2008-09-26

The Great Clock

Bengt has a big clock on his wall. It's an old mechanical clock, powered by a spring. The spring keeps a pendulum wheel in motion. The pendulum wheel is spinning back and forth. The torque on the wheel from the spring is constant in magnitude but with alternating direction.
The wheel, which is made of copper, 1 mm thick, and with 1 dm circumference, moves a quarter of a turn from its equilibrium position in each direction. Each time it reaches one side (but not when it reaches the other) a big gear moves one step. On the same axis as this gear is small gear, which meshes with an identical big gear, moving it ahead one step for every turn of the small gear. This second big gear moves the second hand on the clock, moving with the same angular velocity as it. On the same axis as this second big gear is another small gear, which similarly meshes with a gear for the minute hand. And so on.
Apart from the normal hands, this clock also has a hand which completes a turn in one year. It is one meter long.
"Suppose that the gears and the hands have no mass." thinks Bengt. "And suppose that the clock won't break if I hang in it." So he does.
Bengt jumps up and grabs a hold of the tip of the year hand. It is autumn, so it is pointing to the left, moving upwards.
  1. How big is the torque on the pendulum wheel from the spring?
  2. Even though Bengt hangs in the year hand, the clock doesn't stop. How heavy would he have to be to stop the clock?
  3. Bengt's mass is 100 kg. How much behind would the clock be if he hangs there for 24 hours?
  4. "When spring comes," thinks Bengt, "I'm going to hang in the year hand again, to see if I can make it go twice as fast." How heavy would Bengt have to be to succeed in this?