Let's Give it a Loop

Classical Mechanics Level 2

Figure below shows a loop-the-loop track of radius, R = 10 m. A light electric car starts from a platform with a uniform velocity at a distance h above the top of the loop and goes around the loop without falling off the track.

Find the minimum value of h in metres for a successful looping. Neglect friction.


The answer is 5.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Based on the conservation of mechanical energy:

1 2 \frac{1}{2} m v 2 mv^{2} = m g h mgh

Or, v 2 v^{2} = 2 g h 2gh ... (1)

At the point when the car is at the maximum height in the loop:

N + mg = m v 2 R \frac{mv^{2}}{R}

For h to be mininmum, normal force, N should be zero.

mg = m v 2 R \frac{mv^{2}}{R}

Or, g = v 2 R \frac{v^{2}}{R}

Using (1), we get:

g = 2 g h R \frac{2gh}{R}

Or, h = R 2 \frac{R}{2} ; R = 10 m has been given

Or, h = 10 2 \frac{10}{2} = 5 m

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