https://file.aiquickdraw.com/mathbot/file/1728763913597ad9qc7wu.jpg

The problem in the image describes a flowerpot suspended by three cables: AD, AC, and AB. We are given the force in cable AD and asked to find the force in cable AC by balancing the forces in the x-direction. Let's break this down:

### Given:
- Force in cable \( AD = 472 \, \text{N} \)
- Geometry: 
  - Point D is 1.5 m above point A in the z-direction.
  - Point C is 2 m along the x-axis from point A.
  - Point B is 1.5 m along the y-axis from point A.

### Step-by-Step Solution:

#### Step 1: Force in cable AD
The coordinates of point \( D \) are \( (0, 0, 1.5) \), while point \( A \) is at \( (0, 0, 0) \). Therefore, the vector \( \mathbf{r}_{AD} \) is:
\[
\mathbf{r}_{AD} = \mathbf{D} - \mathbf{A} = (0, 0, 1.5)
\]
The length of this vector is:
\[
|\mathbf{r}_{AD}| = \sqrt{0^2 + 0^2 + 1.5^2} = 1.5 \, \text{m}
\]
The unit vector along cable AD is:
\[
\mathbf{u}_{AD} = \frac{\mathbf{r}_{AD}}{|\mathbf{r}_{AD}|} = (0, 0, 1)
\]
So, the force vector in cable AD is:
\[
\mathbf{F}_{AD} = 472 \, \text{N} \times (0, 0, 1) = (0, 0, 472) \, \text{N}
\]

#### Step 2: Force in cable AC
The coordinates of point \( C \) are \( (2, 0, 0) \), so the vector \( \mathbf{r}_{AC} \) is:
\[
\mathbf{r}_{AC} = \mathbf{C} - \mathbf{A} = (2, 0, 0)
\]
The length of this vector is:
\[
|\mathbf{r}_{AC}| = \sqrt{2^2 + 0^2 + 0^2} = 2 \, \text{m}
\]
The unit vector along cable AC is:
\[
\mathbf{u}_{AC} = \frac{\mathbf{r}_{AC}}{|\mathbf{r}_{AC}|} = (1, 0, 0)
\]

Let the tension in cable AC be \( T_{AC} \). The force vector in cable AC is:
\[
\mathbf{F}_{AC} = T_{AC} \times (1, 0, 0) = (T_{AC}, 0, 0)
\]

#### Step 3: Equilibrium in the x-direction
Since the system is in equilibrium, the sum of forces in the x-direction must be zero. Therefore, the force in the x-direction must satisfy:
\[
\Sigma F_x = 0
\]
The only force contributing in the x-direction is from cable AC. Hence:
\[
T_{AC} = 0
\]

Thus, the tension in cable AC is zero. Therefore, there is no force in cable AC required for equilibrium in this configuration.

Would you like any further clarification on this solution? Here are a few related questions to deepen your understanding:

1. How would the equilibrium change if the force in cable AD was smaller?
2. Can you calculate the force in cable AB if the pot were heavier?
3. What happens to the equilibrium if cable AC were angled in the z-direction?
4. How would changing the position of point D affect the overall forces?
5. Can you explain how to determine the forces in three-dimensional force systems?

**Tip:** In force equilibrium problems, always break forces down into their components and apply equilibrium conditions in all directions (x, y, and z).

Three cables are used to support a flowerpot as shown. If the force developed in cable AD for equilibrium is 472 N, find the magnitude of the force in cable AC.

Solve the problem of determining the force in cable AC when a flowerpot is supported by three cables. The force developed in cable AD for equilibrium is given as 472 N. Learn how to apply force equilibrium in the x-direction and vector resolution.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation