In the roller coaster shown the cars travel at a speed of 9

In the roller coaster shown, the cars travel at a speed of 9 ft/s at the top of the hill, determine the radius r of the loop necessary to induce the feel of weightlessness at B. Assume the weight of the passenger and car are 125 lbs and 130 lbs, respectively, and that the height of hill ht = 55 ft. Ignore friction and the sizes of the passenger and car.

                                                                r = _______________

Solution

Following points need to be kept in consideration so as to answer this question:

1.) Weightlessness at any point is felt only when the acceleration of the body downwards reaches the value of \'g\'

2.) Point B is part of a circular loop, hence the car and the passenger will suffer a net centripetal force given at any point as mv^2/R

c.) The net energy at the points A and B will be same. [Conservation of energy]

We will get the equation for conservation of energy at A and B so as to determine the velocity of the body at B in terms of R

That is, 0.5*255*81 + 255*32.2*(55 - 2R) = 0.5*255*v^2

Now, the velocity v should be such that the person feels weightlessness at B

That is, 125*v^2 / R = 125*g

or, v^2 = Rg, using this in the above equation, we get:

10327.5 + 451605 - 16422R = 4105.5R

R = 461932.5 / 20527.5 = 22.503 Ft

Therfore the required radius for the person to feel weightlessness is 22.503 ft

Note: The velocity that ther person and the car attains is obtained by using up the potential energy and kinetic energy they would have while getting off the top of the hill. That velocity is used to get the required centripetal force which makes the person feel weightless.