A winch of radius 4 ft is used to lift heavy loads. If the winch makes 4 revolutions every 15 s, find the speed at which the load is rising.

To find the speed at which the load is rising, we can follow these steps:

### Step 1: Calculate the Circumference of the Winch
The circumference \( C \) of the winch can be found using the formula:
\[
C = 2\pi r
\]
where \( r \) is the radius of the winch.

Given:
\[
r = 4 \text{ ft}
\]

So,
\[
C = 2\pi \times 4 = 8\pi \text{ ft}
\]

### Step 2: Determine the Distance Covered in One Revolution
Each revolution of the winch lifts the load by the distance equal to the circumference of the winch, which is \( 8\pi \) ft.

### Step 3: Calculate the Number of Revolutions per Second
The winch makes 4 revolutions every 15 seconds, so the number of revolutions per second (angular velocity \( \omega \)) is:
\[
\text{Revolutions per second} = \frac{4 \text{ revolutions}}{15 \text{ seconds}} = \frac{4}{15} \text{ revolutions per second}
\]

### Step 4: Calculate the Speed at Which the Load is Rising
The speed \( v \) at which the load is rising is given by:
\[
v = \text{Circumference} \times \text{Revolutions per second}
\]
Substituting the values:
\[
v = 8\pi \times \frac{4}{15} = \frac{32\pi}{15} \text{ ft/s}
\]

### Step 5: Simplify the Speed
\[
v \approx \frac{32 \times 3.1416}{15} \approx 6.7 \text{ ft/s}
\]

So, the load is rising at approximately **6.7 ft/s**.

Would you like more details on any part of the solution? Here are some related questions:

1. How does changing the radius of the winch affect the speed at which the load is lifted?
2. What happens to the lifting speed if the winch makes more revolutions in the same amount of time?
3. How can the speed be converted from ft/s to another unit, like m/s?
4. What is the significance of the winch’s angular velocity in this problem?
5. How does the circumference of the winch relate to the linear speed of the load?

**Tip:** Remember that the speed of an object in circular motion can be found by multiplying the circumference by the number of revolutions per unit time.

Learn how to calculate the speed at which a load rises using a winch with a radius of 4 ft, making 4 revolutions every 15 seconds. Detailed step-by-step solution provided.

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