Gravitational force and oscillation question

[Natbird]

Part1-A tunnel is bored through the center of a planet, as shown in the figure. (This drawing is NOT to scale and the size of the tunnel is greatly exaggerated.) Assume that the planet is a homogenous sphere with a total mass M = 3.5x 1024 kg and a radius R = 6600 km. A package of mass m = 7.5 kg is dropped into the tunnel. Calculate the magnitude in Newtons of the gravitational force acting on the package when it is a distance r = 2100 km from the center of the planet.


Figure is just a circle with a tunnel through it that's suppose to be a lot smaller then shown

Part-2Continuation: If the tunnel is used to deliver mail from one side of the planet to the other, how long (in s) would it take for a letter to travel through the planet?




Part 1 was easy just F=mM/R^2*(r/R) and the answer is 12.8

ok so part two has a hint "Anything dropped into the tunnel will oscillate, back and forth, through the Earth while executing simple harmonic motion! 2 Write down F=ma and then use ideas from the chapter oscillations to find the period of the oscillations. "

ok so F=ma and I understand that the changing force of gravity will cause it to oscillate
so first I did a=GM/R^2*(r/R) since masses cancel...
then I thought find time it takes to get to the center and multiply that by two..but gravity as the object moves is always going to be changing. What equation relates the period (mentioned in the hint) with the always changing force from gravity so that time can be found

Thanks

 

[DavidWhitbeck]

The simple harmonic oscillator equation is $$a=-\omega^2 x$$

Use N-2 to find the constant of proportionality, the square root of that is your angular frequency. Period is related to angular frequency by

$$T=\frac{2\pi}{\omega}$$

 

[kamerling]

In the chapter oscillations, you probably found an equation for the force in and harmonic oscillator: F = - k x, and also an equation for the frequency of the oscillator.
Does the force of gravity has the same form?

 

[Natbird]

k=F/dx where dx=distance package is away from center at the gravitational Force F found earlier

angular frequency of package=sqrt(k/mass of package)

2pi/angular frequency of package=T=total time

is this correct

 

[kamerling]

k=F/dx where dx=distance package is away from center at the gravitational Force F found earlier

angular frequency of package=sqrt(k/mass of package)

2pi/angular frequency of package=T=total time

is this correct

well yes. can you give an expression for k that involves G, M, m and R