IAstronauts visiting Planet X have a 2.5 -m-long string...

IAstronauts 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?

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Key Concepts

Standing Waves

Standing waves arise when two waves of identical frequency and amplitude travel in opposite directions and interfere, forming fixed nodes and antinodes along the medium. Their formation in strings fixed at both ends is characterized by specific resonant frequencies corresponding to the harmonic modes of vibration.

Wave Propagation on a String

The speed of a wave traveling along a string is determined by the equation v = ?(T/?), where T is the tension in the string and ? is its linear mass density. This concept links the physical properties of the string and applied forces to the resulting behavior of waves.

Harmonic Frequencies

Harmonic frequencies in a vibrating string are the natural resonant frequencies at which standing waves occur. These frequencies are given by f_n = (n/2L) * v for a string of length L, where n is an integer corresponding to the harmonic number and v is the wave speed.

Tension Due to Gravitational Force

In scenarios where the tension in the string is provided by a hanging mass, the tension is equal to the weight of the mass (T = mg). This relationship connects the acceleration due to gravity directly to the dynamics of wave propagation in the string.

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