Problem 23 - University Physics with Modern Physics

Question

A 392-N wheel comes off a moving truck and rolls without slipping along a highway. At the bottom of a hill it is rotating at 25.0 rad/s. The radius of the wheel is 0.600 m, and its moment of inertia about its rotation axis is 0.800MR\(^2\). Friction does work on the wheel as it rolls up the hill to a stop, a height \(h\) above the bottom of the hill; this work has absolute value 2600 J. Calculate \(h\).

Step-by-Step Solution

Answer

The height is 14 meters.

Step 1: Understand the Problem

The problem involves a wheel rolling up a hill, coming to a stop after starting with a certain angular velocity. The work done by friction is known and causes the wheel to stop at a certain height. We need to find that height, taking into account both the kinetic energy changes and work-energy principle.

Step 2: Identify Given Values and Formulas

Given values are: initial angular velocity (\(\omega_i = 25.0 \text{ rad/s}\)), radius (\(R = 0.600 \text{ m}\)), moment of inertia in the form \(I = 0.800MR^2\), weight of the wheel (\(F = 392 \text{ N}\)), and work done by friction (\(W = 2600 \text{ J}\)).We need to calculate height (\(h\)). Use the gravitational potential energy formula \(PE = Mgh\) and work-energy principle.

Step 3: Calculate Mass of the Wheel

The weight of the wheel (\(F = Mg\)) is given. To find mass (\(M\)):\[M = \frac{F}{g} = \frac{392}{9.8} \text{ kg}\]Thus, \(M = 40 \text{ kg}\).

Step 4: Calculate Initial Kinetic Energy of the Wheel

The total kinetic energy (\(KE\)) is given by the sum of translational and rotational kinetic energies. Since the wheel rolls:\[KE = \frac{1}{2} Mv^2 + \frac{1}{2} I\omega^2\]Substitute \(v = R\omega\) and \(I = 0.800MR^2\):\[KE = \frac{1}{2} M(R\omega_i)^2 + \frac{1}{2} (0.800MR^2) \omega_i^2 = \frac{1.8}{2} MR^2 \omega_i^2\]Calculate \(KE\) using \(M = 40\), \(R = 0.6\), and \(\omega_i = 25.0\):\[KE = 0.9 \times 40 \times 0.6^2 \times 25^2 \]

Step 5: Use Work-Energy Principle

According to the work-energy principle:\[\Delta KE + W = 0\]Where the change in kinetic energy is equal to the negative of work done by friction plus the gravitational potential energy gained:\[0 - KE + Mgh = W\]Substitute \(KE\) and \(W\) values to find \(h\). Rearrange for \(h\):\[h = \frac{KE - W}{Mg}\]

Step 6: Calculate the Height

Substitute the known values:\[KE = 0.9 \times 40 \times 0.36 \times 625 = 8100 \text{ J}\]\[h = \frac{8100 - 2600}{392} \]Calculate \(h\):\[h = 14 \text{ m}\]

Key Concepts

Rolling Motion

Rolling motion occurs when an object like a wheel moves forward while simultaneously rotating around its own axis. This type of motion is prevalent in many daily life scenarios, such as tires on a car or a rolling ball. For effective rolling motion, the object must not slip on the surface, which means that the rotational speed of the object matches the translational speed of the surface it rolls over.

• **Rolling without slipping** ensures that every point on the rim of the wheel touches the surface momentarily before moving on.
• In rolling motion, the point of contact between the wheel and the surface is momentarily at rest.
• The total kinetic energy in rolling motion includes both translational and rotational components.

Understanding rolling motion is crucial as it sets the stage for analyzing the kinetic energies involved in problems like this one.

Work-Energy Principle

The work-energy principle is a fundamental concept in physics that relates work to the change in energy. It states that the work done by external forces on an object leads to a change in the kinetic energy of that object. In this exercise, as the wheel moves uphill, friction does negative work on the wheel.

• The work-energy principle can be expressed as \( \Delta KE + W = 0 \) where \( \Delta KE \) is the change in kinetic energy and \( W \) is the work done by friction.
• This principle ties together the initial kinetic energy, the work done, and the final potential energy, helping to solve for unknowns such as height in problems involving both kinetic and potential energies.

Understanding this principle is key to finding out how energy is transformed and conserved in motion scenarios.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. In the context of rolling motion, the wheel has both translational kinetic energy (due to its linear velocity) and rotational kinetic energy (due to its spin).

• Translational kinetic energy is given by \( \frac{1}{2} mv^2 \).
• Rotational kinetic energy is computed through \( \frac{1}{2} I\omega^2 \), where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.
• Total kinetic energy for a rolling object thus becomes the sum of these two forms: \( KE = \frac{1}{2} mv^2 + \frac{1}{2} I\omega^2 \).

In the exercise, calculating the wheel's initial kinetic energy is crucial to apply the work-energy principle properly.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy stored in an object due to its height above the ground. When the wheel rolls up the hill, it loses kinetic energy but gains gravitational potential energy. As the wheel gains height, the potential energy increases accordingly.

• Gravitational potential energy is calculated using \( PE = mgh \), where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height.
• In the broader picture of energy conservation, the wheel's loss in kinetic energy is balanced by a gain in gravitational potential energy and the work done by friction.

Calculating GPE allows us to determine how much height the wheel reaches before it stops, demonstrating the conversion and conservation of energy.