Problem 80 - Algebra and Trigonometry

Question

A tuning fork is held a certain distance from your ears and struck. Your eardrums' vibrations after \(t\) seconds are given by \(p=3 \sin 2 t .\) When a second tuning fork is struck, the formula \(p=2 \sin (2 t+\pi)\) describes the effects of the sound on the eardrums' vibrations. The total vibrations are given by \(p=3 \sin 2 t+2 \sin (2 t+\pi)\) a. Simplify \(p\) to a single term containing the sine. b. If the amplitude of \(p\) is zero, no sound is heard. Based on your equation in part (a), does this occur with the two tuning forks in this exercise? Explain your answer.

Step-by-Step Solution

Answer

The equation simplifies to \(p = \sin 2t\). There are certain times when \(p=0\), which corresponds to periods of no sound.

Step 1: Simplifying the equation

The problem provides the equation \(p=3 \sin 2 t+2 \sin (2t + \pi)\). This should be simplified. The second term in the equation can be expanded using the sine of a sum formula (\(\sin (a+b) = \sin a \cos b + \cos a \sin b\)). This gives us: \(2 \sin (2t + \pi) = 2 \sin 2t \cos \pi + 2 \cos 2t \sin \pi\). \(\sin \pi = 0\) and \(\cos \pi = -1\) simplifies the equation to \(2 \sin 2t \cdot -1\), or just \(-2 \sin 2t\). Now substituting this back into the original equation given, \(p = 3 \sin 2t + (-2 \sin 2t)\), simplifies down to \(p = \sin 2t\).

Step 2: Checking if the amplitude can be zero

Looking at the simplified equation \(p = \sin 2t\), we know that the sine function oscillates between -1 and 1 for all values of t. Hence, there are certain values of t at which the sine function, and therefore the value of \(p\), would indeed be zero. Hence, there will be times when no sound is heard.