Freefall chord through the Earth -- calculus proof

[Orodruin]

I would suggest you need to work more methodically rather than in short snippets. You have to answer the following questions:

1. What is the magnitude of the force on a mass $m$ at a distance $r$ from the center?
2. What are its components in the different directions?
3. Can you write down Newton's second law for each component of the motion?

You should note that when you write down 3, you obtain equations of motion for each component that is independent of the other components. This is key. The resulting differential equation for each component should be easily solvable if you are familiar with basic ODEs.

 

[achap01]

That is dimensionally wrong, so cannot be right.
There should be an x in there. Check your working. If you can’t find the error, post it.

sorry i mean 9.8x/R=a

 

[haruspex]

sorry i mean 9.8x/R=a

Ok, nearly there!
So write the differential equation relating $x$ to $\ddot x$.

But it is better to leave it as "g". If you substitute a numeric value you should include units.

 

[achap01]

Ok, nearly there!
So write the differential equation relating $x$ to $\ddot x$.

But it is better to leave it as "g". If you substitute a numeric value you should include units.

would ẍ just be the second derivative of x with respect to time, so ẍ=a=9.8x/R?

 

[haruspex]

would ẍ just be the second derivative of x with respect to time, so ẍ=a=9.8x/R?

Almost, but watch the signs. If we take x as going from $x=-L$ to $x=+L$ then when $x<0$ the acceleration is positive. Correcting that, do you recognise the form of the ODE?

Please use "g", not "9.8".

 

[achap01]

so the ODE would be ẍ=a=-gx/R? Im a little confused

 

[haruspex]

so the ODE would be ẍ=a=-gx/R? Im a little confused

Yes, or to put it in its more standard form, $\ddot x+\frac gRx=0$.
Do you recognise that form?

 

[achap01]

Yes, or to put it in its more standard form, $\ddot x+\frac gRx=0$.
Do you recognise that form?

harmonic motion, so angular frequency (rate of oscillation) would be g/R, which have a derivative of 0 with respect to theta?

 

[haruspex]

harmonic motion, so angular frequency (rate of oscillation) would be g/R, which have a derivative of 0 with respect to theta?

Bingo.

 

[kuruman]

Not quite. The angular frequency is $\omega =\sqrt{g/R}.$

 

[achap01]

Bingo.

is there an online or a video proof of this that i can get

 

[kuruman]

What exactly do you want to prove?