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Two identical wheels are moving on horizontal surfaces. The center of mass of each has the same linear speed. However, one wheel is rolling. while the other is sliding on a frictionless surface without rolling. Each wheel then encounters an incline plane. One continues to roll up the incline, while the other continues to slide up. Eventually they come to a momentary halt, because the gravitational force slows them down. Each wheel is a disk of mass \(2.0 \mathrm{kg} .\) On the horizontal surfaces the center of mass of each wheel moves with a linear speed of \(6.0 \mathrm{m} / \mathrm{s}\). (a) What is the total kinctic energy of each wheel? (b) Determine the maximum height reached by each wheel as it moves up the incline.

Step by step solution

01

Calculate the Kinetic Energy of the Sliding Wheel

The sliding wheel only has translational kinetic energy because it slides without rolling. The translational kinetic energy \( KE_t \) is given by:\[KE_t = \frac{1}{2} m v^2\]Substitute \( m = 2.0 \, \text{kg} \) and \( v = 6.0 \, \text{m/s} \) into the formula:\[KE_t = \frac{1}{2} \times 2.0 \, \text{kg} \times (6.0 \, \text{m/s})^2 = 36 \, \text{J}\]

02

Calculate the Kinetic Energy of the Rolling Wheel

The rolling wheel has both translational and rotational kinetic energy. The total kinetic energy \( KE_{total} \) is given by:\[KE_{total} = KE_t + KE_r = \frac{1}{2} mv^2 + \frac{1}{2} I \omega^2\]For a disk, the moment of inertia \( I \) is \( \frac{1}{2} m r^2 \) and \( \omega = \frac{v}{r} \). Substitute these into the formula:\[KE_r = \frac{1}{2} \left( \frac{1}{2} m r^2 \right) \left( \frac{v}{r} \right)^2 = \frac{1}{4} m v^2\]Therefore, the total kinetic energy is:\[KE_{total} = \frac{1}{2} mv^2 + \frac{1}{4} mv^2 = \frac{3}{4} mv^2\]Substitute the values \( m = 2.0 \, \text{kg} \) and \( v = 6.0 \, \text{m/s} \):\[KE_{total} = \frac{3}{4} \times 2.0 \, \text{kg} \times (6.0 \, \text{m/s})^2 = 54 \, \text{J}\]

03

Calculate Maximum Height for Sliding Wheel

The maximum height \( h \) can be found using the conservation of energy:\[KE = mgh\]For the sliding wheel:\[36 \, \text{J} = 2.0 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times h\]Solve for \( h \):\[h = \frac{36}{19.62} \approx 1.83 \, \text{m}\]

04

Calculate Maximum Height for Rolling Wheel

Use the conservation of energy again, this time considering the total initial kinetic energy:\[54 \, \text{J} = 2.0 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times h\]Solve for \( h \):\[h = \frac{54}{19.62} \approx 2.75 \, \text{m}\]

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Key Concepts

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

Rolling Motion

Rolling motion is a fascinating combination of both translational and rotational motion. When an object like a wheel rolls, every point on its surface traces a circular path. This means the wheel not only moves forward but also spins.
This dual movement involves two types of kinetic energy:

• Translational Kinetic Energy: Comes from the motion of the center of mass of the wheel moving linearly forward.
• Rotational Kinetic Energy: Comes from the wheel spinning around its axis.

Together, these energies explain why a rolling object behaves differently from one that slides without rolling.

Linear Speed

Linear speed refers to how fast a point on an object's surface moves in a straight line. For both rolling and sliding wheels, this refers to the speed at which the center of mass is moving. In our example:

• The initial linear speed of each wheel's center of mass is given as 6.0 m/s.
• It’s crucial for determining the translational kinetic energy of the wheel.

While both wheels have the same linear speed, their energies differ due to one having rotational energy as well.

Moment of Inertia

Moment of inertia is like the rotational equivalent of mass. It measures how much torque is needed for an object to rotate. A disk, such as our wheel, has a specific moment of inertia given by \( I = \frac{1}{2} m r^2 \).
This formula shows that:

• Mass (m): More mass means more resistance to rotation.
• Radius (r): Larger radius increases the moment of inertia.

For the rolling wheel, its moment of inertia is crucial in calculating rotational kinetic energy, which adds to its total kinetic energy.

Conservation of Energy

The principle of conservation of energy tells us that energy can neither be created nor destroyed, only transformed from one type to another. In our scenario:

• Initial Kinetic Energy: Each wheel starts with a certain amount of kinetic energy (36 J for sliding and 54 J for rolling).
• Potential Energy at Height: As the wheels rise on the incline, their kinetic energy transforms into potential energy.

The maximum height reached by each wheel is determined by equating their initial kinetic energy to their gravitational potential energy at the peak of their climb: \( mgh \), where \( g \approx 9.81 \text{ m/s}^2 \). This energy transformation illustrates how different motions result in different heights.