]> Exercise 3

Math Animated™
Mathematical Introduction for Physics and Engineering
by Samuel Dagan (Copyright © 2007-2020)

Chapter 1: Differentiation; Section 1: Real Numbers; page 4

Presentation, Exercise 3

Question

A simple pendulum is swinging. In principle the time period of a swing T could depend on the following parameters: the swinging mass m, the length of the pendulum l and the constant acceleration on the surface of the earth g.

Find how T depends from the parameters by using physical dimensions!

Notation

We'll follow the notation for the 3 basic physical dimensions already used.

Part 1

By use of this notation, the dimensions of the variables from this exercise are

Part 2

We assume that the period T is expressed by the other variables by

T=c m α l β g γ , where c is a dimensionless constant.

Part 3

From part 1 and part 2 one obtains

Θ= Μ α Λ β ( Λ Θ −2 ) γ .

Part 4

The comparison of the powers of part 3 gives:

  • α=0
  • β+γ=0
  • −2γ=1

Part 5

The solution of part 4 is

  • α=0
  • β= 1 2
  • γ=− 1 2

and finally

T=c l g

Note

For oscilations with small amplitude c=2π .

Score

Parts 1 and 2 are worth 1 point each.

Parts 3 and 5 are worth 3 points each.
Part 4 is worth 2 points.