To solve this problem, we need to analyze the circular motion of the marble and break down the forces acting on it into vertical and horizontal components. Here’s how to approach it:

Part (a): Finding the Speed of the Marble

1. Identify Forces: The marble experiences two forces:

   • Tension $T$ in the string, which has both vertical and horizontal components due to the angle.
   • Gravity $mg$, acting vertically downward.

2. Determine the Components of Tension: Since the string makes an angle of $37.0^\circ$ with the vertical:

   • Vertical component of tension: $T \cos(37^\circ)$
   • Horizontal component of tension (which provides the centripetal force): $T \sin(37^\circ)$

3. Set Up Equilibrium in the Vertical Direction: In the vertical direction, the upward component of tension must balance the gravitational force: $T \cos(37^\circ) = mg$

4. Express the Centripetal Force in the Horizontal Direction: The horizontal component of the tension provides the centripetal force needed for circular motion: $T \sin(37^\circ) = \frac{mv^2}{r}$ where $v$ is the speed of the marble, and $r$ is the radius of the circular path.

5. Calculate the Radius $r$: The radius $r$ of the circle can be found as: $r = L \sin(37^\circ)$ where $L = 44.0 \, \text{cm} = 0.44 \, \text{m}$ is the length of the string.

6. Solving for Speed $v$: Substitute $T$ from the vertical equilibrium equation into the centripetal force equation to find $v$ in terms of $g$, $L$, and the angle.

Part (b): Expression for Tension $T$

1. Use the vertical force balance equation: $T = \frac{mg}{\cos(37^\circ)}$
2. Substitute this expression back into the centripetal force equation if needed.

Would you like a step-by-step calculation based on these equations?

Related Questions

1. How does changing the angle $\theta$ affect the speed of the marble?
2. What happens to the tension if the marble's mass increases?
3. How would the speed change if the length of the string is doubled?
4. Why is it necessary to resolve the tension into components?
5. How would the problem change if the circle was vertical rather than horizontal?

Tip: In circular motion problems, always resolve forces into components aligned with the direction of motion (radial and tangential directions).