Question

A cart starts from rest at the top of a roller coaster that is 40 meters high. It descends down the first hill all the way to ground level before heading into a vertical circular loop with diameter 20 meters. What is the minimal H(height of the initial hill) needed in order for the cart to just barely make it over the top of the loop?

Answer 1

At the top of the circular loop, Normal force N = 0,

let v be the velocity at the top of the circular loop and m br the mass of the cart

so, Centrifugal force , Fc = mv²/R

and the F_c will be equal to the sum of weight of the cart, W and normal force ,

F_c = W + N

  mv²/R = mg + 0

so, mv² = mgR

so, Kinetic energy required to reach at the top of the circular loop, KE_top = 1/2 *mv² = 1/2 * mgR

so, total energy required by the cart to reach at the top of the circular loop,

TE_top = KE_top + PE_top

= 1/2 * mgR + mg* 2R

Now, let H be the height of the initial hill,

Now, Apply the law of conservation of energy,

Energy at the top of the initial hill = Energy at the top of the circular loop.

so, mg*H + 0 =  1/2 * mgR + mg* 2R

H = 5/2 * R = 5/2 * 10 = 25 m

so, minimum height needed of the initial hill, H = 25 m