NIMCET 2014 Mathematics PYQ — Constant forces and act on a particle. The work done when the par… | Mathem Solvex | Mathem Solvex
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NIMCET 2014 — Mathematics PYQ
NIMCET | Mathematics | 2014
Constant forces P=2i^−5j^+6k^ and Q=−i^+2j^−k^ act on a particle. The work done when the particle is displaced from A whose position vector is 4i^−3j^−2k^, to B whose position vector is 6i^+j^−3k^, is:
Choose the correct answer:
A.
10 units.
B.
−15 units.
(Correct Answer)
C.
−50 units.
D.
25 units.
Correct Answer:
−15 units.
Explanation
Solution
Concept:
If two points A and B have position vectors A and B respectively, then the vector AB=B−A.
For two vectors A and B at an angle θ to each other:
Dot Product is defined as: A⋅B=∣A∣∣B∣cosθ.
Resultant Vector is equal to A+B.
Work: The work (W) done by a force (F) in moving (displacing) an object along a vector D is given by: W=F⋅D=∣F∣∣D∣cosθ.
Calculation:
Let's say that the forces acting on the particle are P=2i^−5j^+6k^ and Q=−i^+2j^−k^.
The resulting force acting on the particle will be F=P+Q.
⇒F=(2i^−5j^+6k^)+(−i^+2j^−k^)
⇒F=i^−3j^+5k^
Since the particle is moved from the point 4i^−3j^−2k^ to the point 6i^+j^−3k^, the displacement vector D will be:
D=AB=B−A
=(6i^+j^−3k^)−(4i^−3j^−2k^)
⇒D=2i^+4j^−k^
And finally, the work done W will be:
W=F⋅D=(i^−3j^+5k^)⋅(2i^+4j^−k^)
⇒W=(1)(2)+(−3)(4)+(5)(−1)
⇒W=2−12−5=−15 units.
Correct Option: 2. (-15 units.)
Explanation
Solution
Concept:
If two points A and B have position vectors A and B respectively, then the vector AB=B−A.
For two vectors A and B at an angle θ to each other:
Dot Product is defined as: A⋅B=∣A∣∣B∣cosθ.
Resultant Vector is equal to A+B.
Work: The work (W) done by a force (F) in moving (displacing) an object along a vector D is given by: W=F⋅D=∣F∣∣D∣cosθ.
Calculation:
Let's say that the forces acting on the particle are P=2i^−5j^+6k^ and Q=−i^+2j^−k^.
The resulting force acting on the particle will be F=P+Q.
⇒F=(2i^−5j^+6k^)+(−i^+2j^−k^)
⇒F=i^−3j^+5k^
Since the particle is moved from the point 4i^−3j^−2k^ to the point 6i^+j^−3k^, the displacement vector D will be:
