A block hangs from a string wrapped on a disc of radius 20 cm free to rotate about its axis which is fixed in a horizontal position. If the angular speed of the disc is 10 rad/s at some instant, with what speed is the block going down at that instant?

Explanation

To solve the problem, we need to relate the angular speed of the disc to the linear speed of the block that is hanging from it. Here’s a step-by-step solution:

### Step 1: Understand the Relationship Between Angular and Linear Speed
The linear speed \( v \) of the block is related to the angular speed \( \omega \) of the disc by the formula:
\[ v = r \cdot \omega \]
where \( r \) is the radius of the disc.

### Step 2: Identify Given Values
From the problem, we have:
- Radius of the disc \( r = 20 \) cm = 0.2 m (since we need to convert cm to meters for standard SI units)
- Angular speed \( \omega = 10 \) rad/s

### Step 3: Substitute the Values into the Formula
Now we can substitute the values of \( r \) and \( \omega \) into the formula:
\[ v = 0.2 \, \text{m} \cdot 10 \, \text{rad/s} \]

### Step 4: Calculate the Linear Speed
Now, perform the multiplication:
\[ v = 0.2 \cdot 10 = 2 \, \text{m/s} \]

### Final Answer
Thus, the speed at which the block is going down at that instant is:
\[ \boxed{2 \, \text{m/s}} \]

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