help with a physics question-about a rollercoaster and its loop?

Question

A block of mass ( m ) slides from rest on a frictionless loop-the-loop track, as illustrated in the diagram. What is the minimum release height, ( h_i ), required for the block to maintain contact with the track at all times and successfully complete the loop? Please express your answer in terms of the radius of the loop, ( r ). Additionally, does the mass of the block affect this minimum height? Thank you in advance for your assistance!

Anonymous · Accepted Answer

Find minimum speed at the top of the loop

ΣFc=mg=mv²/r

mass cancels

g=v²/r

v=√(gr)

use mechanical energy to solve for height

Potential energy at h turns into potential energy and kinetic energy (Replace second h with 2r

PE=PE+KE

mgh=mg2r+½mv²

mass cancels

gh=g2r+½v²

replace v²=gr

gh=g2r+½gr

g cancels

h=2r+½r

h=5r/2

Anonymous · Answer

Blood…is wrong.  If it starts at the height of the top of the loop with zero velocity, it will arrive at the top of the loop with zero velocity and fall off.
The centripetal force between the block and the track is mV^2/r.  At the top of the loop this force must be greater than the weight of the block, mg, or the block will fall.
If it has this velocity, V, then its kinetic energy is (1/2)mV^2, and it got this velocity from its gravitational potential at the start:  (1/2)mV^2 = mgh where h is the starting height distance above the top of the loop.
The first condition is mV^2/r > mg  or V^2 > r*g
The second condition is mgh = (1/2)mV^2 > (1/2)m*r*g   or h > r/2
If this is measured from the bottom of the loop, h(bot) = 5r/2.

Beth Kautzer · Answer

Mass doesn’t matter.The roller-coaster has to be released at at least the height of the top of the loop without any acceleration.

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help with a physics question-about a rollercoaster and its loop?

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