Systems and Center of Mass
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AP Physics 1 › Systems and Center of Mass
A magnet attracts a steel cart on a low-friction track. The magnet is mounted to a second cart; the carts pull toward each other and eventually collide. Define the system as both carts (including the magnet). The magnetic forces between the carts are internal. Assume external horizontal forces are negligible. Initially both carts are at rest. What is the correct description of the center-of-mass motion?
It accelerates toward the cart with smaller mass because it speeds up more.
It moves in the direction of whichever cart is initially closer to the center of mass.
It remains at rest while the carts accelerate toward each other.
It moves toward the magnet because magnetic forces are stronger than contact forces.
Explanation
This question tests the concept of center-of-mass motion for a system of particles. The motion of the center of mass of a system is determined solely by the net external force acting on the system, as if all the mass were concentrated at the center of mass and all external forces acted there. Internal magnetic forces between the carts cancel in pairs and do not produce net force on the system. With negligible external horizontal forces and initial rest, the center of mass remains at rest as the carts approach. Choice A is incorrect because magnetic forces are internal and cannot move the center of mass without external influence. To solve similar problems, always identify the system boundary and calculate the net external force to determine the center-of-mass acceleration.
A rocket in deep space ejects exhaust gases backward. Define the system as rocket + exhaust gases that have been expelled. Forces between the rocket and the exhaust are internal to this system, and external forces are negligible. Initially the system is at rest. As fuel burns and exhaust is expelled, which statement about the center of mass is correct?
It remains at rest because there is no net external force on the system.
It moves forward because the rocket speeds up.
It must stay at the rocket’s geometric center because the rocket is symmetric.
It moves backward because the exhaust has greater total momentum.
Explanation
This question tests the concept of center-of-mass motion for a system of particles. The motion of the center of mass of a system is determined solely by the net external force acting on the system, as if all the mass were concentrated at the center of mass and all external forces acted there. Internal forces between the rocket and exhaust gases cancel out, preserving the system's total momentum. In deep space with negligible external forces and initial rest, the center of mass remains at rest. Choice B is incorrect because the rocket's forward motion is balanced by the exhaust's backward momentum within the system. To solve similar problems, always identify the system boundary and calculate the net external force to determine the center-of-mass acceleration.
Two identical pucks slide on frictionless ice. They collide and stick together. Choose the system as both pucks together; the contact forces during the collision are internal. There are no external horizontal forces. Before the collision, puck 1 moves east and puck 2 moves west with equal speed. Which statement about the center-of-mass motion is correct?
The center of mass moves with the stuck-together pucks because internal forces create net momentum.
The center of mass moves east because puck 1 hits first.
The center of mass remains at rest before, during, and after the collision.
The center of mass reverses direction at the instant they stick.
Explanation
This question tests the concept of center-of-mass motion for a system of particles. The motion of the center of mass of a system is determined solely by the net external force acting on the system, as if all the mass were concentrated at the center of mass and all external forces acted there. Internal forces during the collision cancel out and do not affect the center-of-mass velocity. With no external horizontal forces and initial total momentum zero, the center of mass remains at rest throughout. Choice D is incorrect because internal forces cannot create net momentum; the stuck pucks stop, but the center of mass stays put. To solve similar problems, always identify the system boundary and calculate the net external force to determine the center-of-mass acceleration.
A firework explodes into two fragments while moving horizontally in midair; for the two fragments as the system during the explosion, what happens to the center-of-mass horizontal motion?
It stops because the fragments move in opposite directions after the explosion.
It changes because the explosion provides a large internal impulse.
It moves toward the larger fragment because the center of mass must lie inside the larger piece.
It continues at constant velocity because external horizontal forces on the system are negligible.
Explanation
This question tests understanding of center-of-mass motion for systems with only internal forces. When a firework explodes into fragments, the explosion forces are internal to the fragment system—they're action-reaction pairs between pieces. The center of mass of a system responds only to net external forces, not internal forces. During the brief explosion in midair, external horizontal forces (like air resistance) are negligible compared to the huge internal explosion forces, so the center of mass continues at constant horizontal velocity through the explosion. Choice C incorrectly assumes opposite fragment motions cause the center of mass to stop, but the center of mass depends on the mass-weighted average position, not individual velocities. To analyze explosive separations, recognize that violent internal forces redistribute momentum within the system while total momentum—and thus center-of-mass velocity—remains constant.
Two astronauts push off each other in deep space, far from other forces; for the two-astronaut system, how does the center of mass move?
It accelerates toward the astronaut who pushes harder because internal forces can change system motion.
It moves toward the astronaut with smaller mass because that astronaut moves faster.
It remains at rest or continues at constant velocity because no external forces act on the system.
It moves in a circle because the astronauts move away from each other in opposite directions.
Explanation
This question tests understanding of center-of-mass motion for systems with only internal forces. When two astronauts push off each other in deep space, the push forces are internal to the two-astronaut system—they form Newton's third law pairs. The center of mass of any system accelerates only in response to net external forces. Since the astronauts are far from other forces in deep space, no external forces act on the system, so the center of mass remains at rest or continues at constant velocity. Choice A incorrectly suggests one astronaut pushing harder creates unbalanced internal forces, but Newton's third law guarantees the forces are always equal and opposite, regardless of who initiates the push. Remember: the absence of external forces means the system's center of mass maintains its initial state of motion, even as components move dramatically relative to each other.
