Canoe and Man Motion: Finding the Relationship Between Distance Traveled

[alex.pasek]

Homework Statement

A 55 kg man is standing on a 65 kg and 4.0 m length canoe that floats without friction on water.
The man walks from a 0.75m from one end of the canoe to another point 0.75 from the other end of the canoe.
What distance does the canoe move?

Homework Equations

Center of mass equation.

The Attempt at a Solution

A tried looking for the ΔX_CM, the change on the center of mass, but I do not know how to compute the canoe's CM.

Mod note: Fixed misspelling of "canoe."

 

[Orodruin]

I do not know how to compute the canoe's CM.

The center of mass for the canoe does not matter, only its displacement does.

 

[alex.pasek]

So; I should compute CM_o = CM_f?
CM_o=(1/M)⋅∑x_n ⋅ m_n
CM_f=(1/M)⋅∑x_n ⋅ m_n

And calculate ΔCM=C_f - C_o

With this I get ΔCM= -0.33 but I do not know how to proceed forward.

 

[Orodruin]

So; I should compute CM_o = CM_f?
CM_o=(1/M)⋅∑x_n ⋅ m_n
CM_f=(1/M)⋅∑x_n ⋅ m_n

And calculate ΔCM=C_f - C_o

With this I get ΔCM= -0.33 but I do not know how to proceed forward.

Well, first of all, you should define all of your quantities, I cannot read your mind to know what C_f and C_o are and what sets them apart from CM_o and CM_f.

Second, the difference in the center of mass is a length and must be given in terms of a length unit.

 

[alex.pasek]

Sorry Here goes the complete reasoning;

CM_o=(1/M)⋅∑xn ⋅ mn= (1/120 kg)⋅[(55⋅0.75 kg⋅m)+(65⋅2 kg⋅m)] = 1.43 m
CM_f=(1/M)⋅∑xn ⋅ mn= (1/120 kg)⋅[(55⋅ 3.25 kg⋅m)+(65⋅2 kg⋅m)] = 2.57 m

And calculate ΔCM=CM_f-CM_o= (2.57-1.43)m = 1.14 m

Would be this the displacement of the boat?

 

[Orodruin]

Sorry Here goes the complete reasoning;

CM_o=(1/M)⋅∑xn ⋅ mn= (1/120 kg)⋅[(55⋅0.75 kg⋅m)+(65⋅2 kg⋅m)] = 1.43 m
CM_f=(1/M)⋅∑xn ⋅ mn= (1/120 kg)⋅[(55⋅ 3.25 kg⋅m)+(65⋅2 kg⋅m)] = 2.57 m

And calculate ΔCM=CM_f-CM_o= (2.57-1.43)m = 1.14 m

Would be this the displacement of the boat?

What you have computed is the change in the center of mass in the boat frame. How does this relate to how far the boat moves? What assumptions go into that relation?

 

[alex.pasek]

So, If I applied the reference frame, I guess I should undo it. Then by substracting it to each center of mass I should be given the result; so:
d_canoo= (x^o_canoo-ΔCM)= (2 - 1.14)m = 0.86 m
Would it be correct?

 

[Orodruin]

So, If I applied the reference frame, I guess I should undo it. Then by substracting it to each center of mass I should be given the result; so:
d_canoo= (x^o_canoo-ΔCM)= (2 - 1.14)m = 0.86 m
Would it be correct?

No. What is true about the center of mass of both man and boat (assuming that there is no transfer of momentum to the water)?

 

[alex.pasek]

No. What is true about the center of mass of both man and boat (assuming that there is no transfer of momentum to the water)?

I think the momentum is conserved, no externa forces are acting on the system.

 

[Orodruin]

So, assuming that there is no momentum to begin with, what happens with the center of mass?

 

[alex.pasek]

So, assuming that there is no momentum to begin with, what happens with the center of mass?

The conoe does not move?

 

[Orodruin]

No, the center of mass does not move. You have computed that the center of mass moves by 1.14 m (1.15 m with correct rounding) it a frame that is attached to the canoe. How far must the canoe move for this to happen at the same time as the CoM does not move in the lake frame?

 

[alex.pasek]

What if I tried a different approach, as CM_o=CM_f=1.43 m And calling the displacement "x";
1.43 ⋅ 120 m⋅kg = 55 ⋅ 3.25 m⋅kg + 65 ⋅ x kg ; solving for x i get x ≈-0.11 m
And this would be the displacement of the canoe, I think.

 

[Orodruin]

What if I tried a different approach, as CM_o=CM_f=1.43 m And calling the displacement "x";
1.43 ⋅ 120 m⋅kg = 55 ⋅ 3.25 m⋅kg + 65 ⋅ x kg ; solving for x i get x ≈-0.11 m
And this would be the displacement of the canoe, I think.

No, this is not correct. Why switch methods when you are basically done?

 

[alex.pasek]

I do not understand what is actually happening within the lake's reference frame. I do not know where is the origin for that frame :(

 

[Orodruin]

I do not understand what is actually happening within the lake's reference frame. I do not know where is the origin for that frame :(

You do not need an origin (you can pick one wherever you like).

It may be beneficial to start looking in the lake frame, where the center of mass does not move. What does this tell you about the displacements of the man and boat, respectively?

 

[alex.pasek]

So, if in the lake's frame the CM of the man and the boat does not move even if they do. This may suggest the displacement of the man is somehow proportional to the boat's?
I mean, if the man walks to the right, then the boar moves to the left?

 

[Orodruin]

So, if in the lake's frame the CM of the man and the boat does not move even if they do. This may suggest the displacement of the man is somehow proportional to the boat's?
I mean, if the man walks to the right, then the boar moves to the left?

Correct, but how much?

 

[alex.pasek]

The distance the man walks minus the change in the center of mass (in the boat's frame)?

 

[Orodruin]

The distance the man walks minus the change in the center of mass (in the boat's frame)?

I suggest you only look at the lake frame. What is the relation between how far the canoe moves and how far the man moves in that frame?