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 remains at rest while the carts accelerate toward each other.
It accelerates toward the cart with smaller mass because it speeds up more.
It moves toward the magnet because magnetic forces are stronger than contact forces.
It moves in the direction of whichever cart is initially closer to the center of mass.
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 moves backward because the exhaust has greater total momentum.
It remains at rest because there is no net external force on the system.
It must stay at the rocket’s geometric center because the rocket is symmetric.
It moves forward because the rocket speeds up.
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 east because puck 1 hits first.
The center of mass reverses direction at the instant they stick.
The center of mass moves with the stuck-together pucks because internal forces create net momentum.
The center of mass remains at rest before, during, and after the collision.
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.
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 moves toward the cart with greater mass because the center of mass must be inside the heavier object.
It accelerates in the direction of the larger cart’s motion because internal forces are unbalanced.
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.
Two blocks $m_1$ and $m_2$ rest on a rough horizontal floor and are connected by a light string. A student pulls on block $m_1$ with a constant horizontal force to the right. Take the system boundary to include both blocks and the string. Friction from the floor on each block is external to the system. Which statement about the center-of-mass acceleration is correct?
It is zero because internal forces (tension) cancel in pairs.
It is determined by the net external horizontal force on the two-block system.
It depends only on the tension in the string, since tension is the largest force.
It must point toward the more massive block because the center of mass is closer to it.
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, such as the tension in the string, do not affect the net force on the system. The net external horizontal force includes the pulling force and friction on both blocks, which determines the center-of-mass acceleration. Choice C is incorrect because internal forces like tension do cancel, but the acceleration is due to external forces, not zero unless those are balanced. To solve similar problems, always identify the system boundary and calculate the net external force to determine the center-of-mass acceleration.
Two pucks connected by a taut string slide right on frictionless ice. The string snaps due to internal tension. What happens to the center-of-mass velocity of the two-puck system?
It remains unchanged because the snapping involves only internal forces.
It becomes zero because the pucks separate.
It increases because stored energy becomes kinetic energy.
It decreases because the string’s tension opposes the motion.
Explanation
This question examines how internal forces affect center-of-mass motion. The center of mass of a system moves according to the net external force only—internal forces have no effect on it. The string tension between the pucks is an internal force within the two-puck system. When the string snaps, this internal force disappears, causing the pucks to change their individual velocities relative to each other. However, on frictionless ice with no external horizontal forces, the center-of-mass velocity must remain unchanged. The pucks may separate or change speeds individually, but their center of mass continues moving right at the same velocity. Choice B incorrectly suggests that energy conversion affects center-of-mass motion. Remember: internal forces can change relative positions and velocities within a system but never alter the center-of-mass velocity.
A bowling ball and a tennis ball are dropped together (no air resistance). For the system of both balls, what is true about the center-of-mass acceleration?
It equals $g$ downward because the only significant external force is gravity.
It is less than $g$ because internal forces between the balls reduce the net force.
It is zero because both balls accelerate equally.
It is greater than $g$ because the heavier ball pulls the lighter ball down.
Explanation
This problem tests understanding of center-of-mass acceleration under gravity. The center-of-mass acceleration equals the net external force divided by total mass, regardless of internal forces or individual object motions. For the two-ball system, gravity acts as an external force on both balls, giving each ball a downward acceleration of g. The center of mass, being a weighted average position, must also accelerate downward at g. Any forces between the balls (like air pressure differences) are internal and don't affect center-of-mass acceleration. Choice C incorrectly assumes internal forces can reduce the effect of external forces on the center of mass. When analyzing multi-object systems under gravity, the center of mass always falls with acceleration g (neglecting air resistance), regardless of the objects' masses or interactions.
A student tosses a ball straight up while standing on a skateboard. For the system (student+skateboard+ball), which statement about center-of-mass motion is correct?
The center of mass moves upward at constant velocity until the ball reaches its peak.
The center of mass accelerates upward while the ball is rising.
The center of mass accelerates downward due to the external gravitational force.
The center of mass remains fixed in space because internal forces cancel.
Explanation
This question examines center-of-mass motion under external forces. The center of mass of any system accelerates according to Newton's second law: the net external force divided by the total mass. For the student-skateboard-ball system, gravity is an external force acting downward on all parts of the system. Therefore, the center of mass must accelerate downward at g, regardless of the internal motions (ball going up, student/skateboard recoiling down). The tossing forces between student and ball are internal and cannot affect center-of-mass acceleration. Choice D incorrectly assumes the center of mass follows the ball's motion, but it actually follows a parabolic path like any projectile. When analyzing multi-object systems, remember that internal forces never affect center-of-mass motion—only external forces matter.
A firework rises vertically and explodes into two equal fragments that fly apart horizontally. Neglect air resistance. How does the center-of-mass motion change at the explosion?
It accelerates upward because chemical energy is released.
It changes to horizontal motion because the fragments move horizontally.
It continues as if no explosion occurred, since the explosion forces are internal.
It suddenly stops because the fragments move in opposite directions.
Explanation
This question tests understanding of center-of-mass motion during internal explosions. The center of mass of a system follows a trajectory determined solely by external forces, regardless of internal interactions. Before the explosion, the firework's center of mass follows a parabolic path under gravity (neglecting air resistance). The explosion forces are internal to the firework system—they're forces between parts of the firework itself. These internal forces cause the fragments to fly apart but cannot change the center-of-mass trajectory. After explosion, the center of mass continues on the same parabolic path it would have followed if no explosion occurred. Choice A incorrectly assumes that opposite motions of fragments affect the center of mass. Remember: internal forces can rearrange mass within a system but never change the system's center-of-mass motion.
Two carts on a frictionless track form a system. Cart A pushes Cart B with a spring, then they separate. What happens to the system’s center-of-mass velocity?
It changes direction to point toward the larger-mass cart.
It stays constant because only internal forces act horizontally on the system.
It becomes zero because the carts move apart in opposite directions.
It increases because the spring force adds momentum to the system.
Explanation
This question tests understanding of center-of-mass motion for systems with internal forces. The center-of-mass velocity of a system changes only when there's a net external force acting on the system. Here, the spring force between the carts is an internal force—it's part of the system itself. Since the track is frictionless, there are no external horizontal forces acting on the two-cart system. The spring may cause the individual carts to accelerate in opposite directions, but these are internal accelerations that don't affect the center of mass. Choice A incorrectly suggests internal forces can add momentum to a system, which violates conservation of momentum. To solve center-of-mass problems, always identify whether forces are internal or external to your chosen system.