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To determine the upward velocity \(v\) of the scaffold, we can use the following approach based on the information provided:

### Given:
- Diameter of the drum, \(D = 200 \, \text{mm}\).
- Rotational speed, \(n = 40 \, \text{rev/min}\).

### Formula to find upward velocity:
The upward velocity of the scaffold is related to the linear velocity at the surface of the drum, which can be calculated by the formula:

\[
v = \text{Circumference of the drum} \times \text{Revolutions per second}
\]

The circumference of the drum is given by:

\[
\text{Circumference} = \pi \times D
\]

Next, convert the revolutions per minute to revolutions per second (since velocity will be in mm/s):

\[
n_{\text{rev/sec}} = \frac{40 \, \text{rev/min}}{60} = \frac{2}{3} \, \text{rev/sec}
\]

Now, calculate the upward velocity:

\[
v = \pi \times D \times n_{\text{rev/sec}}
\]

Substituting the values:

\[
v = \pi \times 200 \, \text{mm} \times \frac{2}{3} \, \text{rev/sec}
\]

\[
v \approx 3.1416 \times 200 \times \frac{2}{3} = 418.7 \, \text{mm/s}
\]

Thus, the upward velocity \(v\) of the scaffold is **418.7 mm/s**.

Would you like more details on any part of this solution?

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Here are 5 related questions:
1. How would the velocity change if the diameter of the drum increased?
2. What would happen to the velocity if the rotation speed was halved?
3. Can this method be used for any diameter and speed, or are there exceptions?
4. How would the velocity be affected if we used a non-cylindrical drum?
5. What are some potential factors that could cause errors in this calculation?

**Tip:** Always make sure to use consistent units when performing calculations. Converting units at the beginning can prevent errors later on.

Determine the upward velocity of a scaffold raised by power winches with rotating drums. Given a drum diameter of 200 mm and a rotation speed of 40 rev/min, this solution uses kinematic principles and rotational motion equations to find the upward velocity of the scaffold as 418.7 mm/s. Ideal for high school physics students.

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