We consider a roller coaster with a lift hill of height h = 30 m and a circular looping with radius R = 10 m. We assume, that the cars are only accelerated by gravity. If the cars start with zero velocity at point A, how large are the absolute values of the g-force in point B and C?
Additional question: A vertical g-force up to 5g is for the human body easily bearable. Can you modify the roller coaster, so that the maximal g-force is below this value (without removing the looping)?
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The total energy
E = U + T = m g y + 2 1 m v 2 (potential and kinetic energy)
of a car with height y , velocity v and mass m is conserved along the track. The energy in the points A, B and C reads:
E A E B E C = m g h = 2 1 m v B 2 = 2 1 m v C 2 + 2 m g R E A = E B ⇒ E A = E C ⇒ v B = 2 g h v C = 2 g ( h − 2 R )
With the help of the velocities v B and v C we can caculate the centrifugal force F cf = ± m v 2 / R inside the looping. With the gravitional force F g = − m g we get the total force, that acts on the passengers:
F B F C = − m R v B 2 − m g = − ( R 2 h + 1 ) m g = − 7 ⋅ m g = + m R v C 2 − m g = + ( R 2 h − 5 ) m g = + 1 ⋅ m g