Final answer:
The final velocity of the spherical shell can be determined using the principle of conservation of energy. The kinetic energy of the shell at the bottom of the incline is equal to mgh. The time it takes for the shell to reach the bottom can be found using the equation t = sqrt(2d/g). The gravitational potential energy lost by the shell is equal to mgh.
Step-by-step explanation:
a) To determine the final velocity of the spherical shell, we can use the principle of conservation of energy. The initial potential energy is given by mgh, where m is the mass, g is the acceleration due to gravity, and h is the height of the incline. The final kinetic energy is given by (1/2)mv^2, where v is the final velocity. Setting the initial potential energy equal to the final kinetic energy, we have mgh = (1/2)mv^2. Solving for v, we get v = sqrt(2gh).
b) To calculate the kinetic energy of the shell at the bottom of the incline, we can use the equation (1/2)mv^2, where m is the mass and v is the final velocity. Plugging in the values, we have (1/2)(m)(sqrt(2gh))^2. Simplifying, we get (1/2)(m)(2gh), which further simplifies to mgh.
c) To find the time it takes for the shell to reach the bottom, we can use the equation d = (1/2)gt^2, where d is the height of the incline and t is the time. Solving for t, we get t = sqrt(2d/g).
d) The gravitational potential energy lost by the shell is equal to the initial potential energy minus the final potential energy. The initial potential energy is mgh, where m is the mass, g is the acceleration due to gravity, and h is the height of the incline. The final potential energy is zero, since the shell is at the bottom of the incline. Therefore, the gravitational potential energy lost is mgh.