Question

At the Cable Car Museum you can see the four cable lines that are used to pull cable cars up and down the hills of San Francisco. Each cable travels at a speed of 9.55 miles per hour, driven by a rotating wheel whose diameter is 8.5 feet. How fast is the wheel rotating? Express your answer in revolutions per minute.

Short answer

Revolutions per minute = approximately 9.52 rpm

Step-by-step solution

- Convert the Cable Speed to Feet per Minute

First, convert the cable speed from miles per hour to feet per minute. We know there are 5280 feet in a mile and 60 minutes in an hour. o Speed in feet per minute = 9.55 miles/hour * 5280 feet/mile * (1/60) hour/minute Calculate the value: Speed = (9.55 * 5280) / 60

- Use the Diameter to Find the Circumference

Next, convert the wheel's diameter to a circumference. The diameter of the wheel is 8.5 feet, and the formula for the circumference is given by: o Circumference (C) = π * diameter Therefore, C = π * 8.5 feet

- Determine the Rotational Speed in Feet per Minute

Calculate how many feet the circumference travels in one minute. If the wheel rotates once, it travels its circumference distance. Therefore, o Rotational speed = Cable speed in feet per minute / Circumference Calculate: Rotations per minute = Speed in feet per minute / (π * 8.5)

- Put It All Together

Finally, use the values calculated in the previous steps to find the speed in revolutions per minute. o Revolutions per minute = (9.55 * 5280) / (60 * π * 8.5)

Key concepts

unit conversion

Unit conversion is an essential skill in many fields, including mechanics and physics. It involves changing measurements from one unit to another to make calculations easier. In the exercise, you need to convert miles per hour to feet per minute. Here are the steps:

• One mile equals 5280 feet.
• There are 60 minutes in an hour.

Combining these facts, we can change the speed of 9.55 miles per hour:
\[\text{Speed in feet per minute} = 9.55 \times 5280 \times \frac{1}{60}\]
By carrying out this multiplication and division, you convert the speed to a more workable unit for the exercise.

circumference formula

The circumference of a circle is the distance around it. It's essential for understanding how far a wheel travels in one rotation. To find the circumference, you use the formula:
\[\text{Circumference} = \pi \times \text{Diameter}\]
For our exercise, the diameter of the wheel is 8.5 feet. So, the circumference is:
\[\text{Circumference} = \pi \times 8.5 \approx 26.7 \text{ feet}\]
This means each full rotation of the wheel covers a distance of about 26.7 feet. Knowing the circumference helps us to determine the rotational speed of the wheel.

revolutions per minute calculation

To find the rotational speed in revolutions per minute (RPM), you need the speed in feet per minute and the wheel's circumference. The idea is to see how many times the wheel's circumference fits into the distance the cable travels in one minute.
First, you already have the speed converted to feet per minute:
Then, use the formula:
\[\text{RPM} = \frac{\text{Speed in feet per minute}}{\text{Circumference}}\]
Substitute the calculated values:
\[\text{RPM} = \frac{9.55 \times 5280}{60 \times \pi \times 8.5}\]
This formula provides the number of wheel rotations in one minute, solving the problem directly.

cable car mechanics

Cable car systems use rotating wheels and cables to move cars up and down hills. Understanding the mechanics involves knowing the speed of the cables and how the wheels drive them. Here’s a breakdown:

• Cables move at a fixed speed.
• Rotating wheels have a given diameter.
• The rotational speed determines how fast the wheel turns to move the cable.

In our exercise, the wheel connected to the cable rotates to maintain a speed of 9.55 miles per hour. By calculating the revolutions per minute, you get an insight into how often the wheel turns to keep the cable moving at the desired speed. This showcases the interplay between wheel rotation and cable movement in cable car mechanics.