Physics, asked by VishnuPriya2801, 11 months ago

The maximum and minimum resultant of two vectors are in the ratio 4 : 3 . Then what is the ratio of their forces ?

A) 7 : 1
B) 1 : 5
C) 4 : 7
D) 3 : 7​

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Answers

Answered by Anonymous
19

AnswEr :

Option (A) is correct

Explanation :

The ratio of the maximum and minimum resultant forces is 4 : 3

To finD

The ratio of the forces

Law of Cosines

\sf R = \sqrt{A^2 + B^2 + 2AB cos \phi }

For the resultant force to be maximum,∅ = 0° (cos0 = 1)

\sf R = \sqrt{A^2 + B^2 + 2AB \times cos0} \\ \\ \longrightarrow \sf R = \sqrt{(A^2 + B^2 + 2AB} \\ \\ \longrightarrow \sf R_{max} = A + B

For the resultant force to be minimum,∅ = 180° (cos180 = - 1)

\sf R = \sqrt{A^2 + B^2 + 2AB \times cos180} \\ \\ \longrightarrow \sf R = \sqrt{A^2 + B^2 - 2AB} \\ \\ \longrightarrow \sf R_{min} = A - B

\rule{300}{2}

We get the equations,

A + B = 4_______(1)

A - B = 3_______(2)

Adding equations (1) and (2),we get :

2A = 7

» A = 7/2

Putting A = 7/2 in (1),

(7/2) + B = 4

» - B = (-8 + 7)/2

» B = 1/2

Solving the above system of equations,

(A,B) = (7/2,1/2)

\rule{300}{2}

Ratio of A and B :

A : B = (7/2) : (1/2)

» A : B = 7/2 × 2/1

» A : B = 7 : 1

\rule{300}{2}

\rule{300}{2}

Answered by nirman95
9

Answer:

Given:

Maximum and minimum resultant of 2 vectors are in the ratio of 4:3.

To find:

Ratio of forces

Concept:

Since ratio of resultant has been provided , lets consider x as the constant of proportionality.

So the maximum resultant will be 4x

The minimum resultant will be 3x

Calculation:

We should also remember that :

Max resultant is obtained when vectors are directed in the same direction.

 \boxed{ \huge{ \red{  | \vec a |  \:  +  \:   | \vec b  | = 4x}}}

Minimum resultant is obtained when vectors are directed in opposite direction .

 \boxed{ \huge{ \blue{  | \vec a |  \:   -  \:   | \vec b |  = 3x}}}

Adding the 2 Equations , we get :

 \huge{ \green{  =  > 2 | \vec a |  \:    = 7x}}

 \huge{ \green{  =  >  | \vec a |  \:    =  \frac{7}{2} x}}

Putting value of vector a in 2nd equation :

 \huge{ \orange{  =  >  | \vec b |  \:    =  \frac{1}{2} x}}

Therefore ratio :

 \huge{   | \vec a |  \:  :   | \vec b|   =  \frac{7}{2} x :  \frac{1}{2} x}

 \boxed{ \bold{ \huge{   | \vec a |  \:  :   | \vec b|   =  7 : 1}}}

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