# Puzzle #18 – Parks and Reciprocation

A 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.

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.