To determine which of the given force combinations has the same effect as the force \( F \) (40 N at a 40° angle from the horizontal), we need to resolve the force \( F \) into its horizontal and vertical components and then compare those components with the components of the forces in the options provided.
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Calculate the components of the force \( F \):
The horizontal (x) and vertical (y) components of the force \( F \) can be calculated using trigonometric functions.
\[ F_x = F \cos(\theta) = 40 \cos(40°) \approx 40(0.766) = 30.64 , \text{N} \]
\[ F_y = F \sin(\theta) = 40 \sin(40°) \approx 40(0.643) = 25.72 , \text{N} \]
So, the force \( F \) can be approximated in terms of its components as:
- \( F_x \approx 30.64 , \text{N} \) (to the right/east)
- \( F_y \approx 25.72 , \text{N} \) (upward/north)
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Evaluate each pair of forces given:
Now, let's break down the force options into their components and see which pair adds up to the components of \( F \):
i. 26 N west and 31 N north
- \( F_{x} = -26, \text{N} \) (west is negative)
- \( F_{y} = 31, \text{N} \) (north is positive)
Resultant:
- \( R_x = -26 \)
- \( R_y = 31 \)
ii. 31 N east and 26 N south
- \( F_{x} = 31, \text{N} \) (east is positive)
- \( F_{y} = -26, \text{N} \) (south is negative)
Resultant:
- \( R_x = 31 \)
- \( R_y = -26 \)
iii. 26 N east and 31 N south
- \( F_{x} = 26, \text{N} \) (east is positive)
- \( F_{y} = -31, \text{N} \) (south is negative)
Resultant:
- \( R_x = 26 \)
- \( R_y = -31 \)
iv. 31 N west and 26 N north
- \( F_{x} = -31, \text{N} \) (west is negative)
- \( F_{y} = 26, \text{N} \) (north is positive)
Resultant:
- \( R_x = -31 \)
- \( R_y = 26 \)
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Compare results with \( F \):
- We need approximately \( R_x \approx 30.64 \) N (east) and \( R_y \approx 25.72 \) N (north).
- None of the provided combinations match these values in the same direction and magnitude.
However, one observation can be made. The only option that has a positive \( R_y \) (upward) is option i (31 N north) which closely corresponds to the vertical component, but none of the options result in both required components for \( F \).
Hence, there is no correct answer from the choices provided that results in the same effect as the force \( F \) because each provides either a negative horizontal or a negative vertical component where the force \( F \) requires it to be positive relative to its stated direction (with \( F \) going east and north).
However, if we had to assess which pair is closest or has the most similar direction components, option i, with a north force, would be more aligned, albeit incorrectly matched in horizontal. In strict terms though, none equate to 40 N at 40 degrees.
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