Question
Forces of magnitudes
and
act at a point parallel to the sides of an equilateral triangle taken in order. The resultant of these forces, is
- 3P
Hint:
In this question, we have given a equilateral triangle and the force of magnitude is parallel with the sides of triangle. Which is P- Q, P, P + Q. We have to find the resultant force. Find the component of force for each magnitude.
The correct answer is: ![square root of 3 Q](data:image/png;base64,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)
Here,
![](https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/generalfeedback/1723257/1/993275/Picture3352.x-wmf)
Þ
=
…..(i)
![](https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/generalfeedback/1723257/1/993275/Picture3349.x-wmf)
![](https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/generalfeedback/1723257/1/993275/Picture1.png)
Þ ![](https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/generalfeedback/1723257/1/993275/Picture3348.x-wmf)
=
…..(ii)
Squaring and adding (i) and (ii), we get
![](https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/generalfeedback/1723257/1/993275/Picture3345.x-wmf)
![](https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/generalfeedback/1723257/1/993275/Picture3344.x-wmf)
.
Here we have to find the resultant force.
Now, we have 3 magnitudes of force,
a = P, b = P – Q, c = P + Q,
Now the component of force,
At A, θ= 60°
acos θ = P cos 60° =![negative P over 2](data:image/png;base64,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)
asin θ = P sin 60° = ![negative fraction numerator square root of 3 p end root over denominator 2 end fraction](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADoAAAAnCAYAAABe81ttAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAbSkFbcgAAAg1JREFUeNrtmU0oRFEYhm+TNEkiSqJIFpJkQQhJSZKFNIospCxkJRsKYWWHWFJ2lPwthI0kSWNDViJZWijKgkn+3q9eNZ3umDua7p05c996mrnfOfc0753v/F7DsFep4NsmHFUXODISQPugX3eTWeCZn1prAGybxBvBHngFAXAD5kFmvBqVvukLEe8AyUGxcnAQjyYLwCPwRnDPSzwaHQVLEdQvBNcm8UkwBqbBA1NdMiInVoxegBYL9bJBH/tpq0n5OrgDIyADeMAsWLPbUKpJrJRPP+mP+9TJfihEPXkAJUoszc40LwIr/JF1StkMmLPYjvxL7cAPGpQyD0dmVV47jY7TqMyT50rZPaj4R2b4lVhViFVVDTi0O3UHwRf7mqga3IZJ21AKKNfdfJiqhsGEEwOPpNcqvy+CqX+0Ucb+GKwFDlTBkrn3yqlRdxm88V+UubM4TP1j0Mn6Hq6UJN17lXpbNN/E61zG+pyaStLBJ9gEZxbq14NdpurvvNhmUu+JA5WY/QCXvHZUJ2GmiUiVTKMxp0oazYtSe61M05iUL4pt9XIwcuXKwnrTyoHTdxzgypUrV66CVQs2uHl+5xq2R0ejx9xr/h7DyFHJKWPaK5/7zIRQIBFM1jB9tZac6vk5SGkrOe7cAc06myykySKdTcoBmrybSdHZpJwByzuVJENz7Rrhj0S13/BHTT/7DrTei9lwdAAAAIV0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+LTwvbW8+PG1mcmFjPjxtc3FydD48bW4+MzwvbW4+PG1pPnA8L21pPjwvbXNxcnQ+PG1uPjI8L21uPjwvbWZyYWM+PC9tYXRoPnZm1YkAAAAASUVORK5CYII=)
At C, θ= 60°
c cos θ = P+Q cos 60° = ![fraction numerator left parenthesis space minus P plus Q right parenthesis over denominator 2 end fraction](data:image/png;base64,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)
c sin θ = P+Q sin 60° = ![fraction numerator square root of 3 left parenthesis P plus Q right parenthesis end root over denominator 2 end fraction](data:image/png;base64,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)
No component for B because it is at x- axis.
So Resultant force is,
Fi = (
+ P – Q ) i^
Fi = (
- Q/2 + P – Q ) i^
Fi = (
) i^
And
Fj = (
) j^
Fj = (
) j^
Fj = (
) j^
Therefore,
|F| = ![square root of open parentheses open parentheses negative fraction numerator 3 Q over denominator 2 end fraction close parentheses squared plus open parentheses fraction numerator square root of 3 Q end root over denominator 2 end fraction close parentheses squared close parentheses end root](data:image/png;base64,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)
|F| = ![square root of open parentheses 9 Q to the power of 2 over 4 end exponent plus 3 Q to the power of 2 over 4 end exponent close parentheses end root](data:image/png;base64,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)
|F| = ![square root of open parentheses 12 Q to the power of 2 over 4 end exponent close parentheses end root](data:image/png;base64,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)
|F| = ![square root of open parentheses 3 Q squared close parentheses end root](data:image/png;base64,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)
|F| = ![square root of 3 Q end root](data:image/png;base64,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)
Therefore, the resultant force is ![square root of 3 Q end root](data:image/png;base64,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)
The correct answer is
.
Here we have to find the resultant force of the given magnitude. Firstly, find the component of the force which is Fsin θ and Fcos θ.
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