Final answer:
The total kinetic energy of a solid sphere at the bottom of the ramp is 18.375 J. The rotational kinetic energy is 5.25 J, and the translational kinetic energy is 13.125 J.
Step-by-step explanation:
When a 2.5-kg solid sphere with radius 0.10 m rolls down a ramp without slipping, its total kinetic energy at the bottom of the ramp is equal to the gravitational potential energy it had at the top of the ramp due to its height of 0.75 m. The total kinetic energy (Ktotal) is a sum of the translational kinetic energy (Ktrans) and the rotational kinetic energy (Krot). Using the conservation of energy, we can calculate Ktotal = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
To determine Krot, we use the formula Krot = 1/2 I ω2, where I is the moment of inertia and ω is the angular velocity. For a solid sphere, I = 2/5 mR2, where R is the radius. Since the sphere rolls without slipping, ω can be related to the translational velocity v by v = Rω. Ktrans is given by Ktrans = 1/2 mv2.
By substituting I and ω in terms of v and R, we can express Krot in terms of the sphere's mass and velocity. Krot is 2/5 times Ktrans for a solid sphere. We find that Ktotal = Ktrans + Krot, which allows us to calculate the individual energies.
- Ktotal = mgh = (2.5 kg)(9.8 m/s2)(0.75 m) = 18.375 J
- Krot = (2/7) * Ktotal = (2/7) * 18.375 J = 5.25 J
- Ktrans = (5/7) * Ktotal = (5/7) * 18.375 J = 13.125 J