# Dependence (or lack thereof) of forces on frames of reference

Consider a block A on top of block B with a coefficient of friction of say 0.3 in between them and the path providing zero resistance (no friction), with the bottom block moving such that its acceleration is 2m/s$$^2$$. The two blocks move such that the top block shares the acceleration with the bottom block due to the frictional force acting on it being less than the maximum friction value. Now in this instance the top block would have a frictional force of 4N acting on it. Introduce another body C moving at an acceleration of 5m/s$$^2$$ in the same direction ahead of the first two blocks such that their acceleration relative to body C is now "-3m/s$$^2$$" since we're factoring in body C being a non inertial frame of reference.

Now if the blocks A and B move with a relative acceleration of "-3" m/s$$^2$$, the opposing frictional force for the blocks would have a different value (since the blocks are now "moving" with a different acceleration along the negative x axis) which I assume has to be a wrong interpretation since forces are somehow independent of frame of reference but can't fathom how since acceleration isn't. Essentially why wouldn't the value of frictional force change if we factor in pseudo force when changing the entire frame.

• Newton's laws are the same in all inertial frames of reference. May 12, 2022 at 13:53

When you say "blocks A and B move with a relative acceleration of -3 m/s2" you are considering the motion of one block in the reference frame attached to the other block (block C). But the block is accelerated to the (aproximately) inertial frame attached to the ground/earth so this is a non-inertial reference frame (NIRF). Newton's second law is valid only in inertial reference frames so you cannot draw the conclusion about forces in NIRF by using it as is. So the acceleration depends on the reference frame but the force doesn't because it is not just F=ma where F is the net "real" force. In order to use Newton's law in NIRF a "fictious" or "inertia" force is introduced, and this force compensate for the effect you observe in the OP. Using the reference frame attached to block C, with the positive in the direction of its acceeration, the inertial force acting on block B (the one on top) will be $$F_i= -m_A \cdot 5m/s^2$$, in direction opposite to the acceleration of C. So N's second will be $$F_f-m_A \cdot 5m/s^2= -m_A \cdot 3m/s^2$$ From this you can see that the the friction force is the same as in the inertial frame, $$F_f=+m_A \cdot 2m/s^2$$. (in the positive direction).