Wheel RPM Calculation Explained

This solution details how to calculate the rotation per minute (RPM) of a car wheel based on its diameter and the car's speed.

Given Wheel and Speed Data

• Wheel Diameter ($d$): $36$ cm
• Car Speed ($v$): $60$ km/h

Calculating Wheel Rotations Per Minute

To find the RPM, we need to determine how many times the wheel completes a full circle in one minute. This involves converting units and using the circumference of the wheel.

Step 1: Standardize Units

First, let's convert the given measurements into consistent units, typically meters (m) for distance and seconds (s) for time, to match standard physics calculations.

• Diameter in Meters: The diameter is $36$ cm. Since $1$ m = $100$ cm, the diameter in meters is: $d = 36 \text{ cm} = \frac{36}{100} \text{ m} = 0.36 \text{ m}$
• Speed in Meters per Second (m/s): The speed is $60$ km/h. We convert kilometers to meters and hours to seconds ($1$ km = $1000$ m, $1$ hour = $3600$ s): $v = 60 \frac{\text{km}}{\text{h}} = 60 \times \frac{1000 \text{ m}}{3600 \text{ s}} = 60 \times \frac{5}{18} \frac{\text{m}}{\text{s}} = \frac{300}{18} \frac{\text{m}}{\text{s}} = \frac{50}{3} \frac{\text{m}}{\text{s}}$

Step 2: Determine Wheel Circumference

The circumference ($C$) of the wheel is the distance it travels in one full rotation. The formula for circumference is $C = \pi d$.

• Using the diameter in meters: $C = \pi \times 0.36 \text{ m}$

Step 3: Find Distance Traveled in One Minute

We know the car's speed in meters per second. To calculate rotations per minute, we first find the total distance the car travels in one minute (60 seconds).

• Distance per minute ($D_{\text{minute}}$) = Speed ($v$) $\times$ Time ($60$ s) $D_{\text{minute}} = \frac{50}{3} \frac{\text{m}}{\text{s}} \times 60 \text{ s} = 50 \times 20 \text{ m} = 1000 \text{ m}$

Step 4: Compute Rotations Per Minute (RPM)

The RPM is the total distance traveled in one minute divided by the distance covered in one rotation (the circumference).

• $RPM = \frac{\text{Distance per minute}}{\text{Circumference}}$ $RPM = \frac{1000 \text{ m}}{\pi \times 0.36 \text{ m}}$ $RPM = \frac{1000}{0.36\pi}$
• Now, we calculate the numerical value using $\pi \approx 3.14159$: $RPM \approx \frac{1000}{0.36 \times 3.14159} \approx \frac{1000}{1.1309724} \approx 884.19$

Final Result Summary

The calculated value for the rotation per minute (RPM) is approximately $884.19$. Based on the options provided, the closest value is $884$.