Question

In designing circular rides for amusement parks, mechanical engineers must consider how small variations in certain parameters can alter the net force on a passenger. Consider a passenger of mass mriding around a horizontal circle of radius$\mathbf{r}$at speed$\mathbf{v}$. What is the variation$\mathbf{dF}$in the net force magnitude for

(a) a variation$\mathbf{dr}$in the radius with vheld constant,

(b) a variation dvin the speed with$\mathbf{r}$held constant, and

(c) a variation$\mathbf{dT}$in the period with rheld constant?

Short answer

1. Variation in the net force$dF$ for variation in radius$dr$with$v$constant is$-\frac{{\mathrm{mv}}^2}{{\mathrm{r}}^2}\mathrm{dr}$.
2. Variation in the net force$dF$ for variation in speed$dv$with$r$ constant is$\frac{2\mathrm{mv}}{\mathrm{r}}\mathrm{dv}$.
3. Variation in the net force$dF$ for variation in period$dT$ with$r$ constant is $-\left(\frac{{\mathrm{mv}}^3}{{\mathrm{\pi r}}^2}\right)\mathrm{dT}$.

Step-by-step solution

Given data

The passenger of mass m riding around a horizontal circle of radius$r$ at speed $v$.

To understand the concept

The problem deals with Newton’s laws of motion which describe the relations between the forces acting on a body and the motion of the body. Also, it deals with the centripetal force. It is a force that makes a body follow a curved path.

Formula:

Centripetal force is given by,

$\mathrm{F}={\mathrm{mv}}^2/\mathrm{r}$

(a) The variation dF  in the net force magnitude for (a) a variation dr in the radius with v held constant

The centripetal force on passengers is,

$\mathrm{F}={\mathrm{mv}}^2/\mathrm{r}$

The variation of F with respect to r while holding v constant is,

$\mathrm{dF}=-\frac{{\mathrm{mv}}^2}{{\mathrm{r}}^2}\mathrm{dr}$

Thus, variation in the net force$\text{d}\mathrm{F}$ for variation in radius $\text{d}\mathrm{r}$with $v$constant is $-\frac{{\mathrm{mv}}^2}{{\mathrm{r}}^2}\mathrm{dr}$.

(b) The variation dF in the net force magnitude for (b) a variation dv in the speed with r held constant

The centripetal force on passengers is,

$\mathrm{F}={\mathrm{mv}}^2/\mathrm{r}$

The variation of F with respect to v while holding r constant is,

$\mathrm{dF}=\frac{2\mathrm{mv}}{\mathrm{r}}\mathrm{dv}$

Thus, variation in the net force $\text{d}\mathrm{F}$for variation in speed $\text{d}\mathrm{v}$with $r$constant is $\frac{2\mathrm{mv}}{\mathrm{r}}\mathrm{dv}$.

(c) The variation dF in the net force magnitude for (c) a variation dT in the period with r held constant 

The period of the circular ride is,

$\mathrm{T}=2\mathrm{\pi r}/\mathrm{v}$

Thus,

$\begin{array}{c}\mathrm{F}=\frac{{\mathrm{mv}}^2}{\mathrm{r}}\\ =\frac{\mathrm{m}}{\mathrm{r}}{\left(\frac{2\mathrm{\pi r}}{\mathrm{v}}\right)}^2\\ =\frac{4{\mathrm{\pi }}^2\mathrm{mr}}{{\mathrm{T}}^2}\end{array}$

Hence the variation of F with respect to T while holding r constant is,

$\begin{array}{l}\mathrm{dF}=-\frac{8{\mathrm{\pi }}^2\mathrm{mr}}{{\mathrm{T}}^3}\mathrm{dT}\\ \mathrm{dF}=-8{\mathrm{\pi }}^2\mathrm{mr}{\left(\frac{\mathrm{v}}{2\mathrm{\pi r}}\right)}^3\mathrm{dT}\\ \mathrm{dF}=-\left(\frac{{\mathrm{mv}}^3}{{\mathrm{\pi r}}^2}\right)\mathrm{dT}\end{array}$

Thus, variation in the net force$\text{d}\mathrm{F}$ for variation in period$\text{d}\mathrm{T}$with$r$constant is$-\left(\frac{{\mathrm{mv}}^3}{{\mathrm{\pi r}}^2}\right)\mathrm{dT}$.