Problem 56 - College Physics

Question

Your car's speedometer works in much the same way as its odometer (see the previous problem), except that it converts the angular speed of the wheels to a linear speed of the car, assuming standard-size tires and no slipping on the pavement. (a) If your car has standard 24 -in.-diameter tires, how fast are your wheels turning when you are driving at a freeway speed of \(55 \mathrm{mph} ?\) (b) How fast are you going when your wheels are turning at 500 rpm? (c) If you put on undersize 20 -in.-diameter tires, what will the speedometer read when you are actually traveling at \(50 \mathrm{mph} ?\)

Step-by-Step Solution

Answer

(a) 400 RPM; (b) 35.7 mph; (c) 42 mph.

Step 1: Convert diameter to radius

The radius of a tire can be found by dividing the diameter by 2. For a 24-inch diameter tire, the radius is \[ r = \frac{24}{2} = 12 \text{ inches} \].

Step 2: Convert speed to inches per hour

Since we want the tire's angular speed, we'll first express the linear speed in inches per hour. There are 63,360 inches in a mile, so \[ 55 \text{ mph} = 55 \times 63,360 = 3,484,800 \text{ in/hr} \].

Step 3: Calculate angular speed for 24-inch tire

The formula for angular speed \(\omega\) is \( \omega = \frac{v}{r} \), where \(v\) is the velocity and \(r\) is the radius.Convert the radius to feet (12 inches = 1 foot), and 1 revolution is \(2\pi r\) feet.Convert total inch/hr to feet/hr and then to revolutions per hour:\[ \frac{3,484,800}{63,360} = 55 \text{ rev per hour on a radius 1 foot} \].

Step 4: Calculate speed from RPM for 24-inch tire

Given that the wheels turn at 500 rpm, convert this to mph. First, calculate inches per minute, then convert to mph: \[ \text{Miles per hour} = \left( \frac{500 \times 2 \times \pi \times 12}{63,360} \right) \times 60 = 35.7 \text{ mph} \].

Step 5: Convert speed to RPM for 20-inch tire

For a smaller tire (20-inch diameter), with actual speed at 50 mph:\( r = 10 \text{ in} \), \[ 50 \text{ mph } = 50 \times 63,360 \text{ in/hr} \], and calculate revolutions per hour: \[ \frac{50 \times 63,360}{2\pi \times 10} = 50\times \frac{63,360}{62.8} = \text{ 5064 revolutions per hour}\].Convert to RPM \[ RPM = \frac{5064}{60} \approx 84.4 \text{ RPM} \].

Step 6: Adjust formula for speedometer reading

Since the speedometer will calculate based on the tire's angular speed for 20-inch tires as on standard 24-inch tires, use the revolution per hour equivalence:\[ \text{Speedometer reads } = \frac{84.4 \times 2\pi \times 12}{60 imes 63,360}\times 60= 42 \].

Key Concepts

Angular Speed

Angular speed refers to how fast an object rotates or spins around an axis. In physics, it is often denoted by the symbol \( \omega \). Angular speed is measured in units like revolutions per minute (RPM) or radians per second. This concept is crucial in understanding how the rotational motion of an object translates into other types of motion.

To find angular speed, you will need to know both the speed at which a point on the edge of a rotating object is moving (linear speed) and the radius of the rotation (the distance from the center to the edge). The basic formula to calculate angular speed is:

• \( \omega = \frac{v}{r} \)

where \(v\) is the linear speed and \(r\) is the radius.

One thing to note is that angular speed is independent of the size of the rotating object. Whether you have a small wheel or a large one, if the speed and radius are the same, the angular speed will be the same.

Linear Speed

Linear speed is how fast an object is moving along a path. Essentially, it is the rate of change of position over time. This speed is often measured in units like miles per hour (mph) or meters per second (m/s).

When dealing with rotating objects, linear speed can be converted into angular speed. This conversion comes in handy when we want to relate rotation to travel over a distance. You'll often find the need to convert a car's speed (linear speed) into terms of the tire rotation (angular speed) so that you get an accurate reading from speedometers.

The relationship between linear speed \(v\) and angular speed \(\omega\) is encapsulated by the formula:

• \( v = \omega \times r \)

This means that to achieve a certain linear speed, the angular speed has to increase as the radius decreases, and vice versa.

Tire Diameter

Tire diameter is an important factor when you consider how rotations translate to distance travelled. The diameter is simply the width of the tire across its center. It plays a key role in determining the radius \( r \), which is half of the diameter.

For example, a 24-inch tire has a radius of 12 inches. This radius is essential when calculating both angular and linear speeds because it is part of the formulas used to find these speeds.

Changing the tire diameter, such as switching from 24-inch to 20-inch tires, will directly affect the speed readings on your speedometer. A smaller tire will make more revolutions than a larger tire over the same distance, which can lead to discrepancies in speed readings if not adjusted.

Revolutions Per Minute (RPM)

Revolutions Per Minute (RPM) measures how many complete turns an object makes in one minute. It is a direct indicator of angular speed and is widely used to express the rotational speed of mechanical components like car engines and wheels.

Knowing the RPM of your car's tires can help you determine your car's speed when you take the tire diameter into account. For instance, converting from RPM to linear speed involves multiplying the number of revolutions per minute by the tire's circumference. For a given circumference:

• The formula to find linear speed from RPM: \( \,\text{Linear Speed} = RPM \times \text{Circumference} \times \frac{1}{63,360} \, \text{ (to convert to miles per minute)} \)

Where the circumference \( = 2\pi \times r \). This information is important when setting the speedometer reading in a car. If the tires are changed, the RPM conversion needs to be recalibrated.