https://file.aiquickdraw.com/mathbot/file/1728346853469lhnqwny2.jpg

This is a physics problem involving circular motion and forces. We are asked to find the angle \( \theta \) between the string and the vertical direction, given a stone connected to a string, which is traveling in a horizontal circle.

### Given Data:
- Mass of the stone, \( m = 0.1 \, \text{kg} \)
- Speed of the stone, \( v = 1.88 \, \text{m/s} \)
- Radius of the circular path, \( R = 0.36 \, \text{m} \)

### Forces Acting on the Stone:
1. **Gravitational force**: The weight of the stone, \( mg \), acts vertically downward.
2. **Tension in the string**: The tension in the string has two components:
   - A vertical component \( T \cos \theta \), which balances the gravitational force.
   - A horizontal component \( T \sin \theta \), which provides the centripetal force for circular motion.

### Centripetal Force:
The centripetal force required for circular motion is provided by the horizontal component of the tension:
\[
T \sin \theta = \frac{mv^2}{R}
\]

### Vertical Equilibrium:
In the vertical direction, the vertical component of the tension must balance the weight of the stone:
\[
T \cos \theta = mg
\]

### Solving for \( \theta \):
Dividing the two equations:
\[
\frac{T \sin \theta}{T \cos \theta} = \frac{\frac{mv^2}{R}}{mg}
\]
This simplifies to:
\[
\tan \theta = \frac{v^2}{Rg}
\]
Substitute the values:
\[
\tan \theta = \frac{(1.88)^2}{(0.36)(9.8)}
\]
\[
\tan \theta = \frac{3.5344}{3.528} \approx 1.002
\]
Thus:
\[
\theta = \tan^{-1}(1.002) \approx 45^\circ
\]

### Final Answer:
The angle \( \theta \) is approximately **45°**, which corresponds to option (d).

---

Would you like a more detailed breakdown of any part of the solution? 

Here are five related questions to extend your understanding:
1. How does the tension in the string change if the speed of the stone increases?
2. What would happen to the angle \( \theta \) if the radius of the circular path were smaller?
3. How would the angle \( \theta \) change if the mass of the stone were doubled?
4. Why is the horizontal component of the tension responsible for the centripetal force?
5. What role does the gravitational force play in determining the angle \( \theta \)?

**Tip**: Always remember to separate forces into components when solving problems involving circular motion.

A 0.1 kg stone connected to a string travels in a uniform, horizontal circle at a constant speed. The radius of this circular path is 0.36 m. If the stone is traveling at 1.88 m/s, what is the angle θ between the string and the vertical direction?

Solve a physics problem involving a 0.1 kg stone traveling in a horizontal circle with a radius of 0.36 m and a speed of 1.88 m/s. Learn how to calculate the angle between the string and vertical direction using centripetal force, tension, and trigonometric concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation