Please provide a clear explanation to your work, thank you. You are designing a

Question

Please provide a clear explanation to your work, thank you.

You are designing a new roller coaster on another planet where the acceleration of gravity is equal to half the Earth's acceleration of gravity. A fully- loaded 735kg cart will be launched horizontally along a track on the ground by an ideal spring which is fixed to a wall on the side opposite to the cart. The spring, with a spring constant of 1.31 x 104N/m is compressed a distance, d, and when released, it launches the car on a frictionless track. The track then completes 1 loop (circle) with a radius of 15.2m. What is the minimum compression of the spring such that the car can complete the loop without falling (neglect the size of the car)? d 4.57m

Explanation / Answer

given acceleration due to gravity = g/2
m = 735 kg
spring constat, k = 1.31*10^4 N/m
compression = d
r = 15.2 m

so speed of the cart at the top of the loop = v
mv^2/r = mg/2
v = 8.6345 m/s

hence Total mechanical energy o fcart = 0.5mv^2 + m(g/2)*2r = 136996.65 J
from conservation of energy
136996.65 = 0.5kd^2
d = 4.5733 m

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