To find the height h in terms of R, we can use the conservation of energy principle. Since the ball is rolling without slipping, the total energy at the highest point Q will be equal to the total energy at point H.
At point Q, the total energy is only potential energy given by mgh, where m is the mass of the ball and g is the acceleration due to gravity.
At point H, the total energy is a combination of potential and kinetic energy. The potential energy is still mgh, but the kinetic energy is given by 1/2mv^2, where v is the velocity of the ball at point H.
Since the ball is rolling without slipping, the velocity at point H can be expressed as v = ωR, where ω is the angular velocity of the ball and R is the radius of the loop.
Now, equating the total energy at points Q and H, we get mgh = mgh + 1/2m(ωR)^2. Simplifying this equation, we get h = R(ω^2/2g).
Since R is the ratio of the loop, we can express it as R = 2πr, where r is the radius of the loop.
Substituting this into our equation for h, we get h = 2πr(ω^2/2g).
Now, we can express ω in terms of the time it takes for the ball to complete one revolution, T, and the radius of the loop, r. We know that ω = 2π/T, so substituting this into our equation for h, we get h = 2πr(2π/T)^2/2g.
Simplifying further, we get h = (4π^2r/T^2)/g.
Therefore, the height h in terms of R is given by the equation h = (4π^2r/T^2)/g, where R = 2πr and T is the time for one revolution.
I hope this helps you solve the problem and find the height h needed for the ball to reach point Q. Remember to always check your units and don't forget to include the acceleration due to gravity, g, in your calculations. Good luck!
