- Dale E. Gary
- NJIT Physics Department
9.1 Acceleration without Rotation - It is sometimes useful to work in non-inertial frames, but in order to do that we have to make some changes to the equations we developed for inertial frames up to now.
- We will call our inertial frame So, and consider a second, non-inertial frame S that, by definition, is undergoing some acceleration.
- To start, let’s consider our frame S to be accelerating without rotation, at some acceleration A, which need not be constant. We could consider So to be the ground, for example, and S to be a railroad car moving relative to the ground (we are ignoring any non-inertial motions of the Earth for the moment).
- We will write Now assume someone on the car is playing catch with a ball of mass m. Relative to the inertial frame So, of course, you know that the equation of motion for the ball is:
- where ro is the position of the ball relative to So, and F is the net force on the ball. F is the vector sum of all forces, e.g. gravity, the passenger’s hand on the ball, air resistance, etc.
Acceleration without Rotation-2 - In the non-inertial frame (at least for non-relativistic mechanics), the ball’s position is r, and its velocity is, by addition of velocities,
- Differentiating and rearranging, we have
- Notice that this looks like Newton’s 2nd Law, except there is a new force mA that appears due to the acceleration of the frame S relative to So.
- We will find in general that we can continue to use Newton’s Law for a non-inertial frame, so long as we add an extra force-like term, called the inertial force. It can be considered a “fictitious” force in the sense that it disappears in an inertial frame (to be replaced by a true force of the same magnitude). E.g., the true centripetal force in an inertial frame becomes the “fictitious” centrifugal force in the non-inertial frame. Rather than use the misleading term “fictitious,” we will call it an inertial force.
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