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Question

${{\sqrt 3 } \over 4}N$

${\sqrt 3 }$ N

0.5 N

1.5 N

Solution

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Step-by-Step Solution Step 1: Understand the Problem Setup We have three forces: 1 N, 2 N, and 4 N. They are oriented such that some lie at angles to the x-axis. We need to find the magnitude of an additional force that, when applied along the +x axis, makes the net force along the x-axis equal to zero, leaving only a net y-axis component. Step 2: Resolve the Forces Along the x-axis 1. The 1 N force has a component along the x-axis of $1 \times \cos 60^\circ = 1 \times \frac{1}{2} = 0.5 \text{ N}$. 2. The 2 N force has a component along the x-axis (assuming a 90° difference from the 1 N force’s orientation) of $2 \times \sin 30^\circ = 2 \times \frac{1}{2} = 1 \text{ N}$. Therefore, the total x-axis component of the 1 N and 2 N forces is $0.5 + 1 = 1.5 \text{ N}$ (directed along +x). Step 3: Resolve the 4 N Force Along the x-axis If the 4 N force has a component along the -x direction equal to $4 \times \sin 30^\circ = 4 \times \frac{1}{2} = 2 \text{ N}$ (directed along -x). Step 4: Determine the Additional Force Required Let the unknown force required along +x be $F_x$. To nullify the net force along the x-axis, we want the sum of all x-components to be zero: $\,(\text{x-component of 1 N and 2 N}) - (\text{x-component of 4 N}) + F_x = 0$ Substitute the known values: $\,1.5 \,(-2) + F_x = 0$ $\implies\, 1.5 - 2 + F_x = 0$ $\implies\, -0.5 + F_x = 0$ $\implies\, F_x = 0.5 \text{ N}.$ Step 5: Conclusion The force of 0.5 N along the +x axis balances out the net x-component of the existing forces, so that the resultant has no x-component and lies only along the y-direction. Hence, the correct answer is 0.5 N.

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Parent Quiz 802591

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