Certainly! Let's solve this step-by-step.
### Step 1: Understand the Problem
We are given:
- The linear velocity of a point on the rim of the wheel is [tex]\( 110 \, \text{m/s} \)[/tex].
- The diameter of the wheel is [tex]\( 1.25 \)[/tex] meters.
We are asked to find the revolutions per minute (RPM) of the wheel.
### Step 2: Find the Circumference of the Wheel
The circumference [tex]\( C \)[/tex] of a circle (or wheel) is given by the formula:
[tex]\[ C = \pi \times \text{diameter} \][/tex]
Given the diameter [tex]\( \text{d} = 1.25 \)[/tex] meters, we can calculate:
[tex]\[ C = \pi \times 1.25 \approx 3.927 \, \text{meters} \][/tex]
### Step 3: Calculate the Revolutions per Second (RPS)
To find the number of revolutions per second, we use the relationship:
[tex]\[ \text{Revolutions per second} = \frac{\text{Linear velocity}}{\text{Circumference}} \][/tex]
Given the linear velocity [tex]\( v = 110 \)[/tex] m/s:
[tex]\[ \text{Revolutions per second} = \frac{110 \, \text{m/s}}{3.927 \, \text{meters}} \approx 28.011 \, \text{revolutions per second} \][/tex]
### Step 4: Convert Revolutions per Second to Revolutions per Minute (RPM)
Since there are 60 seconds in a minute, we convert revolutions per second to revolutions per minute by multiplying by 60:
[tex]\[ \text{Revolutions per minute (RPM)} = \text{Revolutions per second} \times 60 \][/tex]
So:
[tex]\[ \text{Revolutions per minute (RPM)} = 28.011 \, \text{revolutions per second} \times 60 \approx 1680.676 \, \text{RPM} \][/tex]
### Summary
- The circumference of the wheel is approximately [tex]\( 3.927 \)[/tex] meters.
- The revolutions per second is approximately [tex]\( 28.011 \)[/tex].
- The revolutions per minute is approximately [tex]\( 1680.676 \)[/tex].
So the wheel completes approximately [tex]\( 1680.676 \)[/tex] revolutions per minute.
