The skater need to start 7.525 meter high.
Let,
The skater need to start on a previous incline at height = h.
Mass of the skater = m
Given, height of the loop = 6.1 meter.
Radius of the loop; r = 6.1/2 meter = 3.05 meter.
Let to make it up and around a loop that is 6.1meters high, the man required minimum velocity v on the highest point of the loop.
So, centripetal force at highest point = weight of the man
⇒ mv²/r = mg
⇒ v = √(gr)
Then, potential energy of the skater at height h is = mgh.
And, minimum energy of the skater at the highest point of the loop is = potential energy + minimum kinetic energy.
= mg(2r) + 1/2 mv²
= 2mgr + 1/2 mgr
= 5/2 mgr
According to conservation of energy,
potential energy of the skater at height h = minimum energy of the skater at the highest point of the loop
⇒ mgh = 5/2 mgr
⇒ h = 5/2 r =( 5/2 )× 3.05 meter = 7.525 meter.
Hence, required height is 7.525 meter.
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