Step 1: Identify the given number.
The given quantity is \( 5.6200 \, \text{J} \). We need to count the number of significant figures in this decimal number.
Step 2: Rule for significant figures.
All non-zero digits are significant. Zeros between non-zero digits are significant. Also, trailing zeros in a decimal number are significant.
Step 3: Analyze the digits in 5.6200.
The digits are: 5, 6, 2, 0, 0.
Here, both zeros are trailing zeros after the decimal point, so they are significant.
Step 4: Count the significant figures.
Thus, total significant figures are:
\[
5, 6, 2, 0, 0 \Rightarrow 5 \text{ significant figures}
\]
Step 5: Final verification.
Since no leading zeros are present and all digits contribute to precision, the count remains unchanged.
Step 6: Final conclusion.
Therefore, the number of significant figures is 5.
Final Answer:
\[
\boxed{5}
\]