Question

You are riding a ferris wheel while sitting on a scale. A ferris wheel with radius 9.7 m and a period 32s. Find the scale reading for a 60kg person at the bottom of the ferris wheel and the top of the ferris wheel, assuming it moves at a constant rate.

Answer 1

Answer:

The scale reading at the top = 565.8N

The scale reading at the bottom = 610.44N

Explanation:

We are given:

t = 32seconds

R = 9.7meters

mass (m) = 60Kg

We take g as gravitational field = 9.8

Therefore, to find the scale reading (N) at the top, let's use the formula:

$N_t = m (g - w^2 R)$

Where w = 2π/t

w = 2π/32 = 0.197

Substituting the figures into the equation, we have

$N = 60 (9.8 - 0.197^2 * 9.7)$

N = 60 (9.8 - 0.37)

N = 60 * 9.43

Na = 565.8N

To find scalar reading at the bottom, we use;

$N_b = m(g + w^2 R)$

= 60 (9.8 + 0.37)

= 60 * 10.17

$N_b = 610.44$

Answer 2

Answer:

The scale reading at the top is $z_{top} = 565.6\mu N$

The scale reading at the bottom is $z_{bottom} = 610.356\ N$

Explanation:

From the question

       The radius is $r = 9.7 m$

       The period is $T = 32sec$

      The mass is $m =60kg$

Generally the mathematical representation for angular velocity of the wheel is

                  $\omega = \frac{2 \pi}{T}$

                    $= \frac{2*3.142}{32}$

                    $= 0.196 \ rad/sec$

The velocity at which the point scale move can be obtained as

                     $v = r\omega$

                        $= 9.7* 0.196$

                       $= 1.9 m/s$

Considering the motion of the 60kg mass as shown on the first and second uploaded image

Let the z represent the reading on the scale which is equivalent to the normal force acting on the mass.

          Now at the topmost the reading of the scale would be

                       $mg - z_{top} =\frac{mv^2}{r} =m\omega^2r$

Where mg is the gravitational force acting on the mass and $\frac{mv^2}{r}$ is the centripetal force keeping the mass from spiraling out of the circle

Now making z the subject of the formula

                    $z_{top} = mg - \frac{mv^2}{r}$

                       $= m(g- \frac{v^2}{r})$

                       $= 60(9.8 - \frac{1.9^2}{9.7} )$

                      $= 565.6\mu N$

Now at the bottom the scale would be

                      $z_{bottom}-mg = \frac{mv^2}{r} = m \omega^2r$

This is because in order for the net force to be in the positive y-axis (i.e for the mass to keep moving in the Ferris wheel) the Normal force must be greater than the gravitational force.  

Making  z the subject

                  $z_{bottom}= mg +m\omega^2r$

                  $z =m(g+\omega^2r)$

                  $z= 60(9.8 + (0.196)^2 *(9.7))$

                     $= 610.356\ N$