Astronauts visiting Planet X have a 2.5 -m-long string whose...

Astronauts visiting Planet X have a 2.5 -m-long string whose mass is $5.0 \mathrm{g}$. They tie the string to a support, stretch it horizontally over a pulley $2.0 \mathrm{m}$ away, and hang a $1.0 \mathrm{kg}$ mass on the free end. Then the astronauts begin to excite standing waves on the string. Their data show that standing waves exist at frequencies of $64 \mathrm{Hz}$ and $80 \mathrm{Hz}$, but at no frequencies in between. What is the value of $g$, the free-fall acceleration on Planet X?

Key Concepts

Wave Speed in a String

The speed at which waves propagate along a string is determined by the tension in the string and its linear mass density according to the relation v = ?(T/?). When a mass is hung on the string, the tension is provided by the gravitational force acting on that mass, thus linking the wave speed to the gravitational acceleration.

Tension from a Hanging Mass

In systems where a mass is suspended from a string, the tension in the string is provided by the weight of the hanging mass. This tension is given by T = mg, where m is the mass and g is the gravitational acceleration. In problems where waves on a string are excited, the tension directly influences the wave speed, making it a key factor in the analysis of the system's resonant frequencies.

Standing Waves

Standing waves are the result of the interference between waves traveling in opposite directions along a medium, such as a stretched string. They are characterized by fixed points called nodes and maximum oscillation points called antinodes. The conditions for standing waves depend on the boundary conditions and the dimensions of the medium, with specific discrete (resonant) frequencies at which the pattern is sustained.

Linear Mass Density

The linear mass density is defined as the mass per unit length of a string or wire. It is an essential parameter in wave mechanics because it, together with the tension in the string, determines the speed of transverse waves along the string. In this context, it is calculated by dividing the total mass of the string by its length.

Harmonics and Resonant Frequencies

Harmonics are the integer multiples of the fundamental frequency at which standing waves can form in a medium with fixed boundaries. For a string fixed at both ends, the frequencies are given by f_n = (n/2L)v, where n is a positive integer, L is the length of the string, and v is the wave speed. The spacing between consecutive harmonics is constant and equal to the fundamental frequency, which is a critical insight in analyzing the observed frequencies.

Transcript

00:01 All right, so i've drawn a diagram and all the pertinent information.

00:07 I'm noticing that the length for vibration is 2 meters, and the frequencies are f, i'm going to write sub n, is 64 hertz, and f sub n plus 1.

00:31 Actually i'm going to use m, m plus 1, is 80 hertz.

00:38 Since there are no frequencies in between, we can just subtract these two.

00:45 80 minus 64 is going to give us the fundamental frequency, which would be 16 hertz.

01:02 All right.

01:03 I also know that the tension is just going to be the hageny, mass times the acceleration due to gravity on the planet.

01:14 We're trying to figure out the acceleration due to gravity on the planet.

01:22 V in the string is the square root of t over mule, which is the square root of capital m g over mu.

01:42 Now, perhaps lastly, i know that the fundamental frequency is v over 2l, which would be the square root of capital m, lower case g, over 2m, times 1 over 2l.

02:15 Mg over mew, 1 over 2l, where we know what l is.

02:21 We know what m is.

02:23 We know what looks like i wrote this is l also.

02:33 I'm going to call this l sub t for total length.

02:40 And so this is going to be l sub t.

02:42 So we do have two different ls here.

02:46 Where did i write mu? okay, there we go...