AP Physics 2 Exam Question - Interference and Standing Waves

A string is fixed at both ends and tied to a vibrator. The vibrator is set to produce a sinusoidal wave of frequency 200 Hz. The length of the string is 1.5 meters. At a certain time, the wave has a maximum displacement of 0.1 meters.

a) Calculate the wavelength of the wave.

b) Determine the speed at which the wave is traveling.

c) Find the amplitude of the wave.

d) Now suppose a node is formed at the midpoint of the string. Calculate the length of the standing wave on the string.

Answer:

a) The relationship between the wavelength ($\lambda$), frequency ($f$), and wave speed ($v$) is given by $v = \lambda \cdot f$.

Rearranging the equation to solve for wavelength, we have $\lambda = \frac{v}{f}$.

Given that the frequency is 200 Hz, we need to determine the wave speed. Since the string is fixed at both ends, the wave speed will be determined by the tension ($T$) in the string and the linear mass density ($\mu$) of the string.

The formula for wave speed on a string is $v = \sqrt{\frac{T}{\mu}}$.

Substituting the given values: $f = 200$ Hz $\mu$ is determined by the properties of the string $T$ is determined by the tension in the string, which is not provided

We proceed to part b) to calculate the wave speed.

b) The formula to calculate the wave speed on a string, $v = \sqrt{\frac{T}{\mu}}$, requires information about the tension ($T$) and linear mass density ($\mu$) of the string.

Since the tension is not provided, we cannot calculate the wave speed at this point.

c) The amplitude ($A$) of a wave is the maximum displacement from the equilibrium position. In this case, the amplitude is given as 0.1 meters.

d) In a standing wave, nodes and antinodes are formed. Nodes are points on the string that do not experience any displacement, while antinodes are points on the string that experience maximum displacement.

Since a node is formed at the midpoint of the string, we can say that the length of the standing wave on the string is twice the distance from the midpoint to either end of the string.

Given that the length of the string is 1.5 meters, the length of the standing wave will be $2 \times \frac{1.5}{2} = 1.5$ meters.

The length of the standing wave on the string is 1.5 meters.