1)A toy train is initially at rest on a track fastened to a bicycle wheel, which is free to rotate. How does the wheel respond when the train moves clockwise? When the train backs up? Does the angular momentum of the wheel-train system change during these maneuvers? How would the resulting motions depend on the relative masses of the wheel and train?
counterclockwise, clockwise, no
The train and the tire move in opposite directions. The angular momentum stays the same IF the tire axle is frictionless. That is probably what they want ou to assume. The larger the (wheel/tire) mass ratio, the lower the (wheel/tire velocity ratio, because the
(MV)train = (MV)tire
When the toy train moves clockwise, it exerts a forward force on the track due to Newton's third law of motion. As a result, an equal and opposite force is exerted on the bike wheel, causing it to rotate in the opposite direction, counterclockwise.
When the train backs up, it exerts a backward force on the track, again due to Newton's third law. This backwards force causes the bike wheel to rotate in the same direction as the train, in this case clockwise.
The angular momentum of the wheel-train system remains constant during these maneuvers. According to the law of conservation of angular momentum, the total angular momentum of a system remains constant unless an external torque acts on it. In this case, since no external torque is acting, the angular momentum remains constant.
The resulting motions depend on the masses of the wheel and train. If the train has a significantly larger mass compared to the wheel, the effect on the wheel's motion will be smaller. On the other hand, if the wheel has a significantly larger mass, the train's motion will have a larger impact on the wheel's rotation. The masses of the wheel and train determine the magnitude of the forces involved, which in turn affects the resulting motion.
To understand how the wheel responds when the train moves clockwise or backs up, let's analyze the situation:
1) When the train moves clockwise:
When the train starts moving clockwise, it exerts a force on the track in that direction. According to Newton's third law, the track exerts an equal and opposite force on the train. Now, from the perspective of the wheel, this force on the track produces a torque (a rotational force) on the wheel, causing it to rotate clockwise. The wheel responds by accelerating in the same direction as the train's motion.
2) When the train backs up:
When the train backs up, it exerts a force on the track in the opposite direction (counter-clockwise). As a result, the track exerts an equal and opposite force on the train. From the wheel's perspective, this force produces a torque in the counter-clockwise direction, causing the wheel to rotate in the opposite sense. The wheel responds by accelerating in the opposite direction to the train's motion.
In both cases, the angular momentum of the wheel-train system remains conserved. Angular momentum is the product of the moment of inertia of the system and its angular velocity. As long as no external torque acts on the system, the angular momentum remains constant.
The resulting motions of the wheel and train depend on the relative masses of the wheel and train. If the mass of the train is much smaller compared to that of the wheel, the effect on the wheel's motion would be more significant. In this case, the wheel would experience a greater change in acceleration and rotational speed. However, if the mass of the train is comparable or larger than the wheel, the effect on the wheel's motion would be relatively smaller.
In summary, when the train moves clockwise or backs up, the wheel responds by rotating in the same or opposite direction, respectively. The angular momentum of the wheel-train system remains conserved, and the resulting motions depend on the relative masses of the wheel and train.