https://file.aiquickdraw.com/mathbot/file/172823827444218x48ilu.jpg

A force of 166N in the yz-plane is directly applied to the parallelepiped at point A (labeled by the red dot). Determine the component of the moment caused by the force about the x-axis. Given parameters of a=923mm , b=755mm and θ=64° . Assume counterclockwise is a positive and round answer to at least 3 significant digits.

Mx:_______N⋅m

To determine the moment about the \( x \)-axis (\( M_x \)) caused by the force \( F \) acting at point \( A \), we'll need to break the force into its components and then calculate the moment using the moment-arm principle. Here's how we can approach this problem step by step:

### Given:
- Force \( F = 166 \, \text{N} \)
- Angle \( \theta = 64^\circ \) (the angle the force makes with the \( y \)-axis in the \( yz \)-plane)
- Distance \( a = 923 \, \text{mm} = 0.923 \, \text{m} \)
- Distance \( b = 755 \, \text{mm} = 0.755 \, \text{m} \)

### Step 1: Resolve the Force into Components
Since the force lies in the \( yz \)-plane, its components will be along the \( y \)-axis and the \( z \)-axis. Using trigonometry, we find:

- \( F_y = F \cos(\theta) \)
- \( F_z = F \sin(\theta) \)

Substitute the given values:

\[
F_y = 166 \cos(64^\circ) = 166 \times 0.43837 \approx 72.77 \, \text{N}
\]
\[
F_z = 166 \sin(64^\circ) = 166 \times 0.89879 \approx 149.20 \, \text{N}
\]

### Step 2: Moment Arm and Moment Calculation
The moment about the \( x \)-axis is caused by the force components \( F_y \) and \( F_z \) acting at point \( A \). 

- The moment arm for \( F_y \) is the distance along the \( z \)-axis, which is \( a \).
- The moment arm for \( F_z \) is the distance along the \( y \)-axis, which is \( b \).

Now, calculate the moment contributions:

- Moment due to \( F_y \): \( M_{x1} = F_y \times a \)
- Moment due to \( F_z \): \( M_{x2} = F_z \times b \)

Substitute the values:

\[
M_{x1} = 72.77 \, \text{N} \times 0.923 \, \text{m} \approx 67.15 \, \text{N} \cdot \text{m}
\]
\[
M_{x2} = 149.20 \, \text{N} \times 0.755 \, \text{m} \approx 112.64 \, \text{N} \cdot \text{m}
\]

### Step 3: Combine the Moments
Since \( F_y \) and \( F_z \) act in opposite directions relative to the \( x \)-axis, the total moment about the \( x \)-axis will be:

\[
M_x = M_{x2} - M_{x1} = 112.64 \, \text{N} \cdot \text{m} - 67.15 \, \text{N} \cdot \text{m} = 45.49 \, \text{N} \cdot \text{m}
\]

Thus, the component of the moment about the \( x \)-axis is approximately:

\[
\boxed{M_x = 45.5 \, \text{N} \cdot \text{m}}
\]

Would you like more details or have any questions?

Here are some follow-up questions to explore further:

1. How would the moment change if the force were applied at a different point along the \( y \)-axis?
2. How do you calculate the total moment when a force has components in all three axes?
3. What effect would reducing the angle \( \theta \) have on the moment about the \( x \)-axis?
4. Can the moment about an axis ever be zero? If so, under what conditions?
5. What is the physical significance of the moment being counterclockwise or clockwise?

**Tip:** Always use the right-hand rule to determine the direction of the moment: curl your fingers in the direction of the applied force, and your thumb will point in the direction of the moment vector.

A force of 166N in the yz-plane is directly applied to the parallelepiped at point A (labeled by the red dot). Determine the component of the moment caused by the force about the x-axis. Given parameters of a=923mm, b=755mm and θ=64°. Assume counterclockwise is a positive and round answer to at least 3 significant digits. Mx:_______N⋅m

Determine the moment about the x-axis for a force of 166N in the yz-plane applied at point A. Learn how to resolve the force into components, use trigonometry, and apply the moment-arm principle with given parameters a=923mm, b=755mm, and angle θ=64°. Ideal for physics or engineering students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation