Question

Consider a roller-coaster with a circular loop. A car starts from rest from the top of a hill which is $5\text{m}$ higher than the top of the loop. It rolls down the hill and through the loop. What must be the radius of the loop, such that the passengers of the car will feel their normal weight at the highest point?

• A. $5\text{m}$
• B. $10\text{m}$
• C. $15\text{m}$
• D. $20\text{m}$

Answer 1

The correct option is A $5\text{m}$

Applying mechanical energy conservation between the points $\text{A}$ and $\text{B}$ and assuming the horizontal level of point $\text{B}$ as the zero potential energy level $(U=0)$, we have

$mg\left(5\right)=0+\frac{1}{2}m{v}^2\Rightarrow v^2=10g........\left(1\right)$


Let the radius of the loop and mass of the car be $r$ and $m$ respectively.

Using Newton's second law at point $\text{B}$,
$N+mg=\frac{m{v}^2}{r}$

Given, $N=mg$

$\Rightarrow 2mg=\frac{m{v}^2}{r}\Rightarrow r=\frac{v^2}{2g}........\left(2\right)$

Using equation $\left(1\right)$ and $\left(2\right)$ we have,

$\Rightarrow r=\frac{10g}{2g}=5\text{m}$

Why this Question ? Tip: In these type of questions, generally two unknowns are present. In this question, $v$ at point $\text{B}$ and $r$ are the unknowns. We needed two equations to solve the value of two unknowns. One equation was generated by applying energy conservation principle and the other by applying Newton's second law in circular motion.