Fun With Fundamentals: Problem 197

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Problem 197 — One and one don’t always make two, as this month’s problem by Jim Irlander of St. Louis shows.

“Thank heavens!” shouted Lucius Bluff, as the rental truck pulled into the driveway. The Lee Key Hydraulic Company was in severe danger of not meeting its special “Double-Price, Double-Service” 24-hr delivery option. Bluff had one hour to deliver two identical sections of round pipe to his fussiest customer, Percy McTwipp.

The truck Bluff rented had a closedtop trailer compartment that was 8 ft high by 8 ft wide. The pipe sections have 60-in. ODs. Compute the maximum diameter of the two pipe sections that can fit inside the trailer section as shown to see if Bluff will be writing a refund check.

Technical consultant, Jack Couillard, Menasha, Wis.

Solution to last month’s problem 196 — You understand the gravity of the situation, if you answered 2.95 hr. Here’s how Gotrocks IX of the 22nd century missed his tennis game:

Let:
G = Gravitational constant, 3.44 × 10^-8 ft⁴ lb^-1 sec^-4
m_p = Mass of planet, slugs
m_s = Mass of satellite, slugs
r_p = Radius of planet, ft
r_s = Radius of satellite’s orbit, ft
d = Specific gravity, given as 10
v = Velocity of satellite, fps
t = Time to complete one orbit, secg = Acceleration due to gravity, 32.2 ft/sec²

The planet’s gravitational force on the satellite is equal to the centrifugal force exerted on the planet by the satellite.

We can rewrite m_p using the planet’s specific gravity, the average density of water at a specific gravity of 1, and the formula for the volume of a sphere:

Also rewrite v in terms of other variables:

Equation (1) then becomes:

Since Marcia set the orbit radius equal to the planet’s diameter, r_s = 2r_p. You can now reduce (2) to:

Solve for t:

Time can also fly when you’re not having fun!

Contest winner — Congratulations to Robert May of Warminster, Pa. who won our June contest by having his name drawn from the 134 contestants who answered correctly out of a total of 138 entrants for that month. A TI-85 calculator is in the mail to him.

The TI-85 Graphing Calculator from Texas Instruments solves for any variable in an equation, can solve 30 simultaneous equations, and finds the roots of a polynomial up to the 30th order. It handles complex numbers in addition to matrices, vectors, lists, and strings. You can perform graphic investigations of almost any type of problem — functions as well as parametric, polar, and differential equations.