Exercises: Systems and Center of Mass

Two carts sit on a lab track. A cart of mass $0.50\ \text{kg}$ is at $x = 0$, and a cart of mass $1.50\ \text{kg}$ is at $x = 0.80\ \text{m}$. Calculate the position of the center of mass of the two-cart system.

Emma pushes a box across a rough floor. She defines the system as "the box only." Classify each of the following forces as internal or external to this system: Emma's push on the box, the friction force from the floor, and the gravitational force from Earth. Then say whether the system would change if she defined it as "box + Emma."

Three point masses lie on the $x$-axis: $2.0\ \text{kg}$ at $x = -1.0\ \text{m}$, $3.0\ \text{kg}$ at $x = 0$, and $1.0\ \text{kg}$ at $x = 4.0\ \text{m}$. Find the $x$-coordinate of the center of mass.

Jayden (mass $60\ \text{kg}$) and his sister Aaliyah (mass $40\ \text{kg}$) stand at opposite ends of a $4.0\ \text{m}$ long boat of mass $100\ \text{kg}$ floating at rest on a calm lake. Treat the boat's center of mass as its midpoint, and take the origin at Jayden's end. (a) Find the center of mass of the system of boat plus two people. (b) Ignoring water resistance, the two people walk toward each other and meet at the boat's midpoint. Explain what happens to the position of the system's center of mass relative to the shore, and why.

A $2.0\ \text{kg}$ ball moves at $3.0\ \text{m/s}$ in the $+x$ direction and a $3.0\ \text{kg}$ ball moves at $2.0\ \text{m/s}$ in the $-x$ direction. No net external force acts on the system. (a) Find the velocity of the center of mass. (b) Describe how this velocity changes when the balls collide with each other, and justify your answer.

A $1.0\ \text{kg}$ block is at $(0, 0)$ and a $3.0\ \text{kg}$ block is at $(4.0\ \text{m}, 2.0\ \text{m})$. A constant external force of $8.0\ \text{N}$ in the $+x$ direction acts on the $1.0\ \text{kg}$ block only. The blocks are connected by a light rod. (a) Find the coordinates of the center of mass. (b) Find the magnitude and direction of the acceleration of the center of mass.

Marcus ($70\ \text{kg}$) stands at one end of a uniform $6.0\ \text{m}$ canoe of mass $30\ \text{kg}$ at rest on still water. The end where he stands is $1.0\ \text{m}$ from a dock, and the canoe's long axis points away from the dock. Marcus walks to the other end of the canoe. Ignore water resistance. (a) Explain why the center of mass of the Marcus-canoe system does not move relative to the dock. (b) Calculate how far the canoe moves, and in which direction relative to the dock. (c) Calculate Marcus's final distance from the dock.

A firework of total mass $1.2\ \text{kg}$ is launched straight up. At the top of its flight, momentarily at rest, it explodes into two pieces. A $0.40\ \text{kg}$ piece moves horizontally east at $30\ \text{m/s}$ immediately after the explosion. (a) Determine the velocity (magnitude and direction) of the other piece immediately after the explosion. (b) Describe the motion of the center of mass of the two pieces after the explosion, ignoring air resistance, and explain how it is consistent with your answer to (a). (c) Explain why the explosion's internal forces cannot change the center of mass's motion, while gravity can.