We are given that the Ferris wheel turns with a constant angular velocity, and we are asked to find the radius of the Ferris wheel given that:

• The person's weight is $476.0 \, \text{N}$.
• The difference between her apparent weight and her true weight at the top of the Ferris wheel is $1.500 \, \text{N}$.
• The angular speed of the Ferris wheel is $\omega = 0.0250 \, \text{rad/s}$.

Step 1: True and Apparent Weight

The apparent weight $W_{\text{app}}$ at the top of the Ferris wheel is less than the true weight $W = 476.0 \, \text{N}$ by the magnitude of the centripetal force. The centripetal force is provided by the difference between the gravitational force and the normal force (apparent weight) at the top of the Ferris wheel.

At the top of the Ferris wheel: $W_{\text{app}} = W - F_{\text{centripetal}}$

Step 2: Centripetal Force Expression

The centripetal force $F_{\text{centripetal}}$ acting on the person at the top is due to circular motion and is given by: $F_{\text{centripetal}} = m \cdot a_c = m \cdot r \cdot \omega^2$ where:

• $m$ is the person's mass,
• $a_c$ is the centripetal acceleration, and
• $r$ is the radius of the Ferris wheel.

The true weight $W$ is related to the mass by: $W = m \cdot g$ where $g = 9.81 \, \text{m/s}^2$, so the mass is: $m = \frac{W}{g}$ Substitute this into the expression for the centripetal force: $F_{\text{centripetal}} = \frac{W}{g} \cdot r \cdot \omega^2$

Step 3: Difference in Weight

We are told that the apparent weight differs from the true weight by $1.500 \, \text{N}$: $W - W_{\text{app}} = 1.500 \, \text{N}$ Using the relation $W_{\text{app}} = W - F_{\text{centripetal}}$, we get: $F_{\text{centripetal}} = 1.500 \, \text{N}$

Substitute the expression for centripetal force: $\frac{W}{g} \cdot r \cdot \omega^2 = 1.500$

Step 4: Solve for Radius $r$

Now solve for $r$: $r = \frac{1.500 \cdot g}{W \cdot \omega^2}$

Substitute the values $W = 476.0 \, \text{N}$, $g = 9.81 \, \text{m/s}^2$, and $\omega = 0.0250 \, \text{rad/s}$: $r = \frac{1.500 \cdot 9.81}{476.0 \cdot (0.0250)^2}$

Now let's calculate the value of $r$.The radius of the Ferris wheel is approximately $49.46 \, \text{m}$.

Would you like further details or have any questions about this solution?

Here are five related questions for further exploration:

1. How does the angular velocity affect the apparent weight at the bottom of the Ferris wheel?
2. What would happen to the person's apparent weight if the Ferris wheel's angular velocity doubled?
3. How would the radius change if the difference in apparent weight were increased to 2.0 N?
4. What is the centripetal force acting on the person at the bottom of the Ferris wheel?
5. How does the mass of the person influence the apparent weight at different points of the ride?

Tip: The apparent weight is always different from the true weight when an object undergoes circular motion because of the additional centripetal acceleration.