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The problem involves a box in equilibrium on a smooth horizontal floor, with four children pulling it using ropes, as shown in the diagram. The given forces are:

- \( 102 \, \text{N} \) (upward),
- \( 86 \, \text{N} \) (to the right),
- \( 150 \, \text{N} \) (downward),
- and a force marked \( T \) (direction and magnitude unknown).

### Solution:

Since the box is in equilibrium, the sum of forces in both the horizontal and vertical directions must be zero.

#### 1. Vertical Force Components:
For vertical equilibrium:
\[
T_y + 102 - 150 = 0
\]
where \( T_y \) is the vertical component of \( T \).

Thus:
\[
T_y = 150 - 102 = 48 \, \text{N}
\]

#### 2. Horizontal Force Components:
For horizontal equilibrium:
\[
T_x - 86 = 0
\]
where \( T_x \) is the horizontal component of \( T \).

Thus:
\[
T_x = 86 \, \text{N}
\]

#### 3. Magnitude of \( T \):
The magnitude of \( T \) can be found using the Pythagorean theorem:
\[
T = \sqrt{T_x^2 + T_y^2} = \sqrt{86^2 + 48^2} = \sqrt{7396 + 2304} = \sqrt{9700} \approx 98.49 \, \text{N}
\]

#### 4. Angle of \( T \):
The angle \( \theta \) that \( T \) makes with the horizontal (positive x-axis) can be found using the tangent function:
\[
\tan \theta = \frac{T_y}{T_x} = \frac{48}{86}
\]
\[
\theta = \tan^{-1} \left( \frac{48}{86} \right) \approx 29.09^\circ
\]

Thus, the force \( T \) has a magnitude of approximately \( 98.49 \, \text{N} \) and makes an angle of approximately \( 29.09^\circ \) with the horizontal (towards the top-left).

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Do you want more details or have any questions about this? Here are some related questions you might want to explore:

1. How does the angle affect the horizontal and vertical components of a force?
2. What would happen if the box were not in equilibrium?
3. How would the solution change if the tension in one of the ropes was unknown?
4. Could we solve this problem using vector addition methods?
5. How does the magnitude of \( T \) compare to the other forces?

**Tip:** When analyzing forces in equilibrium, always check both horizontal and vertical components separately.

A box rests in equilibrium on a smooth horizontal floor. Four children pull the box using inextensible ropes, all in the horizontal plane. The tensions in the ropes are shown in the diagram. Find the magnitude of the force marked T and the angle it makes with the 102 N force.

Explore the solution to a physics problem where a box in equilibrium is pulled by four children using ropes. Learn how to calculate the magnitude and direction of the unknown force T using force components, trigonometry, and the Pythagorean theorem.

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