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Determine the moment of the applied force  F=
70  N
about point  A
. Neglect the thickness of the member.

To determine the moment of the applied force \( F = 70 \, \text{N} \) about point \( A \), we can use the principle of moments. The moment \( M_A \) about point \( A \) is given by:

\[
M_A = F \times d
\]

where \( d \) is the perpendicular distance from point \( A \) to the line of action of the force \( F \).

### Step 1: Analyze the Geometry
Given:
- The force \( F \) has a magnitude of 70 N and is applied at an angle of \( 75^\circ \) from the horizontal.
- The lengths of the segments are:
  - Horizontal segment from \( A \): 150 mm
  - Inclined segment: 500 mm at an angle of \( 30^\circ \) to the horizontal.
  - Horizontal segment where \( F \) is applied: 125 mm.

### Step 2: Determine the Position of the Force Relative to Point A
First, we'll need to find the total horizontal distance from point \( A \) to the point where the force \( F \) is applied.

1. The horizontal distance from \( A \) to the end of the first segment is 150 mm.
2. The horizontal component of the inclined segment is given by \( 500 \times \cos(30^\circ) \).
3. The horizontal distance from the end of the inclined segment to the point of force application is 125 mm.

Thus, the total horizontal distance \( d_{\text{total}} \) from \( A \) to the point where the force is applied is:

\[
d_{\text{total}} = 150 \, \text{mm} + 500 \times \cos(30^\circ) + 125 \, \text{mm}
\]

### Step 3: Calculate the Perpendicular Distance
The perpendicular distance \( d_{\perp} \) is given by:

\[
d_{\perp} = d_{\text{total}} \times \sin(75^\circ)
\]

### Step 4: Calculate the Moment
Finally, the moment about point \( A \) is:

\[
M_A = F \times d_{\perp}
\]

Let's perform these calculations.

1. Calculate \( d_{\text{total}} \):

\[
d_{\text{total}} = 150 \, \text{mm} + 500 \times \cos(30^\circ) + 125 \, \text{mm}
\]

2. Calculate \( d_{\perp} \):

\[
d_{\perp} = d_{\text{total}} \times \sin(75^\circ)
\]

3. Calculate \( M_A \) using the force \( F = 70 \, \text{N} \).

I'll calculate these values now.The moment of the applied force \( F = 70 \, \text{N} \) about point \( A \) is approximately \( 47872.14 \, \text{N-mm} \).

Would you like more details or have any questions?

Here are some related questions you might consider:

1. How would the moment change if the angle of the applied force were different?
2. What would be the effect on the moment if the position of point \( A \) changed?
3. How do moments affect the stability of structures?
4. Can we calculate the moment if the force were applied at a different point on the structure?
5. How does the direction of the force affect the calculation of the moment?

**Tip:** When dealing with moments, always ensure that you calculate the perpendicular distance correctly, as it directly affects the accuracy of your moment calculation.

Learn how to calculate the moment of an applied force of 70 N about point A using the principle of moments. This problem involves analyzing forces and distances, applying vector components, and calculating the perpendicular distance accurately. Suitable for advanced high school students and those interested in mechanics.

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