A 1.0 kg ball swings in a vertical circle on the end of an 90-cm-long string. The tension in the string is 20 N when its angle from the highest point on the circle is θ=30∘. What is the ball's speed when  θ=30∘ ?

To determine the speed of the ball when the angle \(\theta = 30^\circ\), let's analyze the forces acting on the ball and apply Newton’s second law in the direction along the string.

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### Step 1: Identify the forces
At the given point (when the string makes a \(30^\circ\) angle with the vertical), the forces acting on the ball are:  
- **Tension in the string:** \(T = 20 \, \text{N}\)  
- **Gravitational force:** \(mg = (1.0 \, \text{kg})(9.8 \, \text{m/s}^2) = 9.8 \, \text{N}\)

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### Step 2: Break forces into components
We need to analyze the forces in the radial direction, as they determine the centripetal force for circular motion. The gravitational force has two components:  
- **Along the string (radial direction):** \(mg \cos \theta\)  
- **Perpendicular to the string (tangential direction):** \(mg \sin \theta\)

At \(\theta = 30^\circ\):
\[
mg \cos 30^\circ = 9.8 \times \frac{\sqrt{3}}{2} = 8.49 \, \text{N}
\]

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### Step 3: Set up Newton’s second law in the radial direction
The net radial force provides the centripetal force required for circular motion. Using Newton’s second law along the radial direction:
\[
T - mg \cos \theta = \frac{mv^2}{r}
\]

Plugging in the values:
\[
20 - 8.49 = \frac{1.0 \, v^2}{0.90}
\]

\[
11.51 = \frac{v^2}{0.90}
\]

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### Step 4: Solve for the speed \(v\)
\[
v^2 = 11.51 \times 0.90 = 10.36
\]

\[
v = \sqrt{10.36} \approx 3.22 \, \text{m/s}
\]

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### Final Answer
The ball's speed when the angle \(\theta = 30^\circ\) is approximately:
\[
v \approx 3.22 \, \text{m/s}
\]

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Do you have any questions or need further clarification?  

Here are 5 related questions:
1. How does the tension change as the ball moves along the circular path?
2. What would the speed be if the tension at \(\theta = 0^\circ\) (the bottom of the circle) were given?
3. How would the problem change if the string length were doubled?
4. What is the minimum speed needed at the top of the circle to keep the string taut?
5. How is the tension related to the centripetal force at different points on the circle?

**Tip:** In circular motion problems, always resolve forces into components along the radial direction to apply Newton’s second law effectively.

A 1.0 kg ball swings in a vertical circle on the end of an 90-cm-long string. The tension in the string is 20 N when its angle from the highest point on the circle is θ=30∘. What is the ball's speed when θ=30∘?

Learn how to calculate the speed of a 1.0 kg ball swinging in a vertical circle at an angle θ=30° with a string tension of 20 N. This physics problem covers key concepts such as centripetal force, Newton's second law, and circular motion. Ideal for students in Grades 11-12.

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