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ROTATION OF 2 MASSES UNDER MUTUAL Variation of g due to rotation of earth

NEWTON‛S LAW OF VARIATION IN THE VALUE OF WORK DONE IN MOVING OBJECT
GRAVITATION GRAVITATIONAL FORCE OF ATTRACTION
ACCELERATION DUE TO GRAVITY FROM ONE ORBIT TO ANOTHER
m1 F F m2 Gm22 Latitude -
r V1 = Variation due to height ‘h‛ r
P
GM GM
(m1+m2) r
Fc Angle which the line V01 =
r1
V02 =
r2
Gm1m2 m g mgl joining the point to
v2 GM mg
F = g = the centre of earth r1 r2
r2 V2 =
Gm12
h (R + h)2 makes with the
G - Universal gravitational constant (m1+m2) r
r
g gR2 equatorial plane
Value of G r1 COM r2 g =
6.67x10-11 Nm2Kg-2 (SI or MKS) v1
r3
(R + h)2 gl = g - 2
R cos2
T2 = 4 2
6.67x10-8 dyne cm2g-2 (CGS) r G(m1+m2) Re
General Equation
Time period M Note = value of 2
R = 0.034
Dimensional formula [G] Approximate equation
or T2 r3
For poles = 90o gl = g CONCEPT - WORK DONE BY
M-1L3T-2 h<<<<<<R (h < 100 km) EXTERNAL AGENT = CHANGE IN
There is no effect of rotational motion of the
2h earth on the value of g at poles. MECHANICAL ENERGY
use, gl= g 1-
R For equator = 0o gl = g - 2
R
IMPORTANT POINTS ABOUT Note the point GMm 1 1
THREE MASSES(EQUAL) REVOLVING W = E 2 - E1 = -
GRAVITATIONAL FORCE The effect of rotational motion of the earth on 2 r1 r2
UNDER MUTUAL GRAVITATIONAL FORCE If h<<<<R, then decrease in the value
the value of g at the equator is maximum.
1. Gravitational force of g with height
V M When a body of mass m is moved from equator to
* Always attractive in nature 2hg the poles, weight increases by an amount
GM GM Absolute decrease = g = g - gl =
R m (gp - ge) = m
2
R WORK DONE IN MOVING OBJECT
* Independent of the nature of medium V1 = =
l R 3 g g-gl 2h
between masses Fractional decrease = = = FROM SURFACE OF EARTH TO
l l g g R
* Independent of presence or absence R
HEIGHT h ABOVE SURFACE
g g-gl 2h x 100
of other bodies c Time period Percentage decrease = = x 100 =
M
V g g R ORBITAL VELOCITY m
2. Are central forces, acts along the centre T2 l3 Orbit at a height ‘h‛ from the surface h
l M
of gravity of two bodies. V Variation due to depth ‘d‛ m,Vo
m

3. Conservative force g
h
GM gR2 R
r
Vo = =
4. Force between any two masses - d g = g [ 1-
l d
] Re (R+H) (R+H)
gl R
Gravitational force R
r
M

GM gR R
R
Force between earth and any other body - Vo = =
M Re R
Force of gravity FOUR EQUAL MASSES UNDER MUTUAL
-GMm -GMm
GRAVITATIONAL FORCE If orbit is closer to GM gR
W = (U f - Ui
(
= -
Vo = = R R+h
earth‛s surface( neglect ‘h‛) R
g dg Or, R
= g-g =
l
l GM Absolute decrease = (called minimum orbit, velocity-first cosmic velocity) W = mgh
V = (2 2+1) g R
R+h
VECTOR FORM 2 2 l Note - for easy calculations
F21 force on 2 due to 1 g g-gl d Work done to move object W =
mgR
y
l l 1 GM Fractional decrease = = = gR = 8 km/s or GM = 8 km/s = 8x 10 m/s to a height h = R 2
r12 vector from m1 to m2 V = g g R
3
R (2 2+1) R
r12
G m1m2 2 R d mgR
m1
m2
-r12 l Percentage decrease = g x100
GM Work done to move object W =
F21 = x100 = or = 64 x 106 3
r2 r122 T2 l3 Here R = g R R to a height h = R/2
l
r1 2
Very imp graph
G m1m2
x or - r12 The graphical representation of change in KE, PE OR TE FOR AN ORBITING
r123 WORK DONE IN MOVING OBJECT
Similarly the value of g‛ with height and depth SATELLITE
GRAVITY gl m,Vo FROM SURFACE TO CIRCULAR ORBIT
F12 force on 1 due to 2
GMm GMm h
Acceleration due to gravity GMe KE = U =-
G m1m2 G m1m2 2r r
r Vo

F21 = r12 or r12 On the surface of earth g = GMm Re

m
h

r122 r123 m Re2 TE =-

1 2r M
Clearly M - mass of earth gl
Re
Newtons third law F21 = - F12 R - Radius of earth gl
r r2 Relation KE, U & TE
U = 2 x T.E R R
M [Put GMe = g Re2 to solve
K.E = - T.E
Re
problems easily] r
r = R Graph W = Ef - Ei
-GMm GMm
Gravitational force is a two body interaction. energy W = +
gr gR 2 W = Etotal - Ui 2(R+h) R
Force between two particles does not depend for r < R, gl = for r < R, gl =
on the presence or absence of other particles. R r
2

KE
The principle of superposition is valid here. g IN TERMS OF DENSITY OF EARTH TE
“Force on a particle due to a no. of particles distance
U
g=4 G Re g Re

GRAVITATION
is the resultant of forces due to individual
particles.” “If density is mentioned use the PHYSICS
above equation”
WALLAH