Question

A woman rides on a ferris wheel of radius 16m that maintains the same speed throughout its motion. She sits on a bathroom scale (with memory) and sits on it. When she gets off the ride, she uploads the scale readings to a computer.  It showed that the minimum value that the scale read was 510N and the maximum was 666N. What is the woman's mass?

Answer 1

Concepts and reason

The main concepts required to solve this problem are the normal force, centripetal force, and gravitational pulling force.

Initially, write equations for the gravitational force and centripetal force and the normal force. Use these equations and find the mass of the woman.

Fundamentals

The equation for the centripetal force is,

${F_c} = \frac{{m{v^2}}}{r}$

Here, m is the mass of the object, v is the speed of the object and r is the radius of the object.

The equation for the gravitational pulling force is,

${F_g} = mg$

Here, m is the mass of the object and g is acceleration due to gravity.

The normal reaction would be maximum at the bottom of the Ferris wheel, that is,

${N_{{\rm{min}}}} = {F_g} - {F_c}$ …… (1)

The equation for the centripetal force on the woman is,

${F_c} = \frac{{m{v^2}}}{r}$ …… (2)

Here, m is the mass of the woman, v is the speed and r is the radius of the Ferris wheel.

The equation for the gravitational force is,

${F_g} = mg$ …… (3)

Here, m is the mass of the woman and g is the acceleration due to gravity.

Substitute the equations (2), and (3) in above equation (1).

${N_{{\rm{max}}}} = \frac{{m{v^2}}}{r} + mg$ …… (4)

The normal reaction at the top of the Ferris wheel is minimum, that is,

${N_{{\rm{min}}}} = {F_g} - {F_c}$

Substitute the equations (2) and (3) in above equation.

${N_{{\rm{min}}}} = mg - \frac{{m{v^2}}}{r}$ …… (5)

The sum of the maximum and minimum normal forces from the equations (4) and (5) is,

$\begin{array}{c}\\{N_{{\rm{max}}}} + {N_{{\rm{min}}}} = \frac{{m{v^2}}}{r} + mg + mg - \frac{{m{v^2}}}{r}\\\\ = 2mg\\\end{array}$

Rearrange the above equation for m.

$m = \frac{{{N_{{\rm{max}}}} + {N_{{\rm{min}}}}}}{{2g}}$

The equation for the mass of the woman that derived in the above step 1 is,

$m = \frac{{{N_{{\rm{max}}}} + {N_{{\rm{min}}}}}}{{2g}}$

Substitute $666{\rm{ N}}$ for ${N_{{\rm{max}}}}$ , $510{\rm{ N}}$ for ${N_{{\rm{min}}}}$ , and $9.8{\rm{ m/}}{{\rm{s}}^2}$ for g in above equation.

$\begin{array}{c}\\m = \frac{{\left( {666{\rm{ N}}} \right) + \left( {510{\rm{ N}}} \right)}}{{2\left( {9.8{\rm{ m/}}{{\rm{s}}^2}} \right)}}\\\\ = 60{\rm{ kg}}\\\end{array}$

Ans:

The mass of the woman is 60 kg.