Question

A roller coaster is designed such that riders experience “weightlessness” as they go round the top hill whose radius of curvature is 20m. The speed of the car at the top of the hill should be between

& \text{A}\text{.14}m/s\text{ and 15}m/s \\\ & \text{B}.15m/s\text{ and }16m/s \\\ & \text{C}.16m/s\text{ and }17m/s \\\ & \text{D}.13m/s\text{ and 14}m/s \\\ \end{aligned}$$

Answer 1

By drawing a rough diagram we can figure out the forces experienced by the person in the car and then balancing the force we can find the speed of the car. The rider experiences weightlessness means that they don’t experience any upward force acting on them. Once we balance the forces and consider the given situation we can substitute the given values to find the speed.

Formula used:
Centripetal force $F=\dfrac{m{{v}^{2}}}{r}$

Complete answer:
Taking the above picture in consideration, there will be centripetal force acting towards the center of the body and downward force on mg and upward normal force N. The downward force is due to the attractive pull of the earth that is gravity.
Then we can write an expression
$mg-N=F$, where F is centripetal force
Further at the top of the round as rider experiences weightlessness therefore N=0

& \Rightarrow mg=F \\\ & \Rightarrow mg=\dfrac{m{{v}^{2}}}{r} \\\ \end{aligned}$$ Cancelling m from both the sides $$g=\dfrac{{{v}^{2}}}{r}$$ As we have to find speed therefore taking on the left hand side and rest on the right hand side. r is the radius of curvature. $$\begin{aligned} & \Rightarrow {{v}^{2}}=gr \\\ & \Rightarrow v=\sqrt{gr} \\\ \end{aligned}$$ Putting value of g=$$9.8m/{{s}^{2}}$$and r= 20m $$\begin{aligned} & \Rightarrow v=\sqrt{9.8\times 20} \\\ & \Rightarrow v=14m/s \\\ \end{aligned}$$ Hence the speed of the car should be between $$14m/s$$ and $$15m/s$$. **So, the correct answer is “Option A”.** **Note:** The value of g that is acceleration due to gravity is taken $$9.8m/{{s}^{2}}$$ although few takes value of g as $$10m/{{s}^{2}}$$ which can vary the result. Centripetal and centrifugal forces are two different quantities in which people get confused often. Centripetal force is the force acting towards the center of the circular path whereas in centrifugal it acts away from the center.