Problem 59 - College Physics

Question

\(\bullet\) When a toy car is rapidly scooted across the floor, it stores energy in a flywheel. The car has mass \(0.180 \mathrm{kg},\) and its fly- wheel has moment of inertia \(4.00 \times 10^{-5} \mathrm{kg} \cdot \mathrm{m}^{2} .\) The car is 15.0 \(\mathrm{cm}\) long. An advertisement claims that the car can travel at a scale speed of up to 700 \(\mathrm{km} / \mathrm{h}(440 \mathrm{mi} / \mathrm{h}) .\) The scale speed is the speed of the toy car multiplied by the ratio of the length of an actual car to the length of the toy. Assume a length of 3.0 \(\mathrm{m}\) for a real car. (a) For a scale speed of \(700 \mathrm{km} / \mathrm{h},\) what is the actual translational speed of the car? (b) If all the kinetic energy that is initially in the flywheel is converted to the translational kinetic energy of the toy, how much energy is originally stored in the flywheel? (c) What ini- tial angular velocity of the flywheel was needed to store the amount of energy calculated in part (b)?

Step-by-Step Solution

Answer

(a) 35 km/h or 9.72 m/s. (b) 8.50 J in the flywheel. (c) 652 rad/s initial angular velocity.

Step 1: Calculate the Actual Translational Speed

The scale speed is given as 700 km/h. It is defined as the actual speed of the toy car multiplied by the scale factor. The scale factor is the ratio of a real car's length (3.0 m) to the toy car's length (0.15 m). Therefore, \[ \text{Scale Speed} = \text{Actual Speed} \times \frac{\text{Real Car Length}}{\text{Toy Car Length}} \]Thus, the actual translational speed \( v \) of the toy car is given by:\[ v = \frac{\text{Scale Speed}}{\text{Scale Factor}} = \frac{700 \text{ km/h}}{\frac{3.0 \text{ m}}{0.15 \text{ m}}} = \frac{700}{20} \text{ km/h} = 35 \text{ km/h} \]

Step 2: Convert the Speed to Meters per Second

Convert the actual speed from km/h to m/s to make further calculations easier:\[ v = 35 \text{ km/h} = 35 \times \frac{1000 \text{ m}}{3600 \text{ s}} = 9.72 \text{ m/s} \]

Step 3: Determine the Translational Kinetic Energy of the Car

The translational kinetic energy \( KE_t \) of the car is calculated using:\[ KE_t = \frac{1}{2} m v^2 \]where \( m = 0.180 \text{ kg} \) and \( v = 9.72 \text{ m/s} \):\[ KE_t = \frac{1}{2} \times 0.180 \times (9.72)^2 = 8.50 \text{ J} \]

Step 4: Calculate the Initial Rotational Kinetic Energy of the Flywheel

Assume all the kinetic energy initially stored in the flywheel is converted into the translational kinetic energy of the car (8.50 J from Step 3). So, the initial energy stored in the flywheel is 8.50 J.

Step 5: Find the Initial Angular Velocity of the Flywheel

The initial rotational kinetic energy of the flywheel is given by:\[ KE_r = \frac{1}{2} I \omega^2 \]where \( I = 4.00 \times 10^{-5} \text{ kg} \cdot \text{m}^2 \) is the moment of inertia of the flywheel, and \( \omega \) is the angular velocity. Equating this to the energy stored (8.50 J):\[ 8.50 = \frac{1}{2} \times 4.00 \times 10^{-5} \times \omega^2 \]Solve for \( \omega \):\[ \omega^2 = \frac{8.50}{2.00 \times 10^{-5}} = 425000 \]\[ \omega = \sqrt{425000} = 652 \text{ rad/s} \]

Key Concepts

Kinetic Energy

Kinetic energy is all about the energy an object has due to its motion. In our case, the toy car has kinetic energy because it's moving. There are two types of kinetic energy to consider here:

• Translational Kinetic Energy: This is the energy due to the car moving in a straight line. It depends on the mass and speed of the car. We calculate it using the formula: \( KE_t = \frac{1}{2} m v^2 \), where \( m \) is the mass and \( v \) is the velocity of the car.
• Rotational Kinetic Energy: This is the energy stored in the spinning flywheel of the car. This type of energy is dependent on the moment of inertia and the angular velocity of the flywheel, and can be calculated using: \( KE_r = \frac{1}{2} I \omega^2 \).

The problem helps us see how energy can change form. Here, the rotational energy in the flywheel eventually becomes the car's translational kinetic energy.

Moment of Inertia

Moment of inertia is like mass for rotational motion. It tells us how hard it is to change the rotational speed of an object. For the toy car's flywheel, this moment of inertia is given as \( 4.00 \times 10^{-5} \ \text{kg} \cdot \text{m}^2 \).
When a flywheel has a larger moment of inertia, it takes more energy to speed it up or slow it down. Think of it as a measure of how much the flywheel 'resists' changes in its motion.
This property is crucial because it defines how much rotational energy the flywheel stores, which can later be transferred to the car, affecting its speed along the floor.
In our case, to find the initial angular velocity, we used the equation \( KE_r = \frac{1}{2} I \omega^2 \) and related it to the energy calculation to back out the flywheel's rotational speed.

Angular Velocity

Angular velocity represents how fast the flywheel spins, similar to how linear velocity tells us how fast the car is moving in a straight line. It's shown in radians per second (rad/s), and essentially measures how many units of angle the flywheel rotates through in one second.

• Calculating Angular Velocity: We know that the rotational kinetic energy of the flywheel is \( KE_r = \frac{1}{2} I \omega^2 \). From this equation, we can find \( \omega \), the angular velocity, if we know how much energy is initially in the flywheel and its moment of inertia.
• Conversion of Energy: Angular velocity is important because it directly connects to how much rotational energy is availble to be transferred to the car’s motion. Here, solving for angular velocity using known values led us to \( \omega = 652 \text{ rad/s} \).

Understanding angular velocity helps us grasp how the energy stored in rotation translates to the speed we see as the car zooms on the floor.

Scale Speed

Scale speed involves comparing the speed of a toy car to a real car. It's like making things equal by changing scales. In this exercise, the toy car's speed is multiplied by a ratio to get the scale speed. The ratio is based on the size difference between the toy and a real car.
To find scale speed:

• Identify the Scale Factor: This is the ratio of the real car length to the toy car length, here \( \frac{3.0 \text{ m}}{0.15 \text{ m}} = 20 \).
• Adjust the Speed: Multiply the toy car's actual speed by this scale factor to simulate how the toy car's speed would appear if it were car-sized. The exercise shows us this concept using a scale speed of 700 km/h.

It's a fascinating way to see how toy models relate to their real-life counterparts, helping us understand speeds and distances in a different context. This magical multiplication provides a creative and educational look into speed perceptions on different size scales.