https://file.aiquickdraw.com/mathbot/file/1726624257256uc1rf0wv.jpg

To solve this problem, let's analyze the setup based on the diagram and the given conditions:

### Given:
- Mass \( M = 2.0 \, \text{kg} \)
- Angle \( \theta = 30^\circ \) (for the incline of the second string)
- We are asked to find the tension in **String 1**.

### Forces on the mass \( M \):
The weight of the mass \( M \) exerts a downward force due to gravity, which is:
\[
W = M \cdot g = 2.0 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 19.6 \, \text{N}
\]

### Tensions in the strings:
- Let \( T_1 \) be the tension in String 1.
- Let \( T_2 \) be the tension in String 2 (inclined at \( 30^\circ \)).

Since the system is in equilibrium, the forces along both the horizontal and vertical directions must balance.

#### Vertical forces:
In the vertical direction, the upward force from String 1 \( T_1 \) must balance the downward gravitational force from the mass \( M \), and the vertical component of the tension in String 2 (which is \( T_2 \sin 30^\circ \)) must also be taken into account:
\[
T_1 + T_2 \sin 30^\circ = W
\]
\[
T_1 + T_2 \times 0.5 = 19.6 \, \text{N} \quad \text{(Equation 1)}
\]

#### Horizontal forces:
In the horizontal direction, the tension in String 2 has a horizontal component, which must be balanced by the horizontal force from String 1:
\[
T_2 \cos 30^\circ = T_1
\]
\[
T_2 \times \frac{\sqrt{3}}{2} = T_1 \quad \text{(Equation 2)}
\]

### Solving the system of equations:
We now have two equations:
1. \( T_1 + 0.5 T_2 = 19.6 \)
2. \( T_1 = \frac{\sqrt{3}}{2} T_2 \)

Let's solve these equations to find \( T_1 \).

I'll calculate this now.The solution to the system of equations gives:
- The tension in String 1, \( T_1 \), is approximately **12.43 N**.
- The tension in String 2, \( T_2 \), is approximately **14.35 N**.

Thus, the tension in String 1 is closest to the option **11 N**.

Would you like more details on any part of the solution, or help with another question?

Here are five related questions you could explore:
1. How would the tension change if the mass \( M \) increased?
2. What would happen to the tensions if the angle of the second string was different?
3. How does the equilibrium condition affect the choice of equations?
4. What role does the angle \( 30^\circ \) play in splitting the forces?
5. How would the solution change if friction were involved?

**Tip:** When solving systems involving forces, always check for equilibrium in both horizontal and vertical directions!

If M = 2.0 kg, what is the tension in string 1?

Solve for the tension in string 1 when a 2.0 kg mass is held in equilibrium by two strings, with one string making a 30° angle. Learn how to apply Newton's laws and equilibrium conditions to solve this problem. Suitable for students in Grades 10-12.

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