Let's go through this problem step-by-step. Here’s a breakdown of each part:

Given:

• A mass $m$ attached to a string of length $L$.
• The mass is released from a horizontal position and swings down.
• We need to find values related to the tension in the string and the work done as the mass moves to make an angle of $23^\circ$ with the vertical.

Part (a): Tension in the string at $23^\circ$

1. Determine the speed of the mass at $23^\circ$:

   • Use conservation of energy.
   • Initial height $h_i = L$ (when the mass is horizontal).
   • Final height $h_f = L \cos(23^\circ)$.
   • Change in height: $\Delta h = h_i - h_f = L(1 - \cos(23^\circ))$.
   • The potential energy lost converts to kinetic energy: $m g \Delta h = \frac{1}{2} m v^2$ Solve for $v$ (speed at $23^\circ$).

2. Use centripetal force to find tension:

   • The forces acting on the mass at $23^\circ$ are gravity and tension.
   • The component of gravitational force along the radial direction: $mg \cos(23^\circ)$.
   • Apply Newton's second law in the radial direction: $T - mg \cos(23^\circ) = \frac{m v^2}{L}$ Substitute $v$ from the energy equation to find $T$.

Part (b): Total work done to bring the string to $23^\circ$

• Work done by gravity as the mass falls from $L$ to $L \cos(23^\circ)$.
• This work is equal to the change in gravitational potential energy: $W_{\text{total}} = m g \Delta h = m g L (1 - \cos(23^\circ))$

Part (c): Work done by the tension

• Tension is always perpendicular to the direction of motion, so it does no work on the mass.
• Thus, the work done by the tension is zero.

Would you like to see the detailed calculations for each step?

Here are some related questions to explore further:

1. How would the tension change if the mass were increased?
2. What if the angle were different, say $30^\circ$ instead of $23^\circ$?
3. How does the length $L$ affect the speed of the mass at $23^\circ$?
4. What would happen if there were no gravity in this scenario?
5. Can you calculate the period of this pendulum-like motion?

Tip: In circular motion problems, always remember that tension and gravity work together to provide the required centripetal force.