Math Problem Statement

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

Solution

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

MA=F×dM_A = F \times d

where dd is the perpendicular distance from point AA to the line of action of the force FF.

Step 1: Analyze the Geometry

Given:

  • The force FF has a magnitude of 70 N and is applied at an angle of 7575^\circ from the horizontal.
  • The lengths of the segments are:
    • Horizontal segment from AA: 150 mm
    • Inclined segment: 500 mm at an angle of 3030^\circ to the horizontal.
    • Horizontal segment where FF 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 AA to the point where the force FF is applied.

  1. The horizontal distance from AA to the end of the first segment is 150 mm.
  2. The horizontal component of the inclined segment is given by 500×cos(30)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 dtotald_{\text{total}} from AA to the point where the force is applied is:

dtotal=150mm+500×cos(30)+125mmd_{\text{total}} = 150 \, \text{mm} + 500 \times \cos(30^\circ) + 125 \, \text{mm}

Step 3: Calculate the Perpendicular Distance

The perpendicular distance dd_{\perp} is given by:

d=dtotal×sin(75)d_{\perp} = d_{\text{total}} \times \sin(75^\circ)

Step 4: Calculate the Moment

Finally, the moment about point AA is:

MA=F×dM_A = F \times d_{\perp}

Let's perform these calculations.

  1. Calculate dtotald_{\text{total}}:

dtotal=150mm+500×cos(30)+125mmd_{\text{total}} = 150 \, \text{mm} + 500 \times \cos(30^\circ) + 125 \, \text{mm}

  1. Calculate dd_{\perp}:

d=dtotal×sin(75)d_{\perp} = d_{\text{total}} \times \sin(75^\circ)

  1. Calculate MAM_A using the force F=70NF = 70 \, \text{N}.

I'll calculate these values now.The moment of the applied force F=70NF = 70 \, \text{N} about point AA is approximately 47872.14N-mm47872.14 \, \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 AA 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.

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Math Problem Analysis

Mathematical Concepts

Mechanics
Principle of Moments
Vectors

Formulas

Moment formula: M = F * d

Theorems

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Suitable Grade Level

Advanced High School