How Does the Center of Mass Stay Constant When a Child Walks on a Boat?

[to4u]

I'm not the brightest physics student, so I probably just overlooked something obvious...please help!

Problem:

A 40 kg child stands at one end of a 70 kg boat that is 4 meters in length. The boat is initially 3 meters from the pier. The child notices a turtle on a rock near the far end of the boat and proceeds to walk to that end to catch the turtle. Neglecting friction b/w the boat and water...

a. describe motion

I said that the child moves to the right and the boat to the left...but center of mass stays same


b. where is the child relative to the pier when he reaches far end of the boat?

I set...

x(child intial)*m(child)+x(boat initial)*m(boat) / m(child)+m(boat) = x(child final)*m(child)+x(boat final)*m(boat) / m(child)+m(boat)

the denominators cancel so...

40*x(child initial) + 70*x(boat initial) = 40*x(child final) + 70*x(boat final)

so here's my question...how do I solve this without knowing how much the boat moved after the child walked to the other side of the boat? Is the x(child initial)= -2 m (2 m left of center of mass) or 3 m (from pier)? And what about the boat?



Your help is much appreciated!

 

[mjsd]

you need a set of coordinates (of frame of reference) for you system. The pier is basically your origin (if you want), the boat has a center of mass somewhere relative to the pier, the child will have a CM relative to the pier, then the system as a whole (child + boat) will have a CM relative to the pier. movment of the child changes the child's CM and hence boat's CM has to change accordingly so that the CM of whole system remain the same. All these can be done relative to the pier

 

[Moetasim]

Hi there! Don't worry, it's completely normal to overlook something in physics problems. Let's break down the problem and discuss the concept of center of mass motion.

First, let's define what center of mass is. It is the point in a system where the mass is evenly distributed, and the system can be considered as a single point particle with that mass. In this case, the center of mass of the child-boat system will be somewhere between the child and the boat, closer to the heavier object (the boat).

Now, let's look at the motion of the system. Initially, the child and the boat are at rest, so their center of mass is also at rest. When the child starts walking towards the turtle, he exerts a force on the boat and pushes it in the opposite direction. This results in the boat moving to the left, while the child moves to the right. However, the center of mass of the system will remain at the same position since the total mass and the forces acting on the system remain constant.

To solve the problem, we need to consider the initial and final positions of both the child and the boat, relative to the pier. The initial position of the child is 3 meters to the right of the pier, and the boat is 3 meters to the left of the pier. When the child reaches the far end of the boat, the boat would have moved some distance to the left, let's say x meters. At this point, the child and the boat both would be x meters away from the pier. So, the final position of the child would be x+4 meters to the right of the pier, and the boat would be x-3 meters to the left of the pier.

Now, let's plug these values into the equation you have set up:

40*(3m) + 70*(-3m) = 40*(x+4m) + 70*(x-3m)

Solving this equation, we get x = 2.57 meters. This means that the boat moved 2.57 meters to the left, and the child reached a position 6.57 meters to the right of the pier. So, the child is now further away from the pier compared to the initial position.

I hope this explanation helps you understand the concept of center of mass motion and solve the problem. Keep practicing and don't hesitate to ask for help if you need it