Problem 37 - College Physics

Question

A 2000 -kg Ferris wheel accelerates from rest to an angular speed of \(20 \mathrm{rad} / \mathrm{s}\) in \(12 \mathrm{~s}\). Approximate the Ferris wheel as a circular disk with a radius of \(30 \mathrm{~m}\). What is the net torque on the wheel?

Step-by-Step Solution

Answer

The net torque on the Ferris wheel is \( 1,500,000 \text{ N} \cdot \text{m} \).

Step 1: Identify Known Values

First, let's list out the known values from the problem statement:- Mass of the Ferris wheel, \( m = 2000 \text{ kg} \).- Radius of the Ferris wheel, \( r = 30 \text{ m} \).- Final angular speed, \( \omega_f = 20 \text{ rad/s} \).- Time to reach final angular speed, \( t = 12 \text{ s} \).- Initial angular speed, \( \omega_i = 0 \text{ rad/s} \).

Step 2: Calculate Angular Acceleration

To find the angular acceleration, we use the formula: \[ \alpha = \frac{\omega_f - \omega_i}{t} \]Substitute the known values into the formula:\[ \alpha = \frac{20 \text{ rad/s} - 0}{12 \text{ s}} = \frac{20}{12} \text{ rad/s}^2 = \frac{5}{3} \text{ rad/s}^2 \].

Step 3: Compute the Moment of Inertia

The moment of inertia \( I \) for a circular disk is calculated using:\[ I = \frac{1}{2} m r^2 \]Substitute the given values:\[ I = \frac{1}{2} \times 2000 \text{ kg} \times (30 \text{ m})^2 \]\[ I = 900,000 \text{ kg} \cdot \text{m}^2 \].

Step 4: Find the Net Torque Using Angular Acceleration

Net torque \( \tau \) is given by the formula:\[ \tau = I \alpha \]Using the previously calculated values:\[ \tau = 900,000 \text{ kg} \cdot \text{m}^2 \times \frac{5}{3} \text{ rad/s}^2 \]\[ \tau = 1,500,000 \text{ N} \cdot \text{m} \].

Key Concepts

Angular Acceleration

Angular acceleration is an important concept in circular motion. It tells us how quickly an object is speeding up or slowing down its rotation. Imagine you are spinning a toy top. At first, you spin it slow and then a bit faster. Angular acceleration measures how fast this change in spinning speed happens.
To find angular acceleration, we use the formula \( \alpha = \frac{\omega_f - \omega_i}{t} \). Here, \( \omega_f \) is the final angular speed, \( \omega_i \) is the initial angular speed, and \( t \) is the time it takes for this change.
This helps us understand the dynamics involved in rotational systems like Ferris wheels or car tires. When a Ferris wheel starts from a standstill and reaches a certain speed, it's angular acceleration that tells us how quickly this transition occurs.

Moment of Inertia

The moment of inertia is like the mass of objects in linear motion but for rotation. It represents how much torque, or rotational force, is needed to get an object spinning around an axis. Think of it as an indicator of rotational stubbornness.
For a Ferris wheel, which we often approximate as a solid disc, the moment of inertia is calculated using \( I = \frac{1}{2} m r^2 \). Here, \( m \) is the mass and \( r \) is the radius.
- A larger mass or a bigger radius means a larger moment of inertia, making it harder to start or stop the spinning. - Just like how heavier objects are harder to push in a straight line, larger or heavier wheels take more effort to spin.

Circular Motion

Circular motion is when something moves along a circular path. Anything moving in a loop, like a Ferris wheel, is experiencing circular motion. Typically, such objects are subject to forces that keep them on their path.
In circular motion, several unique dynamics come into play:

• Centripetal Force: This keeps the object moving in the circle and prevents it from flying off the path.
• Angular Speed: The rate at which the object travels around the circle.

Learning how these forces work together helps to understand what keeps structures like Ferris wheels safe and functioning.

Ferris Wheel Dynamics

Ferris wheel dynamics combine all principles of physics for circular motion into one structured experience. As the Ferris wheel turns, it offers more than just a thrilling ride. Understanding its dynamics provides insights on how it's engineered for both safety and fun.
Several key principles include:

• Structural Design: Designing a Ferris wheel involves knowing the right materials and design to handle the immense forces.
• Stability: Ensuring rides remain stable, even in strong winds or when fully loaded with people.
• Torque and Acceleration: Calculating how powerful motors need to be to start, stop, and maintain a consistent speed.

By understanding these dynamics, engineers can design Ferris wheels that are both thrilling and reliable.