Final answer:
To stay on the track, the mass must have enough speed to generate centripetal force solely from gravity at the top of the loop. Conservation of energy allows us to calculate this speed, considering kinetic and potential energy. The stability of motion depends on the mass maintaining the minimum speed for stable circular motion.
Step-by-step explanation:
To calculate the minimum speed required for mass m to stay on the track through the loop, we use the concept of centripetal force. At the top of the loop, the gravitational force and the normal force from the track must provide enough centripetal force to keep the mass from falling off the track. Therefore, at minimum speed, the normal force at the top is zero, and gravitational force provides all the needed centripetal force, which is mv2/r. Using conservation of energy, the kinetic energy at the bottom of the loop plus the gravitational potential energy at the top must be equal, leading to the calculation of the minimum speed required at the bottom of the loop.
Regarding the conservation of energy in a frictionless loop-the-loop system, the mechanical energy is conserved. The total mechanical energy is the sum of kinetic energy and gravitational potential energy. As the mass moves around the loop, its kinetic energy and potential energy are transformed into each other while their sum remains constant. This assumes no energy is lost to friction or air resistance.
The stability of the motion through the loop depends on the speed of the mass. If the speed is too low, the mass may not have enough kinetic energy to reach the top and would fall before completing the loop. If moving at or above the minimum speed, the motion will be stable as the centripetal force keeps the mass in circular motion throughout the loop.