Statement-I: Two forces $(\vec{P} + \vec{Q}) a n d (\vec{P} - \vec{Q}) w h e r e \vec{P} ⊥ \vec{Q},$ when act at an angle θ₁ to each other the magnitude of their resultant is $\sqrt[]{3 (P^{2} + Q^{2})},$ when they act an angle θ₂, the magnitude of their resultant becomes $\sqrt[]{2 (P^{2} + Q^{2})} .$ This is possible only when θ₁ < θ₂.

Statement-II: In the situation given above.

θ₁ = 60° and θ₂ = 90°

In the light of the above statements, choose the most appropriate answer from the options given below:-

Option 1 -

Both Statement-I and Statement-II are true

Option 2 -

Both Statement-I and Statement-II are false

Option 3 -

Statement-I is true but Statement-II is false

Option 4 -

Statement-I is false but Statement-II is true

2 Views | Posted 2 months ago

Asked by Shiksha User

1 Answer

Answered by

Vishal Baghel | Contributor-Level 10

2 months ago

Correct Option - 1

Detailed Solution:

$\vec{x} = | \vec{P} + \vec{Q} | = \sqrt[]{P^{2} + Q^{2}}$

=> $\vec{y} = | \vec{P} - \vec{Q} | = \sqrt[]{P^{2} + Q^{2}}$

$| \vec{x} + \vec{y} | = \sqrt[]{x^{2} + y^{2} + 2 x y}$

$\Rightarrow \sqrt[]{3 (P^{2} + Q^{2})} = \sqrt[]{{(\sqrt[]{P^{2} + Q^{2}})}^{2} + {(\sqrt[]{P^{2} + Q^{2}})}^{2} + 2 (P^{2} + Q^{2}) c o s θ_{1}}$

$\Rightarrow θ_{1} = 60 °$

=>Using same formula : θ₂ = 90°

<math> <mrow> <mover accent="true"> <mi>x</mi> <mo>→</mo> </mover> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mover accent="true"> <mi>P</mi> <mo>→</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>Q</mi> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mroot> <mrow> <msup> <mrow> <mi>P</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mi>Q</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow></mrow> </mroot> </mrow> </math>          => <math> <mrow> <mover accent="true"> <mi>y</mi> <mo>→</mo> </mover> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mover accent="true"> <mi>P</mi> <mo>→</mo> </mover> <mo>−</mo> <mover accent="true"> <mi>Q</mi> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mroot> <mrow> <msup> <mrow> <mi>P</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mi>Q</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow></mrow> </mroot> </mrow> </math>  <math> <mrow> <mrow> <mo>|</mo> <mrow> <mover accent="true"> <mi>x</mi> <mo>→</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>y</mi> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mroot> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> </mrow> <mrow></mrow> </mroot> </mrow> </math> <math> <mrow> <mo>⇒</mo> <mroot> <mrow> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mi>P</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mi>Q</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mrow></mrow> </mroot> <mo>=</mo> <mroot> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mroot> <mrow> <msup> <mrow> <mi>P</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mi>Q</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow></mrow> </mroot> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mroot> <mrow> <msup> <mrow> <mi>P</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mi>Q</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow></mrow> </mroot> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mi>P</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mi>Q</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow></mrow> </mroot> </mrow> </math>  <math> <mrow> <mo>⇒</mo> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>6</mn> <mn>0</mn> <mo>°</mo> </mrow> </math> =>Using same formula : θ2 = 90°

0 Upvotes 0 Downvotes

Get authentic answers from experts, students and alumni that you won't find anywhere else

On Shiksha, get access to

65k Colleges
1.2k Exams
688k Reviews
1800k Answers

Community GuidelinesUser Points System

QuestionsDiscussionsTags

1 Answer

Learn more about...

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question