91Ó°ÊÓ

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 flywheel 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 initial angular velocity of the flywheel was needed to store the amount of energy calculated in part (b)?

Step by step solution

01

Calculate the Speed Ratio

The scale speed is based on a ratio between the length of an actual car and the toy car. Calculate this ratio as follows: \[ \text{Speed Ratio} = \frac{\text{Real Car Length}}{\text{Toy Car Length}} = \frac{3.0 \text{ m}}{0.15 \text{ m}} = 20 \]

02

Calculate Actual Translational Speed

The actual speed of the toy car can be found by dividing the scale speed by the speed ratio. Given the scale speed is 700 km/h, the actual speed is calculated as:\[ \text{Actual Speed} = \frac{700 \text{ km/h}}{20} = 35 \text{ km/h} \]

03

Determine Translational Kinetic Energy

Using the formula for translational kinetic energy, \( KE_{\text{trans}} = \frac{1}{2}m v^2 \), where \( m = 0.180 \text{ kg} \) and \( v = 35 \text{ km/h} = \frac{35}{3.6} \text{ m/s} \): \[ KE_{\text{trans}} = \frac{1}{2} \times 0.180 \times \left(\frac{35}{3.6}\right)^2 \approx 3.15 \text{ J} \]

04

Convert Energy to Rotational Energy

Assuming all the energy is converted from the flywheel energy, the initial energy stored in the flywheel \( KE_{\text{rot}} = KE_{\text{trans}} \): \[ KE_{\text{rot}} = 3.15 \text{ J} \]

05

Calculate Initial Angular Velocity

Use the formula for rotational kinetic energy, \( KE_{\text{rot}} = \frac{1}{2}I\omega^2 \), solving for \( \omega \) gives:\[ 3.15 = \frac{1}{2} \times 4.00 \times 10^{-5} \times \omega^2 \]Rearranging gives \( \omega^2 = \frac{2 \times 3.15}{4.00 \times 10^{-5}} \), \[ \omega = \sqrt{\frac{6.3}{4.00 \times 10^{-5}}} \approx 397.5 \text{ rad/s} \]

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Key Concepts

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

Translational kinetic energy

Understanding translational kinetic energy starts with visualizing motion in a straight line. When objects move across a surface, like a toy car scooting across the floor, they possess this type of energy.
The formula to find translational kinetic energy is \( KE_{\text{trans}} = \frac{1}{2}mv^2 \), where:

• \( m \) is the mass of the object.
• \( v \) is the velocity of the object.

Translational kinetic energy depends directly on both mass and speed. In our problem, the car's mass is \( 0.180 \text{ kg} \) and its speed is converted to meters per second to apply the formula effectively. When you plug these numbers in, you calculate how much energy the toy car stores simply by moving across the floor. This principle helps us understand how energy is transferred in motion and is essential in topics ranging from basic physics to complex engineering.

Rotational energy

Rotational energy, also known as the energy of an object due to its rotation, plays a significant role in our flywheel example.
Flywheels are designed to store energy through spinning, based on the object's moment of inertia and its angular velocity.
The formula to calculate rotational energy is \( KE_{\text{rot}} = \frac{1}{2}I\omega^2 \), where:

• \( I \) is the moment of inertia, which measures how much resistance an object has to changes in its rotational state.
• \( \omega \) is the angular velocity, or how fast the object spins around an axis.

In the problem, all the energy stored in the rotating flywheel converts into translational kinetic energy of the toy car as it moves. This conversion illustrates the conservation of energy principle, indicating that energy can change forms but not be created or destroyed. Understanding this relationship helps urban planners, engineers, and physicists alike in developing more efficient systems.

Angular velocity calculation

Finding angular velocity involves understanding how fast an object spins around its axis.
In our problem, we know the energy stored in the flywheel and the moment of inertia, which allows us to find the angular velocity. The formula to calculate angular velocity when rotational kinetic energy is known is derived from \( KE_{\text{rot}} = \frac{1}{2}I\omega^2 \).
Rewriting this equation, we solve for \( \omega \):

• \( \omega^2 = \frac{2 \times KE_{\text{rot}}}{I} \)
• \( \omega = \sqrt{\frac{2 \times 3.15 \text{ J}}{4.00 \times 10^{-5} \text{ kg} \cdot \text{m}^2}} \)

Calculation produces an angular velocity of around 397.5 rad/s. It reflects how rapidly the flywheel spins to store the needed amount of energy that eventually translates into the car's motion. This concept is fundamental in devices like turbines and engines, where rotational motion is converted to power various applications.