A uniform rod of mass and length is held vertically by two strings of negligible mass, as shown below. (a) Immediately after the string is cut, what is the linear acceleration of the free end of the stick? (b) Of the middle of the stick?

Given:

• Mass of the rod, \( m \) = $m$
• Length of the rod, \( L \) = $L$
• Tension in the upper string, \( T_1 \) = $T_1$
• Tension in the lower string, \( T_2 \) = $T_2$
• Linear acceleration of the free end of the stick, \( a \) = $a$
• Linear acceleration of the middle of the stick, \( a_{\text{middle}} \) = $a_{\text{middle}}$

Calculations:

To find the linear acceleration (\( a \)) of the free end of the stick immediately after the string is cut:

\[ T_1 \cdot \frac{L}{2} - T_2 \cdot \frac{L}{2} = 0 \\ T_1 = T_2 \\ T_1 - T_2 = ma \\ 2T_1 = ma \\ a = \frac{2T_1}{m} \]

To find the linear acceleration (\( a_{\text{middle}} \)) of the middle of the stick:

Since the rod is uniform, the middle point will experience the same linear acceleration as the free end.

\[ a_{\text{middle}} = a \]