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 shown. What is the minimum release height, hi, required for the block to maintain contact with the track at all times and successfully complete the loop. Give your answer in terms of the radius of the loop, r. Does the mass matter?thanks guys in advance!

Dr. Reggie Bosco DVM · Accepted Answer

Find minimum speed at the top of the loopΣFc=mg=mv²/rmass cancelsg=v²/rv=√(gr) use mechanical energy to solve for heightPotential energy at h turns into potential energy and kinetic energy (Replace second h with 2rPE=PE+KEmgh=mg2r+½mv²mass cancelsgh=g2r+½v²replace v²=grgh=g2r+½grg cancelsh=2r+½rh=5r/2

Dr. Reggie Bosco DVM · 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*gThe second condition is mgh = (1/2)mV^2 > (1/2)m*r*g or h > r/2If this is measured from the bottom of the loop, h(bot) = 5r/2.

Dr. Reggie Bosco DVM · 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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