Sure, let's work through this problem step by step.
### Step 1: Understand the Problem
We need to find the length of the chain of a swing that guarantees a centripetal acceleration of [tex]\( 21 \, \text{m/s}^2 \)[/tex] with a velocity of [tex]\( 13 \, \text{m/s} \)[/tex].
### Step 2: Recall the Formula for Centripetal Acceleration
The formula for centripetal acceleration is given by:
[tex]\[ a = \frac{v^2}{r} \][/tex]
where:
- [tex]\( a \)[/tex] is the centripetal acceleration,
- [tex]\( v \)[/tex] is the velocity,
- [tex]\( r \)[/tex] is the radius (the length of the chain in this case).
### Step 3: Substitute the Given Values
From the problem, we have:
- [tex]\( a = 21 \, \text{m/s}^2 \)[/tex],
- [tex]\( v = 13 \, \text{m/s} \)[/tex].
### Step 4: Rearrange the Formula to Solve for [tex]\( r \)[/tex]
Rearrange the formula to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{v^2}{a} \][/tex]
### Step 5: Calculate the Radius
Substitute the given values into the rearranged formula:
[tex]\[ r = \frac{13^2}{21} \][/tex]
[tex]\[ r = \frac{169}{21} \][/tex]
### Step 6: Perform the Division
Now, perform the division to find the length of the chain:
[tex]\[ r \approx 8.0476 \, \text{m} \][/tex]
### Step 7: Compare with the Provided Options
Comparing the calculated length with the provided options:
A. [tex]\( 9.5 \, \text{m} \)[/tex]
B. [tex]\( 8.5 \, \text{m} \)[/tex]
C. [tex]\( 9.0 \, \text{m} \)[/tex]
D. [tex]\( 8.0 \, \text{m} \)[/tex]
The length of [tex]\( 8.0 \, \text{m} \)[/tex] is the closest match to our computed value of [tex]\( 8.0476 \, \text{m} \)[/tex].
### Final Answer:
D. [tex]\( 8.0 \, \text{m} \)[/tex]
So the chain on the swing should be approximately [tex]\( 8.0 \)[/tex] meters long.
