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The problem describes a pig of mass \( m = 1.0 \, \text{kg} \) swinging in a conical pendulum motion at the end of a string of length \( L = 2.0 \, \text{m} \), which forms an angle \( \theta = 20^\circ \) with the vertical. We need to determine the speed of the pig's circular motion.

### Step-by-Step Solution:

1. **Vertical and Horizontal Force Components**:
   The forces involved are:
   - Tension in the string, \( T \),
   - Gravitational force, \( mg \), where \( g = 9.8 \, \text{m/s}^2 \).

   Since the string forms an angle \( \theta \) with the vertical, the tension has two components:
   - Vertical: \( T \cos \theta = mg \),
   - Horizontal: \( T \sin \theta \), which provides the centripetal force.

2. **Centripetal Force**:
   The horizontal component of the tension provides the centripetal force that keeps the pig moving in a circular path. The centripetal force is given by:
   \[
   F_{\text{centripetal}} = m \frac{v^2}{r},
   \]
   where \( r \) is the radius of the circular motion, and \( v \) is the speed of the pig.

   From the geometry of the setup:
   \[
   r = L \sin \theta = 2.0 \, \text{m} \times \sin(20^\circ) \approx 0.684 \, \text{m}.
   \]

3. **Relating Tension and Centripetal Force**:
   The horizontal component of the tension must equal the centripetal force:
   \[
   T \sin \theta = m \frac{v^2}{r}.
   \]

4. **Solving for Tension**:
   Using the vertical component of the tension:
   \[
   T \cos \theta = mg \quad \Rightarrow \quad T = \frac{mg}{\cos \theta}.
   \]
   Substituting the known values:
   \[
   T = \frac{1.0 \, \text{kg} \times 9.8 \, \text{m/s}^2}{\cos(20^\circ)} \approx 10.43 \, \text{N}.
   \]

5. **Solving for Speed**:
   Now, substitute \( T \) into the horizontal force equation:
   \[
   10.43 \sin(20^\circ) = 1.0 \frac{v^2}{0.684}.
   \]
   Solving for \( v \):
   \[
   v^2 = \frac{10.43 \sin(20^\circ) \times 0.684}{1.0} \approx 7.11,
   \]
   \[
   v \approx \sqrt{7.11} \approx 2.67 \, \text{m/s}.
   \]

Therefore, the speed of the pig's circular motion is **2.67 m/s**. The correct answer is **(C)**.

Would you like any further explanation, or do you have any questions? Here are a few related questions for practice:

1. How would the speed change if the length of the string were doubled?
2. What is the effect of increasing the angle \( \theta \) on the speed of the pig?
3. How does the mass of the object affect the speed in this conical pendulum motion?
4. What happens to the speed if gravity were weaker, say on the Moon?
5. What would be the tension in the string if the speed of the pig doubled?

**Tip**: For problems involving circular motion, always break the forces into vertical and horizontal components to apply Newton’s laws correctly.

A pig of mass 1.0 kg is swung at the end of a 2.0 m long string in a conical pendulum. The string makes an angle of 20° with the vertical. Find the speed of the pig's circular motion.

Learn how to calculate the speed of a mass in a conical pendulum system. This problem involves a 1.0 kg mass on a 2.0 m string making a 20° angle with the vertical. Explore the tension forces, centripetal force, and trigonometric relationships. Suitable for high school physics students.

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