A block and a spool have the same mass, and are each pulled across a level, frictionless surface by a string, as shown to the right. The string pulling the spool is wrapped around it many times, and the other string is fastened to the center of the block. Both hands pull with the same force, starting at the same instant, and maintain a constant tension.
A) Describe how each object moves on the frictionless surface in terms of both translation and rotation.
B) Does the hand pulling the spool’s string move faster, slower, or at the same speed as the other hand in order to maintain the same tension in the strings? Explain.
C) Which object, if either, passes the finish line first? Explain your reasoning.
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A) In terms of translation, both the block and the spool will move in the same direction as the applied force. However, due to the spool's shape, it will also experience rotation as it moves. This is because the string wrapped around the spool causes a torque on it, resulting in rotational motion. The block, on the other hand, does not experience this torque since the string is attached to its center, so it only undergoes pure translation.
B) The hand pulling the spool's string moves at a slower speed compared to the other hand in order to maintain the same tension in the strings. This is because the spool, being a cylinder, has a larger rotational inertia compared to the block. Therefore, to maintain the same tension, the slower speed is necessary to overcome the rotational inertia of the spool and prevent it from rotating too quickly.
C) In terms of passing the finish line first, the block is likely to be the winner. This is because the block only experiences translation and does not have the additional rotational motion that the spool has. Consequently, the block can move directly towards the finish line, while the spool has the added rotational motion that slows it down. Unless the spool is specifically designed to roll without slipping, it will experience some energy loss due to rotational motion, making the block the faster object.