Question

In many problems in previous chapters, cars and other objects that roll on wheels were considered to act as if they were sliding without friction. (a) Can the same assumption be made for a wheel rolling by itself? Explain your answer. (b) If a moving car of total mass \(1300 \mathrm{kg}\) has four wheels, each with rotational inertia of \(0.705 \mathrm{kg} \cdot \mathrm{m}^{2}\) and radius of \(35 \mathrm{cm},\) what fraction of the total kinetic energy is rotational?

Short answer

Provide an explanation and the steps to calculate the fraction. Answer: When a wheel is rolling by itself, there is some rolling friction, so we cannot assume wheels to be frictionless objects. To calculate the fraction of the total kinetic energy that is rotational in a car, follow these steps: 1. Calculate the translational kinetic energy using the formula \(KE_{translational} = \frac{1}{2}mv^2\). 2. Calculate the rotational kinetic energy of one wheel using the formula \(KE_{rotational-wheel} = \frac{1}{2}(M_{wheel}v^2)\), where \(v = ωr\). 3. Calculate the total rotational kinetic energy by summing up the energy from all four wheels: \(KE_{rotational-total} = 4 × KE_{rotational-wheel}\). 4. Calculate the fraction of rotational kinetic energy using the formula \(Fraction = \frac{KE_{rotational-total}}{KE_{translational} + KE_{rotational-total}}\). To provide a final answer, the linear velocity (v) of the car is needed. With this value, you can substitute it into the formula and calculate the fraction of rotational kinetic energy in the car.

Step-by-step solution

Part (a) - Rolling Wheels and Frictionless Assumption

When a car or an object on wheels is moving, it is assumed that there is no friction between the wheels and the surface at the points where the two surfaces are in contact. However, for a wheel rolling by itself, this assumption does not hold true. The rotational motion of the wheels, which causes the wheel to move, requires some friction between the wheel and ground, known as rolling friction. It is important to note that rolling friction is generally significantly less than sliding friction, but it still exists. In conclusion, we cannot assume rolling wheels to be frictionless objects.

Part (b) - Fraction of the Total Kinetic Energy that is Rotational

In order to find the fraction of the total kinetic energy that is rotational, we need to calculate two separate aspects of kinetic energy in the car: (1) Translational kinetic energy, and (2) Rotational kinetic energy. We can then find the fraction of rotational kinetic energy by dividing rotational kinetic energy by total kinetic energy. Step 1: Calculate the Translational Kinetic Energy The car's translational kinetic energy can be calculated using the formula: \(KE_{translational} = \frac{1}{2}mv^2\) Where \(KE_{translational}\) is the translational kinetic energy, \(m\) is the total mass of the car (\(1300 \mathrm{kg}\)), and \(v\) is the linear velocity of the car (same at the linear velocity of the center of mass of the car). Step 2: Calculate the Rotational Kinetic Energy of One Wheel Rotational kinetic energy of one wheel can be calculated using the formula: \(KE_{rotational-wheel} = \frac{1}{2}Iω^2\) Where \(KE_{rotational-wheel}\) is the rotational kinetic energy of one wheel, \(I\) is the rotational inertia of the wheel (\(0.705 \mathrm{kg} \cdot \mathrm{m}^{2}\)), and \(ω\) is the angular velocity of the wheel. Since \(v = ωr\), we can replace \(ω\) in the formula with \(\frac{v}{r}\), where \(r\) is the radius of the wheel (\(35 \mathrm{cm} = 0.35 \mathrm{m}\)). \(KE_{rotational-wheel} = \frac{1}{2} (\frac{mv^2}{r^2}) r^2 = \frac{1}{2}(M_{wheel}v^2)\) Step 3: Calculate the Total Rotational Kinetic Energy There are four wheels in the car, so the total rotational kinetic energy would be the sum of kinetic energy of all four wheels: \(KE_{rotational-total} = 4 × KE_{rotational-wheel}\) Step 4: Calculate the Fraction of Rotational Kinetic Energy Now, we can find the fraction of rotational kinetic energy by dividing the total rotational kinetic energy by the total kinetic energy (translational kinetic energy plus rotational kinetic energy): \(Fraction = \frac{KE_{rotational-total}}{KE_{translational} + KE_{rotational-total}}\) Substitute the equations for \(KE_{translational}\), and \(KE_{rotational-total}\) found previously, and the fraction can be calculated. In order to provide an answer, we need to know the linear velocity \(v\) of the car. If provided, we can substitute the value of \(v\) in the formula and calculate the fraction of rotational kinetic energy.