Fun With Fundamentals: Problem 214

Carnival of venue
PROBLEM 214 -- There are a number of ways to be all wet, as this month's problem by Steve Wyatt of Beloit, Wis., demonstrates.

It was a cloudy day, and deep within the recesses of the Sneeterville County Courthouse, a heated trial was underway.

"Your honor, I would like to point out that my client suffered severe emotional trauma and humiliation as a result of the negligence of the Fun Unlimited Carnival," intoned Rufus Filibuster, attorney at law, "not to mention damage to a $1,000 business suit.

"Ladies and gentlemen of the jury, the facts of the case are these: My client, Barnaby McSnuff, stepped aboard the Whirl-Lee Bird ride expecting to have an enjoyable experience. Due to the carnival management's inane layout of the ride, the chair in which he sat passed through a nearby artificial waterfall, thus soaking my client to the skin, destroying his apparel, and humiliating him before his colleagues. We therefore are seeking no less than $1,000,000 in damages."

"Why that's nonsense!" roared carnival owner Buford Buff. "He admitted he fell into a pond."

"Order in the court" thundered Judge Dockett as he banged his gavel.

The Whirl-Lee Bird ride consists of one-person seats attached with cables to the top of a revolving vertical pole. The waterfall is 30 ft away from the center of the pole. The ride turns at 10 rpm, and the cable is 40 ft long. Mc- Snuff and the chair together weigh 150 lb. For simplicity's sake, assume the rider and chair occupy a space equivalent to a 4-ft-diameter sphere and that the cable is weightless.

Compute the radius of the chair from the pole while the ride is in motion to see if McSnuff will get his settlement.

SOLUTION TO LAST MONTH'S PROBLEM 213 -- You put the proper spin on it if you answered 2.11 rps.

Here's the solution.

Since the initial energy imparted to the globes is constant, the kinetic energy of rotation is the same for both globes:

ϖ₁ = Angular velocity of first 80- kg globe, given as 4π rad/sec

ϖ₂ = Angular velocity of second 72-kg globe, rad/sec

I₁ = Moment of inertia of first globe, kg-m²

I₂ = Moment of inertia of second globe, kg-m²

The moment of inertia for a solid sphere is defined as:

Equation (1) reduces to:

ϖ₁²m₁ = ϖ₂²m₂ (2)

From the problem, we know that m₁ is 80 kg, and m₂ is 72 kg. Plug these values into (2) to get