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Chapter 9: Problem 25

A car is traveling at a speed of \(101 \mathrm{~km} / \mathrm{h}\) on the highway and has a small stone stuck between the treads of one of its tires. The tires have diameter \(61.0 \mathrm{~cm}\) and are rolling without sliding or slipping. What are (a) the maximum and (b) the minimum speeds of the stone as observed by a pedestrian standing on the side of the highway?

Short Answer

Expert verified
(a) 32.52 m/s, (b) 23.58 m/s.

Step by step solution

01

Convert Speed to m/s

First, we need to convert the car's speed from km/h to m/s. We use the conversion factor that 1 km/h is equal to \( \frac{1}{3.6} \) m/s.\[101 \text{ km/h} \times \frac{1}{3.6} \text{ m/s per km/h} = 28.056 \text{ m/s}\]
02

Calculate the Circumference of the Tire

The circumference of the tire, which is required to determine the rotational speed at the rim, is given by the formula:\[C = \pi d\]where \(d\) is the diameter of the tire. Here, \(d = 0.61 \text{ m}\):\[C = \pi \times 0.61 \text{ m} = 1.9167 \text{ m}\]
03

Determine Angular Velocity of the Tire

Using the car's speed and the circumference of the tire, we find the angular velocity \(\omega\):\[\omega = \frac{v}{C} = \frac{28.056 \text{ m/s}}{1.9167 \text{ m}} = 14.64 \text{ rad/s}\]
04

Find Maximum Speed of the Stone

The maximum speed of the stone relative to the ground is when it is moving in the same direction as the car at the top of the tire:\[v_{\text{max}} = v + \omega r = 28.056 \text{ m/s} + \left( 14.64 \text{ rad/s} \times 0.305 \text{ m} \right) = 32.52 \text{ m/s}\]
05

Find Minimum Speed of the Stone

The minimum speed relative to the ground occurs when the stone is at the bottom of the tire, where it moves opposite to the direction of the car:\[v_{\text{min}} = v - \omega r = 28.056 \text{ m/s} - \left( 14.64 \text{ rad/s} \times 0.305 \text{ m} \right) = 23.58 \text{ m/s}\]

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Key Concepts

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

Linear Speed Conversion
If you wish to perform linear speed conversion from kilometers per hour (km/h) to meters per second (m/s), you need to know a simple conversion factor. Since 1 km contains 1000 meters and 1 hour has 3600 seconds, you have to divide the speed in km/h by 3.6 to get the speed in m/s. For example:
  • Speed in km/h: 101 km/h
  • Convert by dividing by 3.6: \( 101 / 3.6 = 28.056 \, \text{m/s} \)
This conversion is incredibly useful in physical calculations where standard units are required. By understanding this relation, you can easily switch between these two common units of speed, making it simpler to work through physics problems.
Circumference Calculation
The circumference of a circular object, such as a tire, is crucial in understanding its overall motion characteristics. The circumference can be calculated using the formula:
  • \( C = \pi d \)
  • The diameter \( d \) of the tire: 0.61 m
Plug in the diameter to find:
\[ C = \pi \times 0.61 \approx 1.9167 \, \text{m} \]
This value is especially important when determining how far the tire travels with one complete rotation. By calculating the circumference, you have the foundational length needed to connect linear speeds to angular velocities.
Angular Velocity
Angular velocity refers to how fast an object rotates or spins around a point or axis. To find the angular velocity \( \omega \) of a tire, you need both the linear velocity of the car and the tire's circumference:
  • Linear speed of the car: 28.056 m/s
  • Tire circumference: 1.9167 m
The formula to find angular velocity is:
\[ \omega = \frac{v}{C} \]
Substituting the values gives:
\[ \omega = \frac{28.056}{1.9167} \approx 14.64 \, \text{rad/s} \]
Angular velocity helps us understand how fast an object spins, which is essential in determining the speed of the stone at different positions on the tire.
Relative Speed
Relative speed is the speed of an object as observed from a particular frame of reference. In our scenario, the stone stuck in the tire experiences different speeds based on its position鈥攅ither at the top or bottom:
  • **Maximum speed**: Occurs at the top of the tire and adds the tangential speed due to angular motion to the car's speed: \( v_{\text{max}} = v + \omega r \)
  • Substitute car speed (28.056 m/s) and angular speed contributions (\( \omega = 14.64 \times 0.305 \)): \[ v_{\text{max}} \approx 32.52 \, \text{m/s} \]
  • **Minimum speed**: Happens at the bottom where it moves against motion, reducing the speed: \( v_{\text{min}} = v - \omega r \)
  • Using the same values, this computes to: \[ v_{\text{min}} \approx 23.58 \, \text{m/s} \]
Relative speed thus helps identify how fast an object moves from a stationary frame, like that of a pedestrian's view of the car.

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Most popular questions from this chapter

A uniform marble rolls down a symmetric bowl, starting from rest at the top of the left side. The top of each side is a distance \(h\) above the bottom of the bowl. (a) How far up the right side of the bowl will the marble go if the interior surface of the bowl is rough so that the marble rolls without slipping? (b) How high would the marble go if the bowl's surface were frictionless? (c) For which case is the marble moving faster when it reaches the bottom of the bowl?

a car is traveling at a constant speed on the highway. Its tires have a diameter of \(61.0 \mathrm{~cm}\) and are rolling without sliding or slipping. If the angular speed of the tires is \(50.0 \mathrm{rad} / \mathrm{s},\) what is the speed of the car, in SI units?

A \(2.20 \mathrm{~kg}\) hoop \(1.20 \mathrm{~m}\) in diameter is rolling to the right without slipping on a horizontal floor at a steady \(3.00 \mathrm{rad} / \mathrm{s}\). (a) How fast is its center moving? (b) What is the total kinetic energy of the hoop?

BIO The Spinning Eel. American eels are freshwater fish with long, slender bodies that we can treat as a uniform cylinder \(1.0 \mathrm{~m}\) long and 10 \(\mathrm{cm}\) in diameter. An eel compensates for its small jaw and teeth by holding onto prey with its mouth and then rapidly spinning its body around its long axis to tear off a piece of flesh. Eels have been recorded to spin at up to \(14 \mathrm{rev} / \mathrm{s}\) when feeding in this way. Although this feeding method is energetically costly, it allows the eel to feed on larger prey than it otherwise could. A field researcher uses the slow-motion feature on her phone's camera to shoot a video of an eel spinning at its maximum rate. The camera records at 120 frames per second. Through what angle does the eel rotate from one frame to the next? A. \(1^{\circ}\) B. \(10^{\circ}\) C. \(22^{\circ}\) D. \(42^{\circ}\)

A new species of eel is found to have the same mass but one-quarter the length and twice the diameter of the American eel. How does its moment of inertia for spinning around its long axis compare to that of the American eel? The new species has A. half the moment of inertia of the American eel. B. the same moment of inertia as the American eel. C. twice the moment of inertia of the American eel. D. four times the moment of inertia of the American eel.
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