Final answer:
The sphere's total kinetic energy at the bottom of the ramp is found using the conservation of mechanical energy, equating its initial potential energy to its total kinetic energy at the bottom, which is calculated to be 29.43 joules.
Step-by-step explanation:
To solve for the sphere's total kinetic energy at the bottom of the ramp, we can use the principle of conservation of mechanical energy. We know that the sphere transitions from gravitational potential energy at the top of the ramp to both translational and rotational kinetic energy at the bottom because it rolls without slipping.
Initially, at the top of the ramp, the sphere only has gravitational potential energy (U), which can be calculated by U = mgh, where m is the mass, g is the acceleration due to gravity (9.81 m/s2), and h is the height of the ramp. The total mechanical energy (E) of the sphere should remain constant if we ignore friction and air resistance. So the total energy at the top will be equal to the total kinetic energy (both translational and rotational) at the bottom.
Thus, the sphere's total kinetic energy at the bottom of the ramp (KEtotal) is simply equal to its initial potential energy, E = KEtotal = U = mgh = (2.50 kg)(9.81 m/s2)(1.20 m) = 29.43 J.
Therefore, the sphere's total kinetic energy when it reaches the bottom of the ramp is 29.43 J.