Does dropping a Jupiter from the Pisa tower still accelerate at g = 9.81?

[DuctTapePro]

im confused about this thing, because they said Earth's acceleration due to gravity is independent from the mass of the thing dropped

 

[hutchphd]

im confused about this thing, because they said Earth's acceleration due to gravity is independent from the mass of the thing dropped

If Jupiter were the size of a basketball (it must remain external to all of the mass of earth) then yes. Of course someone would need to carry it up the tower: the structural engineering is suspect...

 

[Janus]

im confused about this thing, because they said Earth's acceleration due to gravity is independent from the mass of the thing dropped

The acceleration of the Jupiter will be 9.81 m/sec^2, however the acceleration of the Earth will be 3159.15 m/sec^2 ( assuming we are talking about a small object with the a 1 Jupiter mass which is the only way its center mass mass could be Pisa Tower distance from the surface of the Earth. )
Anytime you drop an object, their closing rate will be due to both the object's acceleration towards the Earth due the Gravitational attraction, and the Earth acceleration. The force acting on both will be equal, but in everyday circumstances, the Earth's acceleration will be negligible.
For example, if you drop a 1 kg weight, the force accelerating it is 9.81 Newtons, yielding an acceleration of 9.81 m/sec^2. The Earth feels that same force, but with a mass of ~6e24 kg, this only produces an acceleration of 1.635e-24 m/sec^2. if we increase the weight to 10 kg, it stills accelerates at 9.81 m/sec^2, and the Earth's acceleration increases to 1.635e-23 m/sec^2. Even though this is 10 times as much it is still immeasurably small compared to 9.81 m/sec.

 

[jbriggs444]

The acceleration of the Jupiter will be 9.81 m/sec^2, however the acceleration of the Earth will be 3159.15 m/sec^2 ( assuming we are talking about a small object with the a 1 Jupiter mass which is the only way its center mass mass could be Pisa Tower distance from the surface of the Earth. )

Of course, the tidal forces from a basketball sized Jupiter would be quite devastating. That's 3159 meters per second squared at the Earth's center, some 6000 kilometers away. Locally at a range of perhaps 6 meters, we are talking about an acceleration that is higher by a factor of a trillion (10¹²).

[The above is a Newtonian calculation. Jupiter's Schwarzschild radius is 2.82 meters. So basketball sized is a bit less than is feasible]

 

[PeroK]

im confused about this thing, because they said Earth's acceleration due to gravity is independent from the mass of the thing dropped

Normally, if you are talking about dropping an object, you are talking about an object that is very small compared to the Earth.

Your example could better be described as dropping Earth towards Jupiter.

But, really, it's a collision between two planets. As others have said, if the planets somehow start at rest relative to each other and very close, then sure enough Jupiter will accelerate at Earth surface $g$ and the Earth will accelerate at the Jupiter surface gravity $g_J$.

Although, actually you would also need to take into account the physical size of the planets. Jupiter's centre of mass would be a long way from the surface of the Earth.

 

[DrClaude]

@Janus provided a very good answer, but I would like to clarity that Jupiter will not accelerate at 9.81 m/s² with respect to the Earth's surface. From the point of view of the tower, it would appear to accelerate much faster.

 

[DuctTapePro]

ohh thanks for the replies.. i had an internet black out so i haven't been online for days... so.. independence from the mass of the second object isn't really true... it's just negligible... i get it now, thank you :)

 

[Ibix]

independence from the mass of the second object isn't really true... it's just negligible...

That depends on your frame of reference. In Newtonian physics the object always accelerates at 9.81ms^-2 according to an inertial observer. How fast the Earth accelerates varies, and is completely negligible in realistic scenarios, yes.

So if you use an inertial frame the acceleration is constant; if you use the Earth's surface frame it varies.

 

[hutchphd]

ohh thanks for the replies.. i had an internet black out so i haven't been online for days... so.. independence from the mass of the second object isn't really true... it's just negligible... i get it now, thank you :)

I think you may misunderstand...it is exactly true, but there are host of additional things you may also need to consider.

 

[PeroK]

That depends on your frame of reference. In Newtonian physics the object always accelerates at 9.81ms^-2 according to an inertial observer. How fast the Earth accelerates varies, and is completely negligible in realistic scenarios, yes.

If the object is large, then its centre of mass is not near the Earth's surface. Jupiter would never get close enough to accelerate at $g$.

 

[Ibix]

If the object is large, then its centre of mass is not near the Earth's surface. Jupiter would never get close enough to accelerate at $g$.

True. I should say that all objects will accelerate at the same rate depending on the altitude of their centre of mass (per a Newtonian inertial frame). For small objects near the surface of the Earth, thus is 9.81ms^-2. Large objects can't get that close, as you say.

 

[jbriggs444]

depending on the altitude of their centre of mass

To be pedantic, what matters is their center of gravity in the non-uniform gravitational field of the Earth.

 

[hutchphd]

To be pedantic, what matters is their center of gravity in the non-uniform gravitational field of the Earth.

But when each object is spherically symmetric there is no difference, yes?.

 

[ZapperZ]

im confused about this thing, because they said Earth's acceleration due to gravity is independent from the mass of the thing dropped

This is why I wrote this:

https://www.physicsforums.com/insights/why-is-acceleration-due-to-gravity-a-constant/
Zz.

 

[jbriggs444]

But when each object is spherically symmetric there is no difference, yes?.

Right you are.

To continue the pedantry, neither Earth nor Jupiter are rigid spheres. They are both "planets". Which, by definition, means they are not rigid.

 

[hutchphd]

Right you are.

To continue the pedantry, neither Earth nor Jupiter are rigid spheres. They are both "planets". Which, by definition, means they are not rigid.

Sir, you have taken pedantry to a new extremum.. ...I confess ambivalence as to the direction traveled.