CENTRAL FORCES
ANGULAR MOMENTUM
Angular momentum is the rotational
equivalent of linear momentum. It is equal to the product of the moment of
inertia and the angular velocity. For a system of n particles, let li
is the angular momentum of ith particle about a point, and then the
total angular momentum of the system is
L ⃗=∑_(i=1)^n▒(l_i
) ⃗
But (l_i ) ⃗=r ⃗_i×P ⃗_i
Where r ⃗_i position vector, P_i= linear momentum
of same particle
L ⃗=∑_(i=1)^n▒〖r ⃗_i×P ⃗_i
〗
LAW OF CONSERVATION OF ANGULAR MOMENTUM
It states that in the absence of external
torque, the total angular momentum of a system is constant.
Proof: on differentiating equation i
w.r.t time
(dL ⃗)/dt=∑_(i=1)^n▒(〖dr ⃗〗_i/dt×P ⃗_i+r ⃗_i×(dP ⃗_i)/dt)
〖dr ⃗〗_i/dt×P ⃗_i=v ⃗_i×m;v ⃗_i=0
r ⃗_i×(dP ⃗_i)/dt=r ⃗_i×F ⃗_i=τ ⃗_i
Where, τ ⃗_i is external torque acting on i^th particle
(dL ⃗)/dt=∑_(i=1)^n▒(τ_i
) ⃗ =τ ⃗_ext
It
is τ ⃗_ext=0, then (dL ⃗)/dt=0
or, L ⃗=constant
Hence in the absence of external torque,
the total angular momentum of the system is constant. Thus is the particle of
conservation of angular momentum.
CENTRAL FORCES
A force which always acts toward or away
from a fixed point and whose magnitude depends only on the distance from that
pint is called central forces.
ORBITAL ENERGY EQUATION
There are certain heavily bodies which revolves around the planets are called satellite like, moon is natural satellite of earth. All satellites resolve under the gravitational attraction exerted by the planet on the satellite.
The velocity of satellite has two
components; are parallel one perpendicular to the position vector r of the
satellite.
∴V=V_r+V_⊥
The kinetic energy then breaks into two
parts
Ke=1/2 m〖V_⊥〗^2+1/2
m〖V_⊥〗^2
The magnitude of angular momentum of the
satellite is given by L=mrV_⊥
We can solve for V_⊥
in terms of l and write kinetic energy
as
k=1/2 m〖V_⊥〗^2+L^2/(2mr^2
)-(GM_m)/r
When a satellite is launched it is placed
upon a rocket which is launched from the earth . After the rocket reaches its
maximum vertical height, a special mechanics gives a thrust to the satellite at
pt. A, producing horizontal velocity v. The total energy of the satellite at a
is
E=1/2 m〖V_⊥〗^2-(GM_m)/r
Where,
m=mass of satellite,
M=mass of earth
R=radius of earth
The orbit will be an ellipse (closed
path), a parabola or an hyperbola depending on whether E is negative, zero or
positive.
ECCENTRICITY
It is defined as the ratio of the
distance between the foci to the major axis of the ellipse. The eccentricity is
zero for a circle.
KEPLER’S LAW
In astronomy, Kepler’s law of planetary
motion is three scientific laws describing the motions of plants around the
sun.
1.
LAW OF ELLIPTICAL ORBITS
Each planet moves in an elliptical orbit
around the sun, the sun being at once of the foci of the ellipses.
2. THE LAW OF AREA
The radius vector of any planet relative
to the sun sweeps out equal areas in equal times that is the areal velocity of
the radius vector is constant.
3. THE HASMORICE LAW
The square of the period of revolution of
any planet around the sun is proportional to the cube of the semi-major areas
of the elliptical orbit.
SATELLITE MANEUVERS
Sometimes we need to correct or change
the orbit of a satellite because of number of forces acts on a satellite to
change its orbits over time. For example, slight asymmetries in universe.