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 calculate the linear speed of the winch cable being wound onto the winch. The steps are as follows:

### 1. **Calculate the Circumference of the Winch:**

The circumference of a circle is given by the formula:

\[
\text{Circumference} = 2\pi \times \text{radius}
\]

Given that the radius \( r = 4 \) feet, the circumference \( C \) is:

\[
C = 2\pi \times 4 = 8\pi \text{ feet}
\]

### 2. **Determine the Number of Revolutions per Second:**

The winch makes 4 revolutions every 15 seconds. To find the revolutions per second:

\[
\text{Revolutions per second} = \frac{4 \text{ revolutions}}{15 \text{ seconds}} = \frac{4}{15} \text{ revolutions per second}
\]

### 3. **Calculate the Linear Speed:**

The linear speed \( v \) at which the load is rising can be found by multiplying the circumference of the winch by the number of revolutions per second:

\[
v = \text{Circumference} \times \text{Revolutions per second} = 8\pi \times \frac{4}{15} = \frac{32\pi}{15} \text{ feet per second}
\]

### 4. **Simplify the Speed:**

Finally, let's compute the value:

\[
v \approx \frac{32 \times 3.1416}{15} \approx \frac{100.5312}{15} \approx 6.702 \text{ feet per second}
\]

So, the load is rising at approximately **6.702 feet per second**.

Would you like more details or have any questions?

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**Related Questions:**
1. How would the speed change if the winch radius was increased to 5 feet?
2. What would be the speed if the winch made 6 revolutions in 15 seconds?
3. How would the speed be affected if the winch had a radius of 3 feet instead of 4 feet?
4. What would be the speed in meters per second, given the current parameters?
5. How does the speed change if the winch takes 10 seconds for 4 revolutions instead of 15 seconds?

**Tip:** Understanding the relationship between angular speed (revolutions per second) and linear speed (feet per second) is key in solving these types of problems.

Learn how to calculate the speed at which a load rises using a winch with a radius of 4 feet making 4 revolutions every 15 seconds. Understand the relationship between angular and linear speed with a step-by-step solution.

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