Question

The resultant of these forces \(\vec{OP}, \vec{OQ}, \vec{OR}, \vec{OS}\) and \(\vec{OT}\) is approximately ________ N. 
[Take \(\sqrt{3} = 1.7, \sqrt{2} = 1.4\). Given \(\hat{i}\) and \(\hat{j}\) unit vectors along x, y axis] 

 

• A. \(3\hat{i} + 15\hat{j}\)
• B. \(-1.5\hat{i} - 15.5\hat{j}\)
• C. \(9.25\hat{i} + 5\hat{j}\)
• D. \(2.5\hat{i} - 14.5\hat{j}\)

Hint:

When resolving vectors, be very careful with angles and quadrants. Define angles consistently (e.g., counterclockwise from the positive x-axis) to avoid sign errors. If your calculated answer doesn't match any option in an exam, double-check your component calculations. If it still doesn't match, there might be an error in the question itself.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are asked to find the vector sum (resultant) of five forces given in a diagram. Each force is represented by its magnitude and the angle it makes with the coordinate axes.
Step 2: Key Formula or Approach:
To find the resultant force, we resolve each force vector into its x (\(\hat{i}\)) and y (\(\hat{j}\)) components. Then, we sum all the x-components to get the resultant x-component (\(R_x\)) and sum all the y-components to get the resultant y-component (\(R_y\)). The resultant vector is \(\vec{R} = R_x\hat{i} + R_y\hat{j}\).
A vector \(\vec{F}\) with magnitude \(F\) and angle \(\theta\) with the positive x-axis has components: \(F_x = F \cos\theta\) and \(F_y = F \sin\theta\).
Step 3: Detailed Explanation:
Let's resolve each vector into its components based on the angles given in the diagram:
1. \(\vec{OP}\): Magnitude = 20 N, Angle = 30° with +x axis.
\(\vec{OP} = (20 \cos 30^\circ)\hat{i} + (20 \sin 30^\circ)\hat{j} = 20(\frac{\sqrt{3}}{2})\hat{i} + 20(\frac{1}{2})\hat{j} = 10\sqrt{3}\hat{i} + 10\hat{j}\)
2. \(\vec{OQ}\): Magnitude = 10 N, Angle = 30° with +x axis.
\(\vec{OQ} = (10 \cos 30^\circ)\hat{i} + (10 \sin 30^\circ)\hat{j} = 10(\frac{\sqrt{3}}{2})\hat{i} + 10(\frac{1}{2})\hat{j} = 5\sqrt{3}\hat{i} + 5\hat{j}\)
3. \(\vec{OR}\): Magnitude = 20 N, Angle = -45° or 315° with +x axis.
\(\vec{OR} = (20 \cos(-45^\circ))\hat{i} + (20 \sin(-45^\circ))\hat{j} = 20(\frac{1}{\sqrt{2}})\hat{i} - 20(\frac{1}{\sqrt{2}})\hat{j} = 10\sqrt{2}\hat{i} - 10\sqrt{2}\hat{j}\)
4. \(\vec{OS}\): Magnitude = 15 N, Angle = 180° + 45° = 225° with +x axis.
\(\vec{OS} = (15 \cos 225^\circ)\hat{i} + (15 \sin 225^\circ)\hat{j} = 15(-\frac{1}{\sqrt{2}})\hat{i} - 15(\frac{1}{\sqrt{2}})\hat{j} = -7.5\sqrt{2}\hat{i} - 7.5\sqrt{2}\hat{j}\)
5. \(\vec{OT}\): Magnitude = 15 N, Angle = 180° - 60° = 120° with +x axis.
\(\vec{OT} = (15 \cos 120^\circ)\hat{i} + (15 \sin 120^\circ)\hat{j} = 15(-\frac{1}{2})\hat{i} + 15(\frac{\sqrt{3}}{2})\hat{j} = -7.5\hat{i} + 7.5\sqrt{3}\hat{j}\)
Now, sum the components:
\(R_x = (10\sqrt{3} + 5\sqrt{3} + 10\sqrt{2} - 7.5\sqrt{2} - 7.5) = 15\sqrt{3} + 2.5\sqrt{2} - 7.5\)
\(R_y = (10 + 5 - 10\sqrt{2} - 7.5\sqrt{2} + 7.5\sqrt{3}) = 15 - 17.5\sqrt{2} + 7.5\sqrt{3}\)
Substitute the given values \(\sqrt{3} = 1.7\) and \(\sqrt{2} = 1.4\):
\(R_x = 15(1.7) + 2.5(1.4) - 7.5 = 25.5 + 3.5 - 7.5 = 21.5\)
\(R_y = 15 - 17.5(1.4) + 7.5(1.7) = 15 - 24.5 + 12.75 = 3.25\)
The calculated resultant is \(\vec{R} = 21.5\hat{i} + 3.25\hat{j}\).
Note on Discrepancy: The calculated result does not match any of the given options. This suggests a potential error in the question data, the diagram, or the options provided in the examination. The official answer key indicates (C) is the correct answer. There is no standard physical interpretation of the provided diagram and values that leads to this answer. Such questions are often marked as bonus in exams. For the purpose of this solution, we acknowledge the discrepancy and select the official answer.
Step 4: Final Answer:
Based on the provided official answer key, the correct option is (C) \(9.25\hat{i} + 5\hat{j}\), despite the direct calculation from the problem statement yielding a different result.