Two carts on a frictionless track collide and stick together; for the two-cart system, which statement about center-of-mass motion is correct?
It must stop after collision because kinetic energy decreases.
It changes abruptly at collision because the carts exert large forces on each other.
It becomes located at the geometric midpoint between the carts after they stick.
It continues with constant velocity because the collision forces are internal to the system.
Explanation
This question tests understanding of center-of-mass motion for systems with only internal forces. During a collision between two carts on a frictionless track, the collision forces are internal to the two-cart system—they're action-reaction pairs between system components. The center of mass of any system accelerates only when acted upon by net external forces. Since the track is frictionless and no external horizontal forces act on the system, the center of mass continues with constant velocity through the collision, regardless of how violently the carts interact. Choice A incorrectly assumes large internal forces can change center-of-mass motion, but Newton's third law ensures internal forces always cancel when considering the whole system. Remember: collisions may dramatically rearrange mass within a system, but they cannot change the system's center-of-mass velocity without external forces.
A student on a low-friction cart pulls a rope attached to a second cart; for the two carts as the system, what is true about center-of-mass motion?
It accelerates toward the student because the student provides an applied force on the system.
It moves toward the cart that is closer because the center of mass is always between objects.
It must move with the faster cart because the center of mass follows the fastest object.
It remains at rest or continues at constant velocity because the rope tension forces are internal.
Explanation
This question tests understanding of center-of-mass motion for systems with only internal forces. When a student on one cart pulls a rope attached to another cart, the rope tension forces are internal to the two-cart system—they act between system components. The center of mass of a system accelerates only when acted upon by net external forces. Since the carts are on a low-friction surface with negligible external horizontal forces, the center of mass remains at rest or continues at constant velocity. Choice A incorrectly identifies the student's pull as an external applied force, but since the student is part of the system (on one cart), all forces they exert on the other cart are internal. To correctly analyze center-of-mass motion, carefully define your system boundaries—forces are only external if they originate from outside the entire system.
A cart on a frictionless track carries a spring-loaded launcher that fires a second cart backward; for both carts as the system, what happens to the center of mass?
It moves toward the cart with the stronger launch because that cart has greater momentum.
It remains at rest or moves at constant velocity because the spring force is internal to the system.
It accelerates in the direction of the cart launched forward because the spring adds energy.
It moves to the physical center of the larger cart because centers of mass lie within objects.
Explanation
This question tests understanding of center-of-mass motion for systems with only internal forces. When a spring-loaded launcher on one cart fires another cart backward, the spring force is internal to the two-cart system—it acts between system components. The center of mass of a system responds only to net external forces, never to internal forces alone. Since the track is frictionless and no external horizontal forces act on the system, the center of mass remains at rest or moves at constant velocity, unchanged by the spring launch. Choice A incorrectly claims the spring adds energy to change center-of-mass motion, but energy addition doesn't violate momentum conservation—internal forces redistribute momentum within the system without changing total momentum. To solve center-of-mass problems, recognize that internal forces can dramatically change individual motions while leaving center-of-mass motion unchanged.
A person stands on a skateboard and throws a ball forward; with person+skateboard+ball as the system, how does the center of mass behave while the ball is in flight?
It moves with constant horizontal velocity because external horizontal forces on the system are negligible.
It accelerates forward because the ball exerts a forward force on the system after release.
It jumps to the ball’s location since the ball is the only moving object.
It remains fixed in space because the center of mass cannot move when objects separate.
Explanation
This question tests understanding of center-of-mass motion for systems with only internal forces. When a person throws a ball while standing on a skateboard, the throwing force is internal to the person+skateboard+ball system. The center of mass of a system moves according to the net external force acting on it. Since there are negligible external horizontal forces (assuming a level surface with minimal friction/air resistance), the system's center of mass maintains constant horizontal velocity throughout the ball's flight. Choice A incorrectly suggests the ball can exert an external force on its own system after release, which is impossible—all ball-person interactions are internal. To analyze center-of-mass motion, first define your system boundaries, then identify which forces cross those boundaries (external) versus which act within them (internal).
Two students on frictionless carts push off each other; considering both carts as the system, how does the center of mass move afterward?
It remains at rest or continues at constant velocity, because only internal forces act within the system.
It oscillates back and forth between the carts as they separate due to the push.
It accelerates in the direction of the larger cart’s motion because internal forces are unbalanced.
It moves toward the cart with greater mass because the center of mass must be inside the heavier object.
Explanation
This question tests understanding of center-of-mass motion for systems with only internal forces. When two students on frictionless carts push off each other, the push forces are internal to the two-cart system—they form an action-reaction pair between system components. According to Newton's laws, the center of mass of a system accelerates only when acted upon by a net external force. Since the system experiences no external horizontal forces (frictionless surface), the center of mass remains at rest or continues at constant velocity. Choice A incorrectly suggests internal forces can accelerate the system's center of mass, which violates conservation of momentum. To solve center-of-mass problems, always identify whether forces are internal or external to your chosen system.