A roller coaster car of mass $100 k g$ is on a track that forms a circular loop of radius $R = 10 m$ in the vertical plane. If the car is to just maintain contact with the track at the top of the loop, what is the minimum value for its velocity at this point? $(g = 10 m / s^{2})$

12 m / s

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10 m / s

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15 m / s

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20 m / s

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Solution

The correct option is C $10 m / s$
We know that at the top:
$F_{n e t} = F_{g} + F_{N}$

Since, $F_{n e t} = m a_{c}$

Compare $F_{n e t}$ equations equal to each other.
$F_{g} + F_{N} = m a_{c}$

[If car just barely makes contact with the track, $F_{N} = 0$ for a slight moment]
$F_{g} + F_{N} = \frac{m v^{2}}{R}$

$m g = \frac{m v^{2}}{R}$
$g R = v^{2}$
$v = \sqrt{10 \times 10}$
$v = 10 m / s$

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