An ordinary clock in a factory is running slow so that the minute hand passes the hour hand at the usual dial position($12$ o'clock, etc.) but only every $69$ minutes. At time and one-half for overtime, the extra pay to which a $\textdollar 4.00$ per hour worker should be entitled after working a normal $8$ hour day by that slow running clock, is

$\textbf{(A) }\textdollar 2.30\qquad\textbf{(B) }\textdollar 2.60\qquad\textbf{(C) }\textdollar 2.80\qquad\textbf{(D) }\textdollar 3.00\qquad \textbf{(E) }\textdollar 3.30$
1. **Understanding the clock's behavior:**
   - The minute hand overlaps with the hour hand every 69 minutes on the slow-running clock.
   - Normally, the minute and hour hands overlap every $\frac{12 \times 60}{11} = 65 \frac{5}{11}$ minutes.

2. **Calculating the actual time worked:**
   - For every hour the worker thinks he works, he actually works for $\frac{69}{65 \frac{5}{11}}$ hours.
   - Simplify $\frac{69}{65 \frac{5}{11}}$:
     \[
     65 \frac{5}{11} = \frac{720}{11}
     \]
     \[
     \frac{69}{65 \frac{5}{11}} = \frac{69}{\frac{720}{11}} = \frac{69 \times 11}{720} = \frac{759}{720} = \frac{253}{240}
     \]

3. **Calculating the overtime:**
   - The worker thinks he works for 8 hours, but actually works:
     \[
     8 \times \frac{253}{240} = \frac{2024}{240} = 8 \frac{104}{240} = 8 \frac{13}{30}
     \]
   - The actual overtime is:
     \[
     8 \frac{13}{30} - 8 = \frac{13}{30} \text{ hours}
     \]

4. **Calculating the overtime pay:**
   - The worker is paid $\$4$ per hour, and overtime is paid at time and a half, so the overtime rate is:
     \[
     1.5 \times 4 = \$6 \text{ per hour}
     \]
   - The overtime pay is:
     \[
     6 \times \frac{13}{30} = \frac{78}{30} = \frac{13}{5} = 2.6
     \]

The final answer is $\boxed{\text{(B)}\$ 2.60}$.