A solid marble, a solid disk and a hoop have the same mass (M = 0.9 kg) and the

Question

A solid marble, a solid disk and a hoop have the same mass (M = 0.9 kg) and the same radius. Each object is held at rest at the same height and then released. At the bottom of the ramp is a loop-the-loop of radius 1 m. 1)What is the minimum height that the marble, disk and hoop can be released so that all three will safely make it around the loop-the loop? 2)At the top of the loop-the-loop, what is the force the track exerts on the hoop when released at the minimum height? 3)At the top of the loop-the-loop, what is the force the track exerts on the marble when released at the minimum height? 4)At the top of the loop-the-loop, what is the force the track exerts on the disk when released at the minimum height?

Explanation / Answer

The minimum linear velocity u at the top of the loop must be such that u^2= rg = 1.3*9.8
u = 3.569 m/s.
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M.I of a sphere is (2/5) mr²
M.I of solid disc is (1/2) mr²
M.I of hoop is mr²

Or in general I = k mr²
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For all objects v = r and the total kinetic energy = ½ (I ² + mv²) = ½ (k mr² * v ²/r ² + mv²)
= ½ mv ² (k +1)
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For all objects
Loss in potential energy = kinetic energy at the top of the loop.
mg ( H - 2.6) = = ½ mu² (k +1)
2g ( H - 2.6) = u² (k +1)
For marble
19.6 ( H - 2.6) = 3.569 ² ((2/5) +1)
H = 3.509 m.
For disc
19.6 ( H - 2.6) = 3.569 ² ((1/2) +1)
H =3.574 m.
For hoop
19.6 ( H - 2.6) = 3.569 ² (1 +1)
H = 3.899 m
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so minimum height= maximum of 3 answers= 3.899 m
B,
mg is the centripetal force and = mg = 0.7*9.8 = 6.86 N
Normal reaction is 6.86 N upward.

C.mg is the centripetal force and = mg = 0.7*9.8 = 6.86 N
Normal reaction is 6.86 N upward.

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