Why does gravity give a constant acceleration and centrifuge?

[scilearner]

Hello everyone,

 

Normally if there is 1 kg object and I apply 10 N force it would accelerate at 10m/s. Here the acceleration is determined by the mass of object. How does gravity in the absense of resistance make objects travel at a fixed acceleration?

 

In a centrifuge a force equivalent to double or triple gravity is used. Does this force give a constant acceleration (9.8m/s2) like gravity too. How is that done?

 

Thanks!!

[Mr Skeptic]

Acceleration is a change in velocity. Velocity is a speed and direction. Changing the direction, even without changing the speed, changes the velocity which requires an acceleration. To create an acceleration, you need a force. This is the centripetal force. The centrifugal force is a pseudoforce; it is simply one way to look at the inertia of an object.

[scilearner]

Thanks for the answer. So in a centifuge with 3g centripetal force does heavy molecules in a test tube move with 3 times 9.8 m/s2 acceleration in the absense of resistance. How can you give an constant acceleration like that. Doesn't mass change acceleration?

[Cap'n Refsmmat]

Well, look at it this way:

 

The force of gravity on an object is mg -- mass times the acceleration due to gravity. The acceleration resulting is:

 

[math]F = ma[/math]

 

[math]mg = ma[/math]

 

[math]\frac{mg}{m} = a[/math]

 

You can see that if we, say, triple the mass, the acceleration is still exactly the same -- the masses cancel out.

[scilearner]

Thanks but you predetermined the acceleration. Isn't the acceleration determined by the mass of the object. If earth exerts gravity on an object and that object exerts gravity on earth. The earth would move less due to mass. So doesn't mass affect acceleration?

[Mr Skeptic]

Nope, a more massive object has a greater gravitational force on it, but it also has a greater mass. As Cap'n said, this results in a constant acceleration. His first formula is the definition of force (also one of Newton's laws), the second one the force due to a constant gravitational acceleration g.

[Cap'n Refsmmat]

An object with more mass has more gravitational force acting upon it, but it also has more inertia to resist motion.

[scilearner]

How is the acceleration due to gravity created? What makes it 9.8 m/s2. I think I'm not understanding gravity properly.

 

My understanding of gravity is that it is the force that attracts 2 objects. If one object is lesss heavy that object would move towards the heavier one at a faster rate(earth). So here isn't the mass affecting acceleration?

[Mr Skeptic]

Like this.

 

Then you also subtract a small bit due to centrifugal force, I think. However, that also distorts the earth under you changing both the effective mass and your distance from it.

 

Eh? I think they use a different radius on google calculator.

[Cap'n Refsmmat]

Indeed. The force due to gravity on two objects is:

 

[math]F=\frac{GmM}{r^2}[/math]

 

where G is the universal gravitational constant and m and M are the two masses. If you put in the mass of the earth as M and the radius of the earth as r, it turns out that

 

[math]F = g m[/math]

 

because [imath]\frac{GM}{r^2} = g[/imath].

 

It only works when you're on the surface of the earth (so the distance is the radius of the Earth).

[scilearner]

Thanks for the replies. So between the two masses what is the direction of gravity. Does the big mass exert 9.8 ms2 on small mass or vise versa. Do they both exert this force on each other. If they do why doens't earth get attracted by the smaller mass?

 

Rechecking the formula I think the bigger mass has less g when you make F=gM. Ok I think I got it now but how is that centripetal force from a centrifuge act as gravity.Basically how come this formula is applicable to that as well.

[DJBruce]

Thanks for the replies. So between the two masses what is the direction of gravity. Does the big mass exert 9.8 ms2 on small mass or vise versa. Do they both exert this force on each other. If they do why doens't earth get attracted by the smaller mass?

 

Like all forces the force of gravity has a equal and opposite reactive force. The force the Earth exerts on an object is the same the force the object exerts on the Earth. The Earth is attracted to the smaller object. However since the Earth is so big the acceleration of the Earth towards the small object is very small.

[scilearner]

Ok I got it Thanks everyone!! Only one question remain the centripetal force in a centrifuge machine. Why is the centripetal force equal to the equation of gravity in this case. Thanks!!

[Mr Skeptic]

The equation for centripetal force is [math]F = \frac{m v^2}{r}[/math]. Note the m there. Since F=ma, you can cancel out the mass and the result is [math]a = \frac{v^2}{r}[/math]. This has nothing to do with gravity, only that the mass of the item in question is in the equation for the force on it. The analogous acceleration for gravity would be [math]a=g=\frac{GM}{r^2}[/math].

Edited November 11, 2009 by Mr Skeptic

[DJBruce]

Ok I got it Thanks everyone!! Only one question remain the centripetal force in a centrifuge machine. Why is the centripetal force equal to the equation of gravity in this case. Thanks!!

 

I am not quite sure what equation you are talking of. The centripetal force is equal to:

 

[math] F=\frac{mv^{2}}{r}[/math]

 

I guess you could express force in terms of the number of "g's" the object is feeling in which case:

 

[math] \frac{F}{g}=\left(\frac{mv^{2}}{r}\right)\left(\frac{1}{g}\right)[/math]

 

In that case the g is simply there to express the force as a multiple of the acceleration of gravity.

[scilearner]

The equation for centripetal force is [math]F = \frac{m v^2}{r}[/math]. Note the m there. Since F=ma, you can cancel out the mass and the result is [math]a = \frac{v^2}{r}[/math]. This has nothing to do with gravity, only that the mass of the item in question is in the equation for the force on it.

 

Hey thanks. This is unique to centripetal force right? I mean a linear single force can not give a constant acceleration. If you don't mind can you show me how centripetal force formula is derived. Thanks

[DJBruce]

Here are two different sites that show the derivation for the equation of centripetal acceleration. I find it hard to explain this proof without a diagram or I would do it myself, so I hope this helps.

[scilearner]

Here are two different sites that show the derivation for the equation of centripetal acceleration. I find it hard to explain this proof without a diagram or I would do it myself, so I hope this helps.

 

Thanks Bruce Centripetal force is bit different to normal linear force right. I mean the acceleration is determined by velocity in c.f as opposed to mass right.

[DJBruce]

Centripetal force is somewhat different than linear force. However, the equation to calculate it is the same. The equation to calculate linear and centripetal force is:

 

[math] F=ma[/math]

 

The acceleration for both linear and circular motion is still:

 

[math]a=\frac{\Delta v}{\Delta t}[/math]

 

However, in uniform circular motion the acceleration is able to be calculated from the equation:

 

[math]a=\frac{v^2}{r}[/math]