A small mass m swings in a horizontal circle of radius r at the end of a string of length Li and
negligible mass, which makes an angle �i with the vertical. The string is slowly shortened (for
example by pulling it through a hole in its support) until the final length is Lf and the string is
making an angle �f with the vertical. Find an expression for Lf in terms of Li; �i and �f .
The height of pivot point above the plane of rotation is h
cosi=h/Li
cosf=h/Lf
Lf=Li•cosi/cosf
To find an expression for Lf in terms of Li, θi, and θf, we can apply the law of conservation of energy.
Initially, when the mass m is at the angle θi, it has gravitational potential energy and kinetic energy due to its circular motion. The total energy at this point is given by:
Ei = mgh + (1/2)mv^2
Where h is the height of the mass from its lowest position and v is the velocity of the mass.
Assuming the mass follows a circular path, we can express the velocity v in terms of the angular velocity ω (omega) and the radius r:
v = ωr
Next, using the relationship between the angular velocity and the angle with the vertical, we have:
v = ωr = √(gr) tan(θi)
Plugging this value of v into the equation for energy, we have:
Ei = mgh + (1/2) m(√(gr) tan(θi))^2
Now, as the string is shortened and the mass moves to an angle θf, the final energy Ef is given by:
Ef = mgh + (1/2) m(√(gr) tan(θf))^2
Since the mass doesn't lose any energy during the process, we can equate Ei and Ef:
mgh + (1/2) m(√(gr) tan(θi))^2 = mgh + (1/2) m(√(gr) tan(θf))^2
Canceling out the common terms, we have:
(√(gr) tan(θi))^2 = (√(gr) tan(θf))^2
Simplifying further, we get:
(gr) tan^2(θi) = (gr) tan^2(θf)
Since we are interested in finding an expression for Lf in terms of Li, θi, and θf, we can relate the length Li to the angle θi using trigonometry:
Li = r cos(θi)
Similarly, the length Lf can be related to the angle θf:
Lf = r cos(θf)
Substituting these expressions into the equation, we have:
(g(r cos(θi))) tan^2(θi) = (g(r cos(θf))) tan^2(θf)
Simplifying further, we get the desired expression:
Lf = Li / (tan^2(θf)/tan^2(θi))
Therefore, the expression for Lf in terms of Li, θi, and θf is:
Lf = Li / (tan^2(θf)/tan^2(θi))