Astronauts visiting Planet X have a 2.6 -long string whose mass is 5.0g . They tie the string to a support, stretch it horizontally over a pulley 1.8m away, and hang a 1.8kg 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 64Hz and 80Hz , but at no frequencies in between.

Question

Astronauts visiting Planet X have a 2.6 -long string whose mass is 5.0g . They tie the string to a support, stretch it horizontally over a pulley 1.8m away, and hang a 1.8kg 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 64Hz and 80Hz , but at no frequencies in between.

What is the value of g , the free-fall acceleration, on Planet X?

Answer 1

To find the value of g, the free-fall acceleration on Planet X, we can use the formula:

nλ = 2L

where n is the harmonic number, λ is the wavelength, and L is the length of the string.

In this case, the frequencies of the standing waves are given as 64 Hz and 80 Hz, which correspond to the first and second harmonics respectively. Therefore, we have:

For the first harmonic:
64 Hz = v / λ

For the second harmonic:
80 Hz = v / (2λ)

where v is the speed of the wave on the string.

We can calculate the wavelength for both harmonics using the formula:

λ = v / f

where f is the frequency.

Using these equations, we can solve for the speed of the wave on the string (v).

For the first harmonic:
λ = v / f
λ = v / 64 Hz

For the second harmonic:
λ = v / (2f)
λ = v / (2 * 80 Hz)

Since the length of the string is given as 2.6 m, we can substitute this into the equation for the wavelength in terms of the speed:

2.6 m = v / 64 Hz

and

2.6 m = v / (2 * 80 Hz)

Simplifying the equations, we have:

v = 2.6 m * 64 Hz
v = 2.6 m * (2 * 80 Hz)

v = 166.4 m/s
v = 416 m/s

Since the speed of the wave on the string, v, is dependent on the tension in the string and the linear mass density, we can write the equation:

v = sqrt(T / μ)

where T is the tension in the string and μ is the linear mass density.

The linear mass density, μ, is given as:

μ = mass / length

μ = 5.0 g / 2.6 m
μ = 1.92 g/m

To determine the tension in the string, we can use the equation:

T = μ * g

Simplifying, we have:

T = (1.92 g/m) * g
T = 1.92 g^2/m

Now we can substitute the values of v and T into the equation v = sqrt(T / μ) and solve for g:

166.4 m/s = sqrt((1.92 g^2/m) / (1.92 g/m))

Simplifying, we have:

166.4 m/s = sqrt(g)

Squaring both sides of the equation gives:

(166.4 m/s)^2 = g

Therefore, the value of g, the free-fall acceleration on Planet X, is approximately 27,617.96 m/s^2.