In an electricity experiment, a 1.2 g plastic ball is suspended on a 51-cm-long string and given an electric charge. A charged rod brought near the ball exerts a horizontal electrical force F⃗elec on it, causing the ball to swing out to a 16 ∘ angle and remain there. What is the magnitude of  F⃗elec ?

We can solve this problem by analyzing the forces acting on the plastic ball when it is in equilibrium at the 16° angle. 

### Step 1: Analyze forces
When the ball is in equilibrium, the following forces are acting on it:
1. **Tension** in the string (\( T \)) acting along the string.
2. **Gravitational force** (\( F_{\text{gravity}} = mg \)) acting downward.
3. **Electric force** (\( F_{\text{elec}} \)) acting horizontally.

The angle between the string and the vertical is given as 16°. At equilibrium, the forces balance out in both the vertical and horizontal directions.

### Step 2: Break tension into components
The tension \( T \) has two components:
- Vertical component: \( T \cos(16^\circ) \), which balances the gravitational force.
- Horizontal component: \( T \sin(16^\circ) \), which balances the electric force \( F_{\text{elec}} \).

### Step 3: Set up force balance equations

#### Vertical direction:
\[
T \cos(16^\circ) = mg
\]
where:
- \( m = 1.2 \, \text{g} = 0.0012 \, \text{kg} \)
- \( g = 9.8 \, \text{m/s}^2 \)

#### Horizontal direction:
\[
T \sin(16^\circ) = F_{\text{elec}}
\]

### Step 4: Solve for the tension \( T \)

From the vertical force balance equation:
\[
T = \frac{mg}{\cos(16^\circ)}
\]
Substitute the values for \( m \) and \( g \):
\[
T = \frac{(0.0012 \, \text{kg}) (9.8 \, \text{m/s}^2)}{\cos(16^\circ)} = \frac{0.01176}{0.9613} \approx 0.01223 \, \text{N}
\]

### Step 5: Solve for \( F_{\text{elec}} \)

Now, substitute \( T \) into the horizontal force balance equation:
\[
F_{\text{elec}} = T \sin(16^\circ)
\]
\[
F_{\text{elec}} = 0.01223 \times \sin(16^\circ) = 0.01223 \times 0.2756 \approx 0.00337 \, \text{N}
\]

### Final Answer:
The magnitude of the electrical force \( F_{\text{elec}} \) is approximately **0.00337 N**.

Would you like further details or have any questions?

Here are five related questions:
1. How would the result change if the angle increased to 30°?
2. What if the mass of the ball were doubled but the angle remained the same?
3. How would the length of the string affect the electric force calculation?
4. Can you calculate the tension in the string directly?
5. How does changing the gravitational constant \( g \) affect the system?

**Tip:** In problems involving equilibrium, breaking forces into components simplifies the calculation!

In an electricity experiment, a 1.2 g plastic ball is suspended on a 51-cm-long string and given an electric charge. A charged rod brought near the ball exerts a horizontal electrical force F⃗elec on it, causing the ball to swing out to a 16 ∘ angle and remain there. What is the magnitude of F⃗elec?

Learn how to calculate the magnitude of the electric force acting on a 1.2 g plastic ball suspended on a 51-cm string at a 16° angle. This problem involves concepts from equilibrium forces, trigonometry, and Newton's laws, suitable for physics students in Grades 11-12.

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