Solveeit Logo
Login

Question

Question: A particle moves from a point \( ( - 2\hat i + 5\hat j) \) to \( (4\hat j + 3\hat k) \) when a force...

A particle moves from a point (−2i^+5j^)( - 2\hat i + 5\hat j) to (4j^+3k^)(4\hat j + 3\hat k) when a force of (4i^+3i^)N(4\hat i + 3\hat i)N is applied. How much work has been done by the force?
(A) 5  J5\;J
(B) 2  J2\;J
(C) 8  J8\;J
(D) 11  J11\;J

Explanation

Solution

The work done by a force on an object is determined by the product of the force with the displacement produced in the object. Force and displacement are both vector quantities. So we can determine the work done by calculating the dot product of the two.

Complete step by step solution
It is given in the question that-
The initial position of the object is,
P⃗i=−2i^+5j^{\vec P_i} = - 2\hat i + 5\hat j
The final position of the object is,
P⃗f=4j^+3k^{\vec P_f} = 4\hat j + 3\hat k
The displacement of the object is,
D⃗=P⃗f−P⃗i\vec D = {\vec P_f} - {\vec P_i}
Upon substituting the values we get,
⇒D⃗=(4j^+3k^)−(−2i^+5j^)\Rightarrow \vec D = (4\hat j + 3\hat k) - ( - 2\hat i + 5\hat j)
⇒D⃗=2i^−j^+3k^\Rightarrow \vec D = 2\hat i - \hat j + 3\hat k
The force acting on the object is given by,
F⃗=4i^+3i^N\vec F = 4\hat i + 3\hat iN
The force acting on the object is equal to the cross product of the displacement of the object with the given force.
In a dot product, all the vectors in the same direction get multiplied, rest are equated to zero. This is equivalent to doing a normal multiplication of numbers in the individual directions, and then adding all the components of the product individually gives the final product of the scalar product.
Putting the values,
W=F⃗.D⃗W = \vec F.\vec D
⇒W=(4i^+3i^).(2i^−j^+3k^)\Rightarrow W = (4\hat i + 3\hat i).\left( {2\hat i - \hat j + 3\hat k} \right)
⇒W=8i^2−3j^2+0k^\Rightarrow W = 8{\hat i^2} - 3{\hat j^2} + 0\hat k
We know that when unit vectors in the same direction are multiplied with themselves, they convert into a scalar with a magnitude of one.
So now the equation becomes,
⇒W=8−3\Rightarrow W = 8 - 3
⇒W=5\Rightarrow W = 5
The amount of work done by the force to move the object through the given displacement is 5  J5\;J .
Thus option (A) is the correct answer.

Note
Dot product of two vectors gives a scalar result. Which means that the work done has only magnitude and no direction, even the quantities that it is derived from (displacement and force) are vectors. A dot product can also be calculated by multiplying the magnitudes of both vectors and the cosine of the angle between them.