Problem 56 - College Physics

Question

A \(\mathrm{A} 392\) -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 \(0.600 \mathrm{m},\) and its moment of inertia about its rotation axis is 0.800\(M R^{2} .\) 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 3500 J. Calculate \(h .\)

Step-by-Step Solution

Answer

The height \( h \) is \( 0.255 \) m.

Step 1: Identify Known Values

Let's list the known values given in the problem:- Weight of the wheel, \(W = 392\, \text{N}\)- Rotational speed at the bottom, \(\omega = 25.0\, \text{rad/s}\)- Radius of the wheel, \(R = 0.600\, \text{m}\)- Moment of inertia, \(I = 0.800MR^2\)- Work done by friction, \(W_f = 3500\, \text{J}\).

Step 2: Calculate the Mass of the Wheel

The mass of the wheel can be calculated using the weight and gravitational acceleration: \[ M = \frac{W}{g} = \frac{392}{9.8} = 40\, \text{kg} \].

Step 3: Determine Rotational Kinetic Energy

Calculate the rotational kinetic energy at the bottom of the hill using the formula:\[ KE_{rot} = \frac{1}{2}I\omega^2 \].

Step 4: Calculate the Moment of Inertia

Find the moment of inertia using the formula provided:\[ I = 0.800MR^2 = 0.800 \times 40 \times (0.600)^2 = 11.52\, \text{kg}\cdot\text{m}^2 \].

Step 5: Compute Initial Rotational Kinetic Energy

Plug the values to find the initial rotational kinetic energy:\[ KE_{rot} = \frac{1}{2} \times 11.52 \times (25.0)^2 = 3600\, \text{J} \].

Step 6: Use Energy Conservation Principle

Utilize conservation of energy, accounting for work done by friction:\[ PE_{top} = KE_{rot} - W_f \], where \(PE_{top}\) is the potential energy at the top of the hill.

Step 7: Calculate Potential Energy at the Top

Find \(PE_{top}\) using:\[ PE_{top} = 3600 - 3500 = 100\, \text{J} \].

Step 8: Calculate Height h

Use the potential energy formula \(PE = Mgh\) to find \(h\): \[ h = \frac{PE_{top}}{Mg} = \frac{100}{40 \times 9.8} = 0.255\, \text{m} \].

Key Concepts

Rotational Kinetic Energy

Rotational kinetic energy is one form of energy in motion, typically observed in objects that spin or rotate. It is the rotational equivalent of linear kinetic energy. In our exercise, it's the energy the wheel possesses due to its rotation around its axis. Here's the formula to understand it better: \[ KE_{rot} = \frac{1}{2} I \omega^2 \]Where:

• \( KE_{rot} \) is the rotational kinetic energy,
• \( I \) is the moment of inertia, and
• \( \omega \) is the angular velocity in radians per second.

The wheel at the bottom of the hill carries this energy, contributing to its ability to ascend until friction and gravity slow it down. Understanding how these variables interact provides insights into rotational dynamics.

Moment of Inertia

The moment of inertia is akin to the "rotational mass" of an object and it tells us how difficult it is to change its rotational motion. It depends on the mass of the object and how that mass is distributed relative to the axis of rotation. For our wheel, the formula provided is:\[ I = 0.800 M R^2 \]This means:

• \( M \) is the mass of the wheel,
• \( R \) is the radius, and
• the 0.800 factor accounts for distribution specifics.

By calculating the moment of inertia, we determine how the wheel's mass and geometry affect its rotational motion. Having larger inertia means it will spin slower for the same amount of energy input, or it will require more energy to achieve the same rotational speed.

Work by Friction

Friction is the resistive force that occurs when two surfaces in contact move past each other. When it comes to the rolling wheel, friction does negative work by removing energy from the system. In our scenario, friction plays a key role as it dissipates some of the wheel's initial kinetic energy, converting it to other forms such as heat. The given work done by friction is:\[ W_f = 3500 \, \text{J} \]Why does this matter? Because it impacts how much energy is left for the wheel to climb uphill. By losing energy to friction, this loss must be considered when calculating potential energy gained by the wheel at the top of the hill.Understanding work by friction allows us to appreciate why rolling objects eventually stop — it's because energy isn't fully conserved in kinetic terms due to these resistive forces.