A sunbather lying on a floating raft that experiences no friction with the water gets up and walks to the western end of the raft. While she is walking, how does the center of gravity of the system composed of her and the raft move? Why? If the mass of the raft is 10 times that of the sunbather's mass, in which direction and about how fast does the raft move in terms of the sunbather's speed?

consider a sytem comprising of Sunbather + raft.

No external force on this system.Therefore thr Center of mass of the system will remain stationary. If she walks to the west, the raft will move to the east by a distance that will keep the CM at the original location.

No external force also means that linear momentum will not change.
Pi=Pf=0
Ms*Vs = - Mr*Vr
Vr = - Ms*Vs/Mr = - Ms*Vs/10*Vs
Vr = - Vs/10

When the sunbather gets up and walks to the western end of the raft, the center of gravity of the system composed of her and the raft will shift towards the eastern end of the raft. This is because the sunbather's movement towards the western end causes her center of mass to move in that direction.

The center of gravity is the point where the entire weight of an object or system can be considered to act. In this case, the center of gravity of the system is initially in the center of the raft, as the sunbather is lying in the middle. When the sunbather walks towards the western end of the raft, her center of mass moves in that direction, causing the overall center of gravity of the system to shift towards the eastern end.

Now, let's consider the motion of the raft in response to the sunbather's movement. We are told that the mass of the raft is 10 times that of the sunbather. According to the principle of conservation of momentum, the total momentum of the system remains constant.

Since momentum is mass times velocity, the momentum of the system composed of the sunbather and the raft is given by:

Total momentum = (Mass of sunbather * Velocity of sunbather) + (Mass of raft * Velocity of raft)

If the sunbather has a positive velocity towards the western end, her momentum is positive. To keep the total momentum constant, the momentum of the raft must be negative and equal in magnitude to the sunbather's momentum. This means the raft moves in the opposite direction to the sunbather, towards the eastern end.

Since the mass of the raft is 10 times that of the sunbather, the velocity of the raft will be approximately one-tenth of the velocity of the sunbather. In other words, the raft will move in the opposite direction at about one-tenth the speed of the sunbather.

To understand how the center of gravity of the system composed of the sunbather and the raft moves, let's break it down step by step:

1. Initially: When the sunbather is lying on the raft, their center of gravity is positioned at the midpoint between the sunbather and the raft, as they are at rest and have equal weights.

2. As the sunbather stands up and starts walking towards the western end of the raft: The sunbather exerts a forward force on the raft, pushing it backward. According to Newton's third law of motion, for every action, there is an equal and opposite reaction. As the sunbather pushes backward on the raft, the raft pushes forward on the sunbather with an equal force. This force causes the sunbather's center of gravity to shift slightly towards the eastern end of the raft.

3. When the sunbather reaches the western end of the raft: At this point, the sunbather's center of gravity is no longer at the midpoint of the system. It has shifted slightly towards the eastern end due to the forces exerted during the sunbather's walk.

Regarding the direction and speed of the raft's movement in terms of the sunbather's speed:

Since the mass of the raft is ten times that of the sunbather's mass, and we know that force equals mass times acceleration (F = m*a), the acceleration experienced by the combined system (sunbather + raft) is much less than that of the sunbather alone. This means that the raft will move in the opposite direction to that of the sunbather's movement but at a slower speed.

In terms of numerical values, if the sunbather's speed is v, then the velocity of the raft (in the opposite direction) would be approximately v/10, assuming there are no other external forces acting on the system.

It's important to note that the exact speed and direction would depend on the specific values of the sunbather's speed and the masses involved.