Two friends, Burt and Ernie, are standing at opposite ends of a uniform log that

Question

Two friends, Burt and Ernie, are standing at opposite ends of a uniform log that is floating in a lake. The log is 3.1 m long and has mass 230 kg . Burt has mass 37.0 kg and Ernie has mass 44.5 kg . Initially the log and the two friends are at rest relative to the shore. Burt then offers Ernie a cookie, and Ernie walks to Burt's end of the log to get it.

Part A

Relative to the shore, what distance has the log moved by the time Ernie reaches Burt? Neglect any horizontal force that the water exerts on the log and assume that neither Burt nor Ernie falls off the log.

Express your answer to two significant figures and include the appropriate units.

Explanation / Answer

There are no horizontal forces on the Burt+Ernie+log system. So the center of mass of the system will remain at the same place even after Ernie walks to the Burt's end.

Let the initial position of Burt be x = 0, Then, the center of mass of log would be located at x = L/2 = 3.1m/2 = 1.55m.

And the position of Ernie would be x = 3.1 m initially, so x-coordinate of center of mas initially is,

X = [MB(0) + (ML)(1.55 m) + ME(3.1m)]/(MB + ML + ME)

or, X = [0 + (230 kg)(1.55 m) + (44.5 kg)(3.1m)]/(37 kg + 230 kg + 44.5 kg)

or, X = 1.587 m

Now let us suppose the log moves a distance d to the right towards Ernie's end.

So, distance moved by Burt = d, distance moved by Ernie is d - 3.1m

So position of Burt with respect to the shore after Ernie is at his end is,

xb = d

position of center of mass log with respect to the shore after Ernie is at his end is,

xl = 1.55m + d

position of Ernie with respect to the shore after he reached Burt's end,

xe = d

So position of center of mass after after Ernie is at Burt's end is,

Xf = [MB(d) + (ML)(1.55 m + d) + ME(d)]/(MB + ML + ME)

or, Xf = [(37 kg)(d) + (230 kg)(1.55 m + d) + (44.5 kg)(d)]/(37 kg + 230 kg + 44.5 kg)  

or, Xf = (311.5d + 356.5)/(311.5) = d + 356.5/311.5

Now, X = Xf, so,

1.587 m = d + 356.5/311.5

or, d = 0.44 m, is the distance by which the log moves towards the Ernie's end after Ernie reaches Burt.

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