Brachistochrone Problem

Question

If a particle falls from rest under the influence of gravity from higher to lower point in the minimum time, what is the curve that the particle will follow?

Solution

Suppose v is the speed of the particle along the curve, then in traversing ds portion of the curve time spent would be \(\frac{{ds}}{v}\) so that total time taken by the particle in moving from highest point 1 to lowest point 2 will be

\[{t_{12}} = \int\limits_1^2 {\frac{{ds}}{v}} \]

Suppose vertical distance of fall upto point 2 be x, then from the principle of conservation of energy of the particle we find that

\(\begin{array}{l}\frac{1}{2}m{v^2} = mgx\\or\\v = \sqrt {2gx} \\then\\{t_{12}} = \int\limits_1^2 {\frac{{\sqrt {d{x^2} + d{y^2}} }}{{\sqrt {2gx} }}} dx\\{\rm{      = }}\int\limits_1^2 {\frac{{\sqrt {1 + {{\dot y}^2}} }}{{\sqrt {2gx} }}} dx\\{\rm{      = }}\int\limits_1^2 {fdx} \end{array}\)

Where,

\( f=\left ( \frac{1+\dot{y}^{2}}{2gx} \right )^{\frac{1}{2}} \)

For \({t_{12}}\) to be minimum equation

\(\frac{d}{{dx}}\left( {\frac{{\partial f}}{{\partial \dot y}}} \right) - \frac{{\partial f}}{{\partial y}} = 0\)

must be satisfied. From expression for \(f\) we find that

\(\begin{array}{l}\frac{{\partial f}}{{\partial y}} = 0\\\frac{{\partial f}}{{\partial \dot y}} = \frac{{\dot y}}{{\sqrt {2gx} \sqrt {1 + {{\dot y}^2}} }}\\\frac{d}{{dx}}\left( {\frac{{\dot y}}{{\sqrt {2gx} \sqrt {1 + {{\dot y}^2}} }}} \right) = 0\\or\\\frac{{{{\dot y}^2}}}{{2gx(1 + {{\dot y}^2})}} = c'\end{array}\)

Where c’ is the constant of integration.

Since c’ is a constant we can also write

\(\frac{{{{\dot y}^2}}}{{x(1 + {{\dot y}^2})}} = c\)

where c is also a constant. On integrating above equation we find

\(\begin{array}{l}\frac{{{{\dot y}^2}}}{c} = x(1 + {{\dot y}^2})\\or,\\{{\dot y}^2}\left( {\frac{1}{c} - x} \right) = x\\{{\dot y}^2}\left( {\frac{x}{c} - {x^2}} \right) = {x^2}\\\dot y = \frac{x}{{\sqrt {\frac{x}{c} - {x^2}} }}\end{array}\)

Putting \(\frac{1}{c} = 2a\) , and on integration we get

\(\int dy=\int \frac{x}{\sqrt{2ax-x^{2}}}dx\)

\( y=acos^{-1}(1-\frac{x}{a})-(2ax-x^{2})^{1/2}+c''\)

where c'' is new constant of integration.
In case c'' is zero then y will be zero for x=0.

As such the equation

\( y=acos^{-1}(1-\frac{x}{a})-(2ax-x^{2})^{1/2}\)

And this y represents an inverted cycloid with its base along y axis and cusp at the origin and is the curve that particle will follow.

Double pendulum

Question 

In case of a double pendulum find the expression for the kinetic energy of the system

Solution

We take a simple case where lengths and masses are same.see below in the figure

Here on being displaced the co-ordinates of two pendulums are

\(\begin{array}{l}{x_1} = l\sin {\theta _1}\\{y_1} = l\cos {\theta _1}\end{array}\)

For the first pendulum where \({\theta _1}\) is the angle through which the first pendulum have been displaced.

For second pendulum

\(\begin{array}{l}{x_2} = l\sin {\theta _1} + l\sin {\theta _2}\\{y_2} = l\cos {\theta _1} + l\cos {\theta _2}\end{array}\)

Where \({\theta _2}\) is the angle through which second pendulum has been displaced.

The total kinetic energy of the system is given by the expression

\(T = \frac{1}{2}m(\dot x_1^2 + \dot y_1^2) + \frac{1}{2}m(\dot x_2^2 + \dot y_2^2)\)

Now

And

Which is the required result