Final answer:
To find the final velocity of the solid sphere rolling down the incline, conservation of mechanical energy is applied, converting potential energy at the top into translational and rotational kinetic energy at the bottom, and solving for velocity.
Step-by-step explanation:
To determine the final velocity of a solid sphere rolling down an incline, we can use the principle of conservation of mechanical energy. The total mechanical energy at the top of the incline must equal the total mechanical energy at the bottom, assuming no energy loss due to friction or air resistance since the sphere is rolling without slipping.
Conservation of Mechanical Energy:
At the top of the incline, the sphere possesses gravitational potential energy (U) due to its height (h) above the bottom. As the sphere rolls down, this potential energy is converted into kinetic energy (K) - both translational and rotational. So, Utop = Kbottom
Since the sphere is released from rest, its initial kinetic energy is zero. The potential energy can be calculated using U = mgh, where g is the acceleration due to gravity (9.8 m/s2), m is the mass of the sphere, and h is the height of the incline. The height can be found from the incline length and angle (19°) using trigonometric relationships. The kinetic energy at the bottom consists of translational kinetic energy (1/2mv2) plus rotational kinetic energy (1/2Iω2), where I is the moment of inertia for a solid sphere (2/5mr2) and ω is the angular velocity related to the linear velocity (v) by ω = v/r. Solving for v gives us the final velocity at the bottom.