mass/volume; density of water is 1000 kg/m(3)

Apply Principle of Archimedes - upward force is equal to the weight of the volume of water displaced. Density (boat) = 120 kg/(3x1x.24)m(3) = 500/3 kg/m(3)
Density of water = 1000kg/m(3). Density (water)/Density of boat = 1000 kg/m(3)/(500/3) kg/m(3) = 6. So density of water is 6 times the density of the boat and so boat obviously floats. Now I reasoned that if the density of the boat with X people aboard floats it will just equal the density of water.

(1) Let X = number of 85-kg people. We have (120 + X 85)/(3x1x.24) = (120 +85X)/.72 = 1000 -> X = 7.1 people. Since we can't have a .1 person, the boat can hold 6 people, each with mass 85 kg. That checks with the answer given (boat can hold 6 people), but not sure logic is correct. Finding the upward force has got me stumped. The answer is 590 N. When I substitute 6-85 kg people into the boat into Equation (1) above, I only get about 520 N. (Note: Professor Wolfson used g = 10 m/sec (2) to simplify the math.) Any comments welcome!

 

[Orodruin]

First of all, the first question is badly stated in my opinion as it does not specify the state of the boat when the force is to be calculated. Is it when the boat is submerged or is it when it is unloaded? Also, which force is asked about? The gravitational force? The buoyancy? The net force?

Second, the correct answer would be 7 persons. You cannot hold 0.1 person but the largest integer smaller than 7.1 is 7, not 6. The value of g is irrelevant to the answer.

 

[haruspex]

When I substitute 6-85 kg people into the boat into Equation (1) above, I only get about 520 N.

As @Orodruin notes, the question is unclear, but I have no idea how you get that answer.
It can't be net force since all situations discussed are static, so that will be zero.
That leaves buoyancy, being the only upward force, either with or without the maximum load of people. But the former of those will be the weight of the boat, 1200N, and the latter 7150N with 7 people or 6300N with 6.
Or maybe the increase in upward force when the people board, 5950N or 5100N.

 

[pbuk]

It depends on what is meant by "swamped". With 7 people the "flatboat" (whatever that is, I'll assume it's a raft) will have less than 2mm of freeboard. I think that probably counts as "swamped". With 6 people there is 3cm of freeboard, perhaps that is enough for it not to be "swamped".

 

[hmmm27]

Contextually, the only relevance of "density" is to find the maximum mass of water the boat can displace without sinking. After that, nobody cares.

Archimedes Principle isn't much help : as long as it's floating, something that weighs 100kg will displace 100kg of water.

You should really try again, from scratch. Note that F=ma, and - as you note - the problem uses 10m/s2 for g.

 

[phinds]

something that weighs 100kg will displace 100kg of water.

So you are saying that a sphere made of something that weights 100kg and is just dense enough to sink will displace 100kg of water AND a very small sphere of depleted uranium will ALSO displace 100kg of water? You might want to rethink that.

OOPS ... I missed the part about it floating (I assume you mean buoyed up by the boat)

 

[pbuk]

(I assume you mean buoyed up by the boat)

Why do you assume that - Archimedes wasn't buoyed up by a boat in his bath?

 

[phinds]

Why do you assume that - Archimedes wasn't buoyed up by a boat in his bath?

Neither was he submerged. The part of him that was submerged displaced an amount of water weighing what that part of him weighed.

Do you REALLY think that a 100Kg sphere of depleted uranium would displace 100Kg of water? Read my whole post.

 

[haruspex]

You should really try again, from scratch.

Why? Are you suggesting that the OP's answer for the number of people is wrong, or that it was obtained by an invalid method?

 

[hmmm27]

Why? Are you suggesting that the OP's answer for the number of people is wrong, or that it was obtained by an invalid method?

The OP's answer for the number of people actually is wrong. Likewise their answer for upwards force, no matter which question you interpret the problem as asking.

 

[pbuk]

Do you REALLY think that a 100Kg sphere of depleted uranium would displace 100Kg of water? Read my whole post.

No, but I don't think it will float either.

 

[haruspex]

The OP's answer for the number of people actually is wrong.

The six is wrong, yes, but along the way the OP got 7.1, then mysteriously rounded it down to six. I don’t see that as a reason to

try again, from scratch