# Help with a Problem: Find H in Terms of R

• mprm86
In summary, the conversation discusses determining the height h (in terms of R) that a ball needs to reach point Q in a loop without slipping. The image attached shows Q as the highest point in the loop, with R being the loop ratio and H being the point on the track. The conversation also mentions finding the minimum kinetic energy required for the ball to complete the loop without falling, and asks if this rings any bells.
mprm86
I have attached an image of the problem. Which should be the height h (expressed in terms of R) if the ball needs to reach the point Q. The ball will roll without slipping.
In the image, Q is the highest point in the loop. R is the ratio of the loop (think of it as a circle). H is a point on the straight secton of the track, and it is measured respect to the ground.

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• Dibujo.GIF
3.7 KB · Views: 420
I'll give you a push but you have to do the work yourself.

Consider the minimum kinetic energy the ball must have to just complete this circular loop without falling. This would be the kinetic energy that corresponds to the tangential speed where the normal force from the track just equals zero.

Does this ring any bells?

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!

## 1. What does "H in Terms of R" mean?

"H in Terms of R" means finding the equation or expression for H (a variable) in relation to R (another variable). This involves solving for H in terms of R, where R is the independent variable and H is the dependent variable.

## 2. Why is it important to find H in Terms of R?

Finding H in Terms of R is important because it allows us to understand the relationship between two variables and make predictions about how one variable (R) will affect the other (H).

## 3. What are the steps for finding H in Terms of R?

The steps for finding H in Terms of R may vary depending on the specific problem, but generally involve isolating H on one side of the equation and simplifying to solve for H. This may require using algebraic techniques such as factoring, distributing, and combining like terms.

## 4. Can you provide an example of finding H in Terms of R?

Sure, let's say we have the equation H = 3R + 5. To find H in terms of R, we would first subtract 5 from both sides to get H - 5 = 3R. Then, we would divide both sides by 3 to isolate R and get H/3 - 5/3 = R. Therefore, H in terms of R is R = H/3 - 5/3.

## 5. What are some real-life applications of finding H in Terms of R?

Finding H in Terms of R has many practical applications in various fields such as physics, engineering, and economics. For example, in physics, finding the acceleration (H) in terms of the force (R) can help us understand the relationship between these two variables and make predictions about the motion of objects. In economics, finding the demand (H) in terms of the price (R) can help businesses determine the optimal pricing for their products.

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