Problem

A man with a mass of 55 kg stands up in a 65-kg canoe of length 4.0 m floating on water. H...

A man with a mass of 55 kg stands up in a 65-kg canoe of length 4.0 m floating on water. He walks from a point 0.75 m from the back of the canoe to a point 0.75 m from the front of the canoe. Assume negligible friction between the canoe and the water. How far does the canoe move?

Step-by-Step Solution

Solution 1

Step #1 of 7

The expression for X coordinate of center of mass for a two-body system is given as follows:

Here, is the x co-ordinate of center of mass, is the x co-ordinate of mass , is the x coordinate of center of .

Step #2 of 7

The following figure represents the motion of the canoe.

Here, l is the length of the canoe, d is the distance moved by canoe, a is the relative position of man with respect to center of mass of the canoe, is the initial position of center of mass of man, is the initial position of center of mass of the canoe, is the final position of center of mass of mass of man and is the final position of center of mass of canoe.

Step #3 of 7

Then center of mass of the canoe and man will not change in our frame of reference. But it will change according to the frame of reference of man. It will move by a distance of as shown in the above diagram. The difference in the position of the center of mass in the two cases will give the distance moved by the canoe.

The center of mass of canoe is at half of the length of canoe.

Calculate the initial position of man from the origin as shown in the diagram.

The man’s initial position is given by following expression.

Substitute 4.0 m for l.

Step #4 of 7

From the figure, the relative position a of the man with respect to center of mass of the canoe is given as follows:

From the figure, the distance to center of mass of canoe is and position of the man is

Substitute and in the above equation to solve for a.

Now, substitute 4.0 m for l.

Here, negative sign represents the position of the man with respect to the center of mass of canoe is to the left.

Step #5 of 7

Then the distance the canoe moves is given by the following formula:

Here, d is the distance moved by the canoe.

Consider the following equation:

Before moving, the center of mass is given by the following formula:

…… (1)

Here, is the initial position of center of mass of man, is the mass of the man, is the initial position of center of mass of the canoe, and is the mass of the canoe.