   Question Answers

# If the force $\left( {3\hat i - 2\hat j + \hat k} \right)N$ , produces a displacement of $\left( {2\hat i - 4\hat j + c\hat k} \right)m$. If the work done is $16J$ then, find the value of $c$.A) $- 1$B) $- 2$C) $1$D) $2$  Hint: We can define the work done as the product of force applied on a body and displacement produced by it. Now, put the values of force and displacement and evaluate the dot product of force and displacement. Also assume the initial position of the particle to be at origin.

Let the force applied on a body be $\vec F$, displacement produced by the body be $\vec s$ and the work done by the body be $\vec W$.
According to the question, it is given that –
$\implies \vec F = \left( {3\hat i - 2\hat j + \hat k} \right)N$
$\implies \vec s = \left( {2\hat i - 4\hat j + c\hat k} \right)m$ and
$\implies \vec W = 16J$
Work done is the transfer of energy for the displacement of an object using the application of force. Work has only magnitude and no direction therefore, it is the scalar quantity. The S.I unit of work is Joule. It transfers energy from one place to another.
Now, we know that the work done can be defined as the product of force applied on a body and displacement done by the body. So, in the vectors, it can be said that work done is the dot product of force applied on a body and displacement done by the body.
Mathematically, the above statement can be represented as –
$\vec W = \vec F.\vec s$
Putting the values of work done, force and displacement in the above equation –
$\implies 16 = \left( {3\hat i - 2\hat j + \hat k} \right).\left( {2\hat i - 4\hat j + c\hat k} \right) \\ \implies 16 = 6 + 8 + c \\$
Now, doing transposition method, we get –
$\implies c = 16 - 14 \\ \implies c = 2 \\$
Hence, the value of $c$ is $2$.
Therefore, the vector of displacement can be expressed as –
$\vec s = \left( {2\hat i - 4\hat j + 2\hat k} \right)m$
Hence, the correct option for this question is (D).

Note: The dot product of any orthogonal vector with itself is always one and the dot product of any orthogonal vector with any other orthogonal vector is always zero.
$\hat i.\hat i = 1 \\ \hat j.\hat j = 1 \\ \hat k.\hat k = 1 \\ \hat i.\hat j = 0 \\ \hat i.\hat k = 0 \\ \hat j.\hat k = 0 \\$
Let there be any two vectors such as –
$\vec A = a\hat i + b\hat j + c\hat k \\ \vec B = x\hat i + y\hat j + z\hat k \\$
Then, $\vec A.\vec B = ax + by + cz$
View Notes
Frictional Force  Balanced Force  Tension Force  Difference Between Force and Power  Magnetic and Electric Force on a Point Charge  What Happens if the Earth Stops Rotating?  Relationship Between Force of Limiting Friction and Normal Reaction  Resultant Force Formula  NCERT Book Class 11 Physics PDF  CBSE Class 11 Physics Waves Formulas  CBSE Class 12 Physics Question Paper 2020  Previous Year Question Paper for CBSE Class 12 Physics  Physics Question Paper for CBSE Class 12 - 2016 Set 1 C  Previous Year Question Paper for CBSE Class 12 Physics - 2018  Previous Year Question Paper for CBSE Class 12 Physics - 2013  Previous Year Question Paper for CBSE Class 12 Physics - 2014  Previous Year Question Paper for CBSE Class 12 Physics - 2019  Previous Year Question Paper for CBSE Class 12 Physics - 2015  Previous Year Physics Question Paper for CBSE Class 12 - 2017  Previous Year Question Paper for CBSE Class 12 Physics - 2016 Set 1 S  NCERT Solutions for Class 9 English Beehive Chapter 11 - If I Were You  HC Verma Class 11 Physics Part-1 Solutions for Chapter 3 - Rest and Motion Kinematics  NCERT Exemplar for Class 8 Science Solutions Chapter 11 Force  Lakhmir Singh Physics Class 9 Solutions Chapter 2 - Force and laws of motion  NCERT Solutions for Class 11 Physics Chapter 11  NCERT Solutions for Class 11 Physics  NCERT Solutions for Class 11 Physics Chapter 2  NCERT Solutions for Class 11 Physics Chapter 8  NCERT Solutions for Class 11 Physics Chapter 14  NCERT Solutions for Class 12 Physics Chapter 11  