newtonian mechanics - Is gravity a force and if so what is its opposite?

1. 
   
   

   For every force there is an equal force in the opposite direction on another body, correct?

   
   


2. 
   

   So when the Suns gravity acts on Earth where is the opposite and equal force?

   
   


3. 
   

   I also have the same question for centripetal force in a planets orbit.

   
   

Answer

As many others said, the Sun feels the same force towards Earth as the Earth feels towards the sun. That is your equal and opposite force. In practice though the "visible" effects of a force can be deduced through Newton's first law, i.e. ${\bf F} = m{\bf a}$. In other words, you need to divide the force by the mass of the body to determine the net effect on the body itself.

So:

${\bf F_s} = {\bf F_e}$

${\bf F_s} = m_s {\bf a_s}$

${\bf F_e} = m_e {\bf a_e}$

therefore,

$m_s {\bf a_s} = m_e {\bf a_e}$

and

${\bf a_s} = {\bf a_s} \frac{m_e}{m_s}$

Now, the last term is $3 \cdot 10^{-6}$! This means that the force that the Earth enacts on the sun is basically doing nothing to the sun.

Another way of seeing this:

$F = \frac{G m_s m_e}{r^2}$

$a_s = \frac{F}{m_s} = \frac{G m_e}{r^2}$

$a_e = \frac{F}{m_e} = \frac{G m_s}{r^2}$

$\frac{a_s}{a_e} = \frac{m_e}{m_s} = 3 \cdot 10^{-6}$

Again, the same big difference in effect.

Regarding the centripetal force, it is still the same force. Gravity provides a centripetal force which is what keeps Earth in orbit.

Note

It's worth pointing out that the mass that acts as the charge for gravity, known as gravitational mass is not, a priori, the same mass that appears in Newtons's law, known as inertial mass. On the other hand it is a fact of nature that they have the same value, and as such we may use a single symbol $m$, instead of two, $m_i$ and $m_g$. This is an underlying, unspoken assumption in the derivation above. This is known as the weak equivalence principle.