Final answer:
The original expression m = (mv²r)(mgr²) seems to be a typo or incorrectly derived. In physics, the mass m in equations for centripetal and gravitational forces often cancels out, leaving an equation for orbital speed v or gravitational acceleration g. Without additional context, the provided expression cannot be simplified because it is not physically meaningful.
Step-by-step explanation:
The student's question involves simplifying a physics equation that contains a typo. The correct equation for centripetal force is F = (mv²)/r, and for gravitational force, it's F = (GmM)/r². If these two forces are set equal to each other for a satellite in orbit, the mass m of the satellite cancels out. This leaves the simplified equation v² = (GM)/r, which is used to determine the orbital speed of an object in space. To simplify the provided mistaken equation m = (mv²r)(mgr²), it appears to have extra factors of m and r and does not seem to represent a physical reality without additional context. If you intended to equate centripetal force with gravitational force, then the correct form without a typo should be m = (v²)/((G)/r) which simplifies to m = v²r/G. However, in your expression m appears both on the left and on the right-hand side, and if you meant to solve for m, both sides would end up canceling each other, which suggests that there has been a mistake in your derivation.