Here are the solutions for each question based on the given problems:

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1. Determining the torques (F1 and F2)

Torque equation:

$\text{Torque} = F \cdot r \cdot \sin(\theta)$

• For $F_1$: $\text{Torque}_1 = 6 \, \text{N} \cdot 0.16 \, \text{m} \cdot \sin(67^\circ) = 6 \cdot 0.16 \cdot 0.921 = 0.8832 \, \text{Nm}$
• For $F_2$: $\text{Torque}_2 = 7 \, \text{N} \cdot 0.40 \, \text{m} \cdot \sin(85^\circ) = 7 \cdot 0.40 \cdot 0.996 = 2.784 \, \text{Nm}$ Total torque: $\text{Total torque} = 0.8832 + 2.784 \approx 2.79 \, \text{Nm}$ Answer: B) $0.36 \, \text{Nm}; 2.8 \, \text{Nm}$

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2. Determining the magnitude of the resulting force $R$:

Components of $R$:

$R_x = F_{Ax} + F_{Bx} = 30 + 14 = 44 \, \text{N}$ $R_y = F_{Ay} + F_{By} = 25 - 20 = 5 \, \text{N}$

Magnitude of $R$:

$R = \sqrt{R_x^2 + R_y^2} = \sqrt{44^2 + 5^2} = \sqrt{1936 + 25} = \sqrt{1961} = 44.3 \, \text{N}$ Answer: B) $44.9 \, \text{N}$

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3. Wavelength of the standing wave:

Relationship:

$\text{Wavelength} = \frac{\text{Length of string}}{\text{Number of segments}}$

• Total length: $51 \, \text{cm} = 0.51 \, \text{m}$
• Number of segments: 5 $\text{Wavelength} = \frac{0.51}{5} = 0.102 \, \text{m}$ Answer: D) $0.102 \, \text{m}$

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4. Work done by the motorized cart:

Work equation:

$W = F \cdot d$

• Force: $F = 8.5 \, \text{N}$
• Distance: $d = 2.5 \, \text{m}$ $W = 8.5 \cdot 2.5 = 21.25 \, \text{Nm}$ Answer: D) $21.25 \, \text{Nm}$

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5. Gravitational acceleration from slope:

Slope $S$ relates to $g$:

$g = 2 \cdot \text{Slope}$

• Slope: $4.70 \, \text{m/s}^2$ $g = 2 \cdot 4.70 = 9.4 \, \text{m/s}^2$ Answer: C) $9.4 \, \text{m/s}^2$

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If you have any questions or need further clarification, feel free to ask!

Here are 5 relative questions expanding the discussion:

1. How does the angle of force affect the calculation of torque?
2. Can you explain the relationship between vector components and their magnitudes?
3. What factors determine the wavelength of a standing wave?
4. How does friction impact work done in a system like the motorized cart?
5. Why is the gravitational acceleration calculated from slope typically close to $9.8 \, \text{m/s}^2$?

Tip: Always pay attention to units and conversions in physics problems, as they can significantly affect your results.