Question

3.89 Three control rods attached to a lever ABC exert on it the forces shown. (a) Replace the three forces with an equivalent force-couple system at B. (b) Determine the single force that is equivalent to the force-couple system obtained in part a, and specify its point of application on the lever.

Answer 1

To solve the problem, we need to handle two parts: finding an equivalent force-couple system at point B from the forces at points A, B, and C, and then reducing that to a single force with a specified point of application. Here are the general steps you would take:

### Part (a): Replace the Three Forces with an Equivalent Force-Couple System at Point B

Assume the forces are given as:
- \( \vec{F_1} \) at point A
- \( \vec{F_2} \) at point B (where we're pivoting)
- \( \vec{F_3} \) at point C

The first step is to find the resultant force \( \vec{R} \) of the force system acting at point B. We need to compute the sum of the forces in both the x and y directions.

1. **Summing Forces**:
\[
R_x = F_{1x} + F_{2x} + F_{3x}
\]
\[
R_y = F_{1y} + F_{2y} + F_{3y}
\]
Therefore, the resultant force \( \vec{R} \) can be defined as:
\[
\vec{R} = (R_x, R_y)
\]

2. **Finding the Resultant Moment about Point B**:
We need to calculate the moment caused by the forces about point B. We will summarize each force’s moment contribution about B:
- For the force at A:
\[
M_A = F_1 \times d_1
\]
- For the force at C:
\[
M_C = F_3 \times d_3
\]
(where \( d_1 \) and \( d_3 \) are the perpendicular distances from point B to the lines of action of forces at A and C).

3. **Summing Moments**:
The total moment about point B is then:
\[
M_B = M_A - M_C \quad (\text{assuming counter-clockwise is positive})
\]

### Equivalent Force-Couple System:
Now we can say the equivalent force-couple system at B consists of:
- A force \( \vec{R} \) at point B
- A couple moment \( M_B \) at point B

### Part (b): Determine the Single Force Equivalent and its Point of Application

1. **Resultant Force**:
From our calculations in part (a), we already found \( \vec{R} \).

2. **Force Location**:
To find the equivalent single force \( \vec{F_{eq}} \) acting at a point (let's say a distance \( d \) from B), we can apply the condition that the moment about B remains the same.

3. **Condition for Equivalence**:
The total moment about B due to this single force must equal the moment from the couple:
\[
F_{eq} \times d = M_B
\]
\[
d = \frac{M_B}{F_{eq}}
\]

### Final Step:
- Calculate \( d \). The point of application of the resultant force \( \vec{R} \) would be at \( d \) along the direction of \( \vec{R} \).

### Summary:
1. Calculate the resultant forces and moments.
2. Combine them to create a force at B and couple.
3. Determine the point of application of the resultant force by using the moment equivalency.

If you had specific values for the forces, distances, and directions, you would substitute them into the equations above for an exact calculation.

Answer 2

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Answer 3

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