Shift in position of plank when a man moves on it (Centre of mass)

In summary: Well I thought that was a useful observation. Since the plank is three times heavier than the man, whatever distance the plank displaces, the man being three times lighter would have to cover thrice the distance to balance out the displacement of the plank.
  • #1
ShaunPereira
40
4
Homework Statement
A uniform wooden plank of mass 150 kg and length 8 m is floating on still water with a man of 50 kg at one end of it . The man walks to the other end of the plank and stops. Than the distance covered by the plank is
Relevant Equations
X(com)=(m1x1 + m2x2)/m1+m2
Lets take the original position of the man to be our origin
The plank is uniform so we can assume its mass to be concentrated at its center i.e. 4m from the origin

Xcom= m1x1+m2x2/m1+m2
=50(0) +150(4) /50+150
=3m

There is no external force on the system so the centre of mass does not move
The man moves to the other side of the plank so the distance form origin becomes 8m
To compensate for this the plank moves a distance d. The original position of the centre of plank was 4m from the origin so its new distance from the origin is (4-d)

Xcom= 3 = 50(8)+ 150(4-d) / 150+50
this gives us d=8/3 m

The answer is incorrect according to the solution provided to me
It should be 2m according to it
 
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  • #2
ShaunPereira said:
The man moves to the other side of the plank so the distance form origin becomes 8m
If the man moves 8 m and the plank moves 8/3 m in the opposite direction, the man will be very wet.
 
  • #3
Orodruin said:
If the man moves 8 m and the plank moves 8/3 m in the opposite direction, the man will be very wet.
Indeed

so if the boat moves a distance d the man moves a distance d too since he is on the boat
the equation becomes

3= (4-d)150 + (8-d)50 /150+50
this gives us d=2m

Is this correct?
 
  • #4
ShaunPereira said:
Homework Statement:: A uniform wooden plank of mass 150 kg and length 8 m is floating on still water with a man of 50 kg at one end of it . The man walks to the other end of the plank and stops. Than the distance covered by the plank is
Relevant Equations:: X(com)=(m1x1 + m2x2)/m1+m2

Lets take the original position of the man to be our origin
The plank is uniform so we can assume its mass to be concentrated at its center i.e. 4m from the origin

Xcom= m1x1+m2x2/m1+m2
=50(0) +150(4) /50+150
=3m
Correct.
ShaunPereira said:
There is no external force on the system so the centre of mass does not move
Correct.
ShaunPereira said:
The man moves to the other side of the plank so the distance form origin becomes 8m
This is not right, as the plank moves.
ShaunPereira said:
To compensate for this the plank moves a distance d. The original position of the centre of plank was 4m from the origin so its new distance from the origin is (4-d)
Okay.

ShaunPereira said:
so if the boat moves a distance d the man moves a distance d too since he is on the boat
I think you mean the man moves a distance ##8-d##. And plank not boat.
ShaunPereira said:
the equation becomes

3= (4-d)150 + (8-d)50 /150+50
this gives us d=2m

Is this correct?
Yes.

Note that as the plank is three times the mass of the man, the man moves three times as far, so ##d + 3d = 8m## and ##d = 2m##.
 
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  • #5
PeroK said:
I think you mean the man moves a distance 8−d. And plank not boat.
Yes. I meant to say that the man moves 8 m say left but the boat moves a distance d towards right .The man is on the boat and hence also moves a distance d towards right otherwise he would fall into the water as Orodruin rightly pointed out.

so the net distance that the man has traveled from the origin becomes 8-d
 
  • #6
PeroK said:
Yes.

Note that as the plank is three times the mass of the man, the man moves three times as far, so d+3d=8m and d=2m.
Ok. Thank you!
 
  • #7
ShaunPereira said:
Ok. Thank you!
Do you see why?
 
  • #8
PeroK said:
Do you see why?
Well I thought that was a useful observation. Since the plank is three times heavier than the man, whatever distance the plank displaces, the man being three times lighter would have to cover thrice the distance to balance out the displacement of the plank.

I reckon this analogy could be used even when we change the masses of the man and the plank. Say the plank is 5 times heavier than the man therefore whatever distance the plank is displaced by the man would have to cover 5 times that

We could thus formulate an expression for this
m1Δx1 =m2Δx2

where Δx = net displacement
man moves 8m to the left. Boat moves Δx2 to right
therefore net displacement of man is Δx1= (8-Δx2)

m1(8-Δx2) =m2Δx2
Here m2=5m1

On solving we get
Δx2= 4/3 m This is the net displacement of the boat
Substituting this value in the original formula we made
m1Δx1 = m2Δx2 we get

Δx1= 20/3 m This is the net distance the man is displaced from the origin.

Is this correct?
 
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  • #9
Yes, precisely. Technically, of course, the displacements are in opposite directions.
 
Last edited:
  • #10
PeroK said:
the displacements are equal in opposite directions.
Hmmm... I didn't get you there. How can the displacements be equal? They are after all dependent on the masses of the two bodies?
 
  • #11
ShaunPereira said:
Hmmm... I didn't get you there. How can the displacements be equal? They are after all dependent on the masses of the two bodies?
Just a typo!
 
  • #12
PeroK said:
Just a typo!
OK. Thanks!
 

Related to Shift in position of plank when a man moves on it (Centre of mass)

1. What causes the shift in position of the plank when a man moves on it?

The shift in position of the plank is caused by a change in the center of mass of the system. When the man moves on the plank, his weight and position on the plank affect the overall distribution of mass, causing the center of mass to shift.

2. How does the weight of the man affect the shift in position of the plank?

The weight of the man plays a significant role in the shift of the plank's position. The heavier the man, the more the center of mass will shift towards his side, causing the plank to tilt in that direction.

3. Can the position of the plank be predicted based on the weight and position of the man?

Yes, the position of the plank can be predicted by considering the weight and position of the man. The further the man is from the center of the plank, the greater the shift in position will be. Additionally, the heavier the man is, the more the plank will tilt towards his side.

4. How does the length of the plank affect the shift in position?

The length of the plank also plays a role in the shift of its position. A longer plank will have a greater area for the man to distribute his weight, resulting in a smaller shift in position. On the other hand, a shorter plank will have a smaller area, causing a larger shift in position.

5. Can the shift in position of the plank be balanced?

Yes, the shift in position of the plank can be balanced by adjusting the position of the man on the plank. If the man moves closer to the center of the plank, the center of mass will shift back towards the center, resulting in a more balanced position for the plank.

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