To solve this problem, we can use the principle of conservation of energy, which states that the total mechanical energy of a system remains constant if there are no non-conservative forces acting on it. In this case, we can assume that there is no frictional force acting on the ball as it rolls up the ramp.
Let's first calculate the initial kinetic energy (KEi) of the ball:
KEi = (1/2) * m * v^2
where m is the mass of the ball and v is its initial speed. We don't know the mass of the ball, but we can use the fact that it is rolling without slipping to relate its linear speed to its rotational speed:
v = R * ω
where R is the radius of the ball and ω is its angular speed. Since the ball is rolling without slipping, we also have:
v = R * f
where f is the frequency of the ball's rotation (i.e., the number of rotations per second).
Combining these two equations, we get:
ω = v / R = f * 2π
Using the formula for the moment of inertia of a solid sphere (I = (2/5) * m * R^2), we can relate the angular speed to the rotational kinetic energy (KErot) of the ball:
KErot = (1/2) * I * ω^2 = (1/5) * m * v^2
Substituting this into the conservation of energy equation, we get:
KEi + PEi = KEf + PEf + KErot
where PEi and PEf are the initial and final potential energies of the ball, respectively. At the bottom of the ramp, the ball has only kinetic energy, so PEi = 0. At the top of the ramp, the ball has only potential energy, so KEf + KErot = 0. Therefore, we can simplify the conservation of energy equation to:
KEi = PEf
Substituting the expressions for KEi and PEf, we get:
(1/2) * m * v^2 = m * g * h
where g is the acceleration due to gravity and h is the height of the ramp. Solving for v, we get:
v = sqrt(2 * g * h) = sqrt(2 * 9.81 m/s^2 * 0.53 m) ≈ 3.06 m/s
So the linear speed of the ball when it reaches the top of the ramp is approximately 3.06 m/s.
If the radius of the ball were increased, the speed found in part (A) would stay the same. This is because the increase in the radius would be offset by a proportional decrease in the rotational speed of the ball, so the total kinetic energy of the ball would remain the same. In other words, the ball would have a greater moment of inertia and a lower angular speed, but the same amount of kinetic energy as before. Therefore, its linear speed would not change.