doubts on gravitation continued

- AravindG

doubts on gravitation continued

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- jamiebookeater

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- AravindG

a hole is drilled from the surface of the earth and a ball is thrown inside .the ball executes SHM why?

- anonymous

Could you draw for the earth and this hole? I think I know what you mean but just to be sure

- AravindG

|dw:1327507100491:dw|

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## More answers

- AravindG

help pls

- TuringTest

The force acting on the falling object acts opposite direction of its displacement|dw:1329322341910:dw|toward the center of mass of the earth (the equilibrium position of the falling object)\[F_g\propto-x\]so there is always a restoring force acting to move the object toward the center of the earth. However when the object passes through the center point of the earth all the potential energy it started with is now kinetic, so it will pass through and gravity will act in the opposite direction. This will continue indefinitely unless a dampening force acts to slow the motion of the object.

- TuringTest

|dw:1329322653275:dw|set up a coordinate axis to define positive and negative direction and it's easy to convince yourself that the force acts with opposite sign as displacement.

- AravindG

i dont get wat the restoring force is

- JamesJ

Yes, as explained here. The gravitational force acting on a mass \( m \) inside a solid like the earth of mass \( M \) and radius \( R \) at a radial distance \( r \)d is
\[ F_g = -\frac{GMm}{R^2}\frac{r}{R} \]
Notice that the first fraction there is a constant. What is varying linearly is the r/R term. Notice also when r=R, that the force is exactly what you expect it to be.
Now, given that what don't you understand?

- AravindG

i havent studied that expression

- JamesJ

In every case, the force acts towards the center of the earth. Hence the force is always in the opposite direction of the displacement from the center of the earth. This is _exactly_ like a spring, where
\[ F = -kx \]

- JamesJ

Well, I'm telling you it right now. This is the only way to answer this question.

- AravindG

hm so gravitational force acts towards centre of earth??

- JamesJ

If you like, write
\[ F_g = -kr \] where \[ k = \frac{GMm}{R^3} \]
===
Yes, by virtue of the negative sign, F_g always points towards the center of the earth.

- AravindG

oh but i thoug the acceleration is maximum on surface of earth

- JamesJ

It is. For r < R, F_g is less that it is on the surface of the earth.
That follows immediately from the equations above, where F_g is proportional to r.

- AravindG

i hav studied this eqn can u relate the answer with this??
|dw:1327512111726:dw|
where d is the depth

- AravindG

turing u can also join he discussion

- TuringTest

James has a handle on it...

- JamesJ

Yes. \[ r = R_e - d \] and \[ g = \frac{GM}{R_e^2} \]
Hence
\[ g' = g(1 - d/R_e) \]
\[ = \frac{GM}{R_e^2} \frac{R_e - d}{R_e} \]
\[ = \frac{GM}{R_e^2} \frac{r}{R_e} \]
\[ = \frac{GM}{R_e^3} r \]

- JamesJ

Therefore the force acting of mass \( m\) at a depth of \( d \), i.e., at a radial distance from the center of the earth of \( r = R_e - d \) is
\[ F_g = mg' \]
\[ = - \frac{GMm}{R_e^3} r \]
I've put the minus sign in here to indicate the direction of the force, which is in the negative r-direction.

- AravindG

i hav a funny doubt :u see the force of gravity is towards the centre,, then y does it decrease from the surface???it should be increasing ryt?

- JamesJ

Suppose you were in the center of the earth. What would gravity there be?

- AravindG

0

- JamesJ

Right, because there is mass all around you, the force of which cancels itself out. Indeed, for any distance below the surface of the earth, the deeper you go, the more cancelation you get from mass 'above' you. That's why it decreases as we head towards the center of the earth until finally we arrive at zero force.

- AravindG

oh k got it

- AravindG

pls continue

- AravindG

can u show the path of the ball in a diag

- AravindG

i mean will the ball move from one end to other end?

- JamesJ

Yes

- AravindG

|dw:1327512744540:dw|

- AravindG

is this wat happens?

- JamesJ

Yes

- AravindG

so wen the ball moves towards centre the foce is towards surface?

- AravindG

gravitational force?

- JamesJ

The force always points towards the center of the earth, exactly the like the force of spring always points towards the relaxed/equilibrium state of the spring:
\[ F_g = -kr, \ \ \ F_{spring} = -kx \]

- AravindG

then the ball is also moving towards centre

- AravindG

both direction same

- JamesJ

Force is not always in the direction of motion, you know that. Acceleration changes direction of motion, that is velocity. But direction of velocity need not equal direction of acceleration, just like with a spring.

- AravindG

cnfused

- JamesJ

With a spring, the mass is moving half the time in the direction of the force; half the time in the opposite direction as the force.

- AravindG

k xplain in this case

- JamesJ

##### 1 Attachment

- TuringTest

^which highlights the same point: acceleration is proportional to negative displacement.

- AravindG

srry guys but i cant catch up with that point

- JamesJ

First you drop the ball, acceleration is toward the center of the earth and the ball moves in that direction.
Then the ball gets the center of the earth.
Then it keeps moving past the center of the earth towards the other side of the earth. But now the acceleration is in the opposite direction, toward the center of the earth. This acceleration slows the balls until ...
...it reaches the other side of the earth and has zero velocity.
Acceleration continues acting on it toward the center of the earth, and now it starts to move towards the center of the earth again, and continues to do so until it reaches the center of the earth ...
and then it moves past the center of the earth and the acceleration is in the opposite direction to motion ... and the whole pattern repeats.

- AravindG

wow i got it ,, but one doubt y does the ball continue moving after reaching centre?

- JamesJ

because exactly at the center it has kinetic energy and momentum, but the force is zero.

- AravindG

k i got that :P thx i hav some more doubts

- JamesJ

so there's nothing to stop it at the center of the earth.

- AravindG

i will post in new tab k?

- JamesJ

I'm going, sorry. I've got my own problems to solve! ;-)

- AravindG

oh

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