91Ó°ÊÓ

One of the new events in the 2002 Winter Olympics was the sport of skeleton (see the photo). Starting at the top of a steep, icy track, a rider jumps onto a sled (known as a skeleton) and proceeds-belly down and head first-to slide down the track. The track has fifteen turns and drops \(104 \mathrm{~m}\) in elevation from top to bottom. (a) In the absence of nonconservative forces, such as friction and air resistance, what would be the speed of a rider at the bottom of the track? Assume that the speed of the rider at the beginning of the run is relatively small and can be ignored. (b) In reality, the best riders reach the bottom with a speed of \(35.8 \mathrm{~m} / \mathrm{s}\) (about \(80 \mathrm{mi} / \mathrm{h}\) ). How much work is done on an \(86.0-\mathrm{kg}\) rider and skeleton by nonconservative forces?

Step by step solution

01

Understanding the Energy Conservation

In the absence of nonconservative forces like friction, the mechanical energy is conserved. The potential energy lost will convert into kinetic energy. We assume the initial speed is negligible.

02

Calculate Potential Energy at the Top

The potential energy at the top of the track is given by \( PE = mgh \), where \( m = 86.0 \, \text{kg} \), \( g = 9.8 \text{ m/s}^2 \), and \( h = 104 \, \text{m} \).

03

Calculate Kinetic Energy at the Bottom

The kinetic energy at the bottom is given by \( KE = \frac{1}{2}mv^2 \). Set the potential energy equal to the kinetic energy at the bottom since they are conserved.

04

Solve for Speed at the Bottom

\[ mgh = \frac{1}{2}mv^2 \]Canceling the mass \( m \) from both sides, we get:\[ gh = \frac{1}{2}v^2 \]\[ v = \sqrt{2gh} \]Substitute \( g = 9.8 \text{ m/s}^2 \) and \( h = 104 \text{ m} \):\[ v = \sqrt{2 \times 9.8 \times 104} \approx 45.28 \text{ m/s} \].

05

Calculate Actual Kinetic Energy at Bottom

The observed speed is \( 35.8 \text{ m/s} \), thus:\[ KE_{\text{actual}} = \frac{1}{2} \times 86.0 \times (35.8)^2 \]

06

Calculate Work Done by Nonconservative Forces

The work done \( W_{\text{nc}} \) by nonconservative forces is the difference between the initial potential energy and the actual kinetic energy at the bottom:\[ W_{\text{nc}} = mgh - KE_{\text{actual}} \]Compute using values: \[ W_{\text{nc}} = (86.0 \times 9.8 \times 104) - \left(\frac{1}{2} \times 86.0 \times (35.8)^2 \right) \approx 20134.24 \text{ J} \].

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Potential Energy

Potential energy is the energy that is stored in an object due to its position. In the context of the skeleton ride, when the rider is at the top of the track, he has a large amount of potential energy because he is elevated at a height of 104 meters above the bottom level.
The potential energy can be calculated using the formula \( PE = mgh \), where \( m \) is the mass of the rider and sled, \( g \) is the acceleration due to gravity, which is approximately \( 9.8 \, \text{m/s}^2 \), and \( h \) is the height.

• If the rider's mass is 86.0 kg and the height is 104 meters, then the potential energy at the top can be calculated as: \( 86.0 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 104 \, \text{m} \).
• This potential energy represents the maximum amount of energy that can be converted into kinetic energy as the rider travels down the track.

Kinetic Energy

Kinetic energy is the energy of motion. When the rider on the skeleton sled moves down the track, the stored potential energy is transformed into kinetic energy.
Kinetic energy depends on the mass of the object and its velocity, defined by the formula \( KE = \frac{1}{2}mv^2 \).

• As the rider descends from the 104m height, theoretically, all the potential energy is converted to kinetic energy if we assume no nonconservative forces are at work.
• If the rider were to reach a speed \( v \), which can be calculated by rearranging and solving the energy conservation equation from potential to kinetic, the rider would ideally attain a speed of about 45.28 m/s.

However, in real-life scenarios, due to factors like friction and air resistance, the actual speed achieved can differ. Such nonconservative forces imply energy loss to the surroundings.

Nonconservative Forces

Nonconservative forces are forces that cause energy dissipation in a system, such as friction and air resistance.
When the rider on the skeleton track moves, nonconservative forces do work on the rider, converting some mechanical energy into thermal energy or sound energy, which is not recovered to system movement.

• This causes the accrued potential energy to be reduced when converting to kinetic energy, resulting in a lower speed \( v = 35.8 \, \text{m/s} \) achieved by the rider at the track's bottom.
• The work done by these forces can be calculated by comparing the initial potential energy with the actual kinetic energy achieved.

This concept is crucial in understanding real-world applications where energy losses are inevitable, unlike ideal situations often studied theoretically.

Work-Energy Principle

The work-energy principle states that the work done by all forces acting on a body will equal the change in the kinetic energy of the body.
For the skeleton ride, this principle can signify how the work done by nonconservative forces influences the speed of the rider at the bottom of the track.

• Initially, the rider has potential energy at the top. As they descend, work done by gravity transforms this into kinetic energy.
• When nonconservative forces (like friction) work on the system, they do negative work, decreasing the total kinetic energy obtained.

To quantify this, one subtracts the kinetic energy at the bottom from the calculated initial potential energy. The resulting value is the work done by nonconservative forces, which is crucial for understanding energy dissipation in mechanical systems.