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Question: A body is moving under the action of two forces \({\vec F_1} = 2\hat i - 5\hat j{\kern 1pt} {\kern 1...

A body is moving under the action of two forces F⃗1=2i^−5j^;F⃗2=3i^−4j^{\vec F_1} = 2\hat i - 5\hat j{\kern 1pt} {\kern 1pt} ;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\vec F_2} = 3\hat i - 4\hat j Its velocity will became uniform under a third force F⃗3{\vec F_3} given by
A. 5i^−9j^5\hat i - 9\hat j{\kern 1pt} {\kern 1pt}
B. −5i^−9j^- 5\hat i - 9\hat j{\kern 1pt} {\kern 1pt}
C. 5i^+9j^5\hat i + 9\hat j{\kern 1pt} {\kern 1pt}
D. −5i^+9j^- 5\hat i + 9\hat j{\kern 1pt} {\kern 1pt}

Explanation

Solution

In order to solve this question, we should know that when a force is written in vector form its simple represents the direction of acceleration of a body and in order to produce uniform velocity which means body has zero acceleration, and zero acceleration implies net force acting on the body will be zero, so here we will add all the forces so that net force acting on the body became zero, so while solving for net zero force we will find the value of F⃗3{\vec F_3} which will cause a uniform velocity to the body.

Complete step by step answer:
According to the question, we have given that two forces which are acting on the body has the value as F⃗1=2i^−5j^;F⃗2=3i^−4j^{\vec F_1} = 2\hat i - 5\hat j{\kern 1pt} {\kern 1pt} ;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\vec F_2} = 3\hat i - 4\hat j and we have to find the value of force F⃗3{\vec F_3} so let us suppose net force acting on the body is F vector which must be zero in order to produce zero acceleration which is uniform velocity.

F⃗=F⃗1+F⃗2+F⃗3\vec F = {\vec F_1} + {\vec F_2} + {\vec F_3} so here, value of vector F be zero and put the value of given vectors F⃗1=2i^−5j^;F⃗2=3i^−4j^{\vec F_1} = 2\hat i - 5\hat j{\kern 1pt} {\kern 1pt} ;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\vec F_2} = 3\hat i - 4\hat j we get,
on solving,
0=  2i^−5j^+3i^−4j^+F⃗30 = \;2\hat i - 5\hat j{\kern 1pt} {\kern 1pt} + 3\hat i - 4\hat j + {\vec F_3}
⇒−F⃗3=5i^−9j^\Rightarrow - {\vec F_3} = 5\hat i - 9\hat j
∴F⃗3=−5i^+9j^\therefore {\vec F_3} = - 5\hat i + 9\hat j

Hence, the correct option is D.

Note: It should be remembered that, force and acceleration are both a vector quantity and acceleration is the rate of change of velocity so if velocity is uniform the, acceleration produced in the body is zero and relation between force and acceleration are written as F⃗=ma⃗\vec F = m\vec a where mm is mass of the body and a is acceleration.