To solve for the speed of the ball at an angle $\theta = 30^\circ$, we can use the following principles:

1. Forces in circular motion: At any point in the circular path, the forces acting on the ball are:

   • The gravitational force ($mg$) acting vertically downward.
   • The tension in the string ($T$).
   • The centripetal force required to keep the ball moving in a circular path, provided by the net radial force (which comes from the combination of tension and the radial component of gravity).

2. Energy conservation: The total mechanical energy (kinetic + potential) is conserved as the ball swings.

Given:

• Mass of the ball, $m = 1.0 \, \text{kg}$
• Length of the string, $r = 90 \, \text{cm} = 0.90 \, \text{m}$
• Tension in the string at $\theta = 30^\circ$, $T = 20 \, \text{N}$
• Gravitational acceleration, $g = 9.8 \, \text{m/s}^2$

Step 1: Analyze forces at $\theta = 30^\circ$

At an angle $\theta = 30^\circ$, the forces acting radially are:

• Radial component of gravity: $mg \cos \theta$
• Tension in the string: $T$

The net radial force is the centripetal force, which is:

$T - mg \cos \theta = \frac{mv^2}{r}$

Where:

• $v$ is the speed of the ball at $\theta = 30^\circ$.
• $r = 0.90 \, \text{m}$ is the radius of the circle.

Step 2: Plug in values

Substitute the known values into the equation:

$20 \, \text{N} - (1.0 \, \text{kg})(9.8 \, \text{m/s}^2)\cos 30^\circ = \frac{(1.0 \, \text{kg}) v^2}{0.90 \, \text{m}}$

First, calculate $mg \cos 30^\circ$:

$mg \cos 30^\circ = 1.0 \times 9.8 \times \frac{\sqrt{3}}{2} \approx 8.49 \, \text{N}$

Now substitute:

$20 - 8.49 = \frac{v^2}{0.90}$

$11.51 = \frac{v^2}{0.90}$

Multiply both sides by 0.90:

$v^2 = 10.36$

Taking the square root:

$v \approx 3.22 \, \text{m/s}$

Conclusion:

The speed of the ball when $\theta = 30^\circ$ is approximately $3.22 \, \text{m/s}$.

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Would you like further details or have any other questions?

Here are 5 related questions to deepen your understanding:

1. How does the speed of the ball change at different points in the circle?
2. How can we calculate the speed of the ball at the highest point in the circle?
3. What would happen to the tension in the string if the speed of the ball increased?
4. How does the angle $\theta$ affect the radial component of the gravitational force?
5. How would the answer change if the length of the string was increased?

Tip: Understanding the relationship between tension, speed, and gravitational forces is crucial when analyzing circular motion.