A rope of mass m and length L hangs from a ceiling. a. Show that the wave speed on the rope a distance y above the lower end is v = 1gy. b. Show that the time for a pulse to travel the length of the string is ∆t = 22L/g.
Mass of rope: m
Length of rope: L
y is an assumed length measured from bottom of the rope.
We know that
Where, v is speed of sound
is tension in the string
and is the linear density of the string.
, where M is mass of string of y length.
g is acceleration due to gravity
Now substituting these values in equation (1) we get,
Now since we have the velocity as a function of length from bottom end of the rope, we can calculate time taken by wave to travel by integrating inverse of velocity over length L i.e.
You have just been pulled over for running a red light, and
the police officer has informed you that the fine will be $250. In
desperation, you suddenly recall an idea that your physics professor recently discussed in class. In your calmest voice, you tell the officer that the laws of physics prevented you from knowing that the light was red. In fact, as you drove toward it, the light was Doppler shifted to where it appeared green to you. “OK,” says the officer, “Then I’ll ticket you for speeding. The fine is $1 for every 1 km/h over the posted speed limit of 50 km/h.< How big is your fine? Use 650 nm as the wavelength of red light and 540 nm as the wavelength of green light.
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