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Wednesday, November 14, 2012

Total energy of earth in its circular orbit around the sun

Question :
Find out the total energy of earth in its circular orbit around the sun in terms of gravitational constant
Answer:
Let R be the total distance between the earth and the sun. If Me and Ms are the mass of earth and sun respectively, the gravitational force of motion of earth and sun is given by
F=GMeMsR2
where G is the gravitational constant. Since the centripetal force balances the gravitational force of attraction, we have
Fc=|FG|,
where
Fc=Mev2R
v being the velocity with which earth is moving. Hence we have
Mev2R=GMeMsR2
or
Mev2=GMeMsR

Therefore kinetic energy of earth in motion is
T=12Mev2=12GMeMsR

As we know that , force in terms of potential energy is
FG=VR
V=FGdR=GMeMsR2dR=GMeMsR
Now total energy of the earth in the orbit around the sun is
E=T+V
E=12GMeMsRGMeMsR

E=12GMeMsR
This is the required expression.

Thursday, November 8, 2012

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 dsv so that total time taken by the particle in moving from highest point 1 to lowest point 2 will be
t12=21dsv
Suppose vertical distance of fall upto point 2 be x, then from the principle of conservation of energy of the particle we find that
12mv2=mgxorv=2gxthent12=21dx2+dy22gxdx=211+˙y22gxdx=21fdx
Where,
f=(1+˙y22gx)12
For t12 to be minimum equation
ddx(f˙y)fy=0
must be satisfied. From expression for f we find that
fy=0f˙y=˙y2gx1+˙y2ddx(˙y2gx1+˙y2)=0or˙y22gx(1+˙y2)=c
Where c’ is the constant of integration.
Since c’ is a constant we can also write
˙y2x(1+˙y2)=c
where c is also a constant. On integrating above equation we find
˙y2c=x(1+˙y2)or,˙y2(1cx)=x˙y2(xcx2)=x2˙y=xxcx2
Putting 1c=2a , and on integration we get
dy=x2axx2dx
y=acos1(1xa)(2axx2)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=acos1(1xa)(2axx2)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
x1=lsinθ1y1=lcosθ1
For the first pendulum where θ1 is the angle through which the first pendulum have been displaced.
For second pendulum
x2=lsinθ1+lsinθ2y2=lcosθ1+lcosθ2
Where θ2 is the angle through which second pendulum has been displaced.
The total kinetic energy of the system is given by the expression
T=12m(˙x21+˙y21)+12m(˙x22+˙y22)
Now






And







Which is the required result
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