Puzzle #18 – Parks and Reciprocation

tower animationA rich physicist (!) wants to build an amusement park ride.

An engine drives a short, solid rod of length a at a constant angular velocity \omega in a vertical plane.

A second, long rod of length b > a is connected by a hinge to the free end of the short rod. The other end of the long rod is connected by a prismatic joint to a vertical beam; the joint is constrained to move vertically.

A platform for the passengers is mounted to the prismatic joint. As the short rod is made to rotate, the platform moves up and down the beam.

Let \varphi be the angle made by the short rod with the vertical direction.

 

tower

Questions:

1) What is the height h(\varphi) of the platform above the ground?

2) Find an expression for the acceleration of the platform at the top and bottom of its oscillation. That is, evaluate \frac{d^2 h}{dt^2} at \varphi = 0 and \varphi = \pi.

3) Find the ratio of the lengths of the short and long rods if the ride is constructed such that both:

  • the passengers on the platform feel 1.5 times their usual weight at the bottom of the oscillation;
  • the passengers feel weightless at the top of the oscillation.

 

Hints:

For part (1), separate the scalene triangle pictured into two right-angled ones.

For part (2), remember to use the fact that \omega\equiv\frac{d\varphi}{dt}.

For part (3), you will need to know the relation between the reaction force experienced by the passengers, and their acceleration.

 

Sorry for the worst play on words yet.

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