Newton’s Shells and the Earth Tunnel

October 28th, 2015

Isaac Newton is best known for his three laws of motion, and for the inverse square law of gravitation. Among the most important of his many other discoveries are two results that Sir Oliver Lodge called Newton’s Superb Theorems. They both have to do with the gravity force produced by a thin spherical shell.

Suppose the shell has a total mass of M and a radius of R. Its thickness is small enough that it can be neglected. If we were to place a test mass in the vicinity of such a shell, we could measure the net gravitational attraction from the shell. This gravity force would be the sum of the attractive force from each tiny piece of the shell. Each piece has its own distance from the test mass, and would add to the total attractive force according to the inverse square law of gravity. Newton wanted to find a formula for the total gravity force from the shell in terms of M and R.

Read the rest of this entry »

We Weigh Less on an Airplane

August 9th, 2015

Some time ago I realized that, for a couple of reasons, an object’s weight while aboard an airline flight would be slightly less than the object’s weight on the ground. Note that I am referring to weight, as measured by a scale, not the mass of the object — mass, of course, stays the same.
In this post, I will detail the two reasons for the weight reduction, and compute the amount of the effect at the typical speed and altitude of a commercial airliner.
I will then describe an experiment I did on an actual flight, where I was able to verify that the weight of a small object was indeed less than its ground weight.

Read the rest of this entry »

Galileo’s Equal Time Ramps

April 24th, 2014

The time it takes for a ball to roll down a slope depends on the slope’s length and steepness. Galileo discovered an ingenious way to vary slope and steepness such that the descent time remains the same. Here, I go through the simple physics required to demonstrate that his method is correct.

Read the rest of this entry »

The Gamma Function – Factorials Unchained

April 8th, 2014

We all encounter factorials in high school math, including in the formulas for combinations, permutations, and the coefficients in the binomial theorem. We do not usually think of the factorial as being an example of a function. With an ordinary function, it is not restricted to having values only for positive integers, as is the case with factorials.

However, mathematicians have extended the concept of the factorial in such a way that we can find a factorial for any number. With this generalization, we can confidently say, for example, that -2.5)! = 2.36327…. It’s a bit like the way we extend the concept of exponents. A exponent that is a positive integer has an obvious meaning, but there is a need to generalize the exponent concept to include fractions and negative numbers.

In this post, I will describe this factorial function (which is actually called the Gamma function), plus a few uses for factorials in post high school math.

Read the rest of this entry »

Fabulous Formulas – the Rocket Equation

March 14th, 2014

Rockets were invented by the Chinese about 800 years ago, and were an early use of another Chinese invention: gunpowder. Considering that long history, it is surprising that it was not until 1903 that a physicist got around to finding the basic equation that lets us compute how a rocket will perform. That physicist was the Russian Konstantin Tsiolkovsky (1857 – 1935), the original rocket scientist.

The equation gives us the rocket’s velocity as a function of time, given four basic numbers that characterize the rocket. It is a pure application of Newton’s laws, and is fundamental to designing a rocket. In spite of its importance and interest, the high school physics textbooks that I have seen do not include it. This may be because the math behind it is not straightforward algebra. In fact, I have only seen it derived using calculus. In this post I will show how we can produce the equation without calculus, although there is a bit of hand waving involved in doing that.

After working out the equation, I will use it compute the final velocity of the Saturn V moon rocket.

Read the rest of this entry »

Standing Waves and String Vibration

January 22nd, 2014

As we all know, when a taunt string is plucked, it vibrates from side to side, producing a musical tone. In 1755, Daniel Bernoulli worked out the mathematics of vibrating strings. His goal was to determine the side-to-side displacement of the string at each point along its length as a function of time. This was a very new kind of problem. Each piece of the string must obey Newton’s laws of motion, but there are an infinite number of pieces, and they are interconnected.

Bernoulli’s solution predicted that there should be two simultaneous waves of displacement along the string, one travelling left, and the other to the right. The resulting string motion is the sum of these two waves.

Read the rest of this entry »