A mass starts at the top of a quarter circular ramp with some radius at a speed

Question

A mass starts at the top of a quarter circular ramp with some radius at a speed of 8 m/s. It then travels on a horizontal surface without friction, around a circular loop without falling off, on another horizontal surface without friction, up a ramp with a length of 22 meters, with a coefficient of kinetic friction of 0.63 which is inclined at 44 degrees. It then ends up on a horizontal frictionless surface moving at a speed of 24 m/s. What is the radius of the quarter circular ramp, in meters, the mass started?

A mass starts at the top of a quarter circular ramp with some radius at a speed of 8 m/s. It then travels on a horizontal surface without friction, around a circular loop without falling off, on another horizontal surface without friction, up a ramp with a length of 22 meters, with a coefficient of kinetic friction of 0.63 which is inclined at 44 degrees. It then ends up on a horizontal frictionless surface moving at a speed of 24 m/s. What is the radius of the quarter circular ramp, in meters, the mass started?

19.86

51.35 ± 10%

Explanation / Answer

conservation of energy

for quarter circular path

mgR1+0.5mv2 =0.5mv21

v1 =sqrt(R2g)

for ramp

height = sin(44) x 22 =15.28m

0.5mv21 =mg(15.28) +0.5mv2

0.5v21 =9.8 x 15.28 +0.5 x 242

v1=29.58m/sec

mgR1+0.5mv2 =0.5mv21

R1 = 46.1m

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