# Find the Upward Force of a Boat with 6 People Aboard

• gjh
In summary: The OP could have made a simple rounding error or mistake in calculations, which can be easily corrected. There is no need to start over from scratch.
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!

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.

erobz
gjh said:
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.

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".

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.

hmmm27 said:
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)

phinds said:
(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?

pbuk said:
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.

Last edited:
hmmm27 said:
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?

haruspex said:
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.

phinds said:
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.

hmmm27 said:
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
hmmm27 said:
try again, from scratch

## 1. What is the upward force of a boat with 6 people aboard?

The upward force of a boat with 6 people aboard is dependent on several factors such as the weight of the boat, the weight of each individual, and the density of the water. In general, the upward force is equal to the weight of the water displaced by the boat, also known as the buoyant force.

## 2. How do you calculate the upward force of a boat with 6 people aboard?

To calculate the upward force of a boat with 6 people aboard, you can use the formula F = ρVg, where F is the buoyant force, ρ is the density of the water, V is the volume of water displaced by the boat, and g is the acceleration due to gravity. You can determine the volume of water displaced by the boat by multiplying the length, width, and height of the boat.

## 3. What is the significance of finding the upward force of a boat with 6 people aboard?

Finding the upward force of a boat with 6 people aboard is important for understanding the stability and safety of the boat. If the upward force is not enough to counteract the weight of the boat and its passengers, the boat may sink. Additionally, knowing the upward force can help determine the maximum weight capacity of the boat.

## 4. How does the upward force of a boat with 6 people aboard change in different bodies of water?

The upward force of a boat with 6 people aboard can vary in different bodies of water due to differences in water density. For example, saltwater is denser than freshwater, so a boat with 6 people aboard will experience a greater upward force in saltwater compared to freshwater.

## 5. Is the upward force of a boat with 6 people aboard affected by the shape or size of the boat?

Yes, the shape and size of the boat can affect the upward force. A boat with a larger surface area will displace more water and experience a greater upward force. Additionally, the shape of the boat can also impact the distribution of weight and the stability of the boat.

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