A sphere of mass M is supported by a string that passes over a pulley at the end of a horizontal rod of length L. The string makes an angle è with the rod. The fundamental frequency of the standing waves in the portion of the string above the rod is f. Find the mass of the portion of the string above the rod.
To find the mass of the portion of the string above the rod, we can use the equation for the fundamental frequency of a string:
f = (1/2L) * sqrt(T/μ)
Where:
f is the frequency of the standing wave
L is the length of the string
T is the tension in the string
μ is the linear mass density of the string
In this case, the string is only supporting the sphere, so we can assume that the tension in the string is equal to the weight of the sphere:
T = Mg
Where:
M is the mass of the sphere
g is the acceleration due to gravity
The linear mass density of the string (μ) is given by:
μ = M/L
Substituting these values into the equation for the fundamental frequency gives:
f = (1/2L) * sqrt(Mg/(M/L))
Simplifying the equation:
f = (1/2L) * sqrt(gL)
To find the mass of the portion of the string above the rod, we rearrange the equation:
M = f^2 * 4L / g
So, the mass of the portion of the string above the rod is given by:
M = f^2 * 4L / g