A force F⃗=2 î+b ĵ+k̂ is applied on a particle and it undergoes a displacement î-2 ĵ-k̂ What will be the value of b, if work done on the particle is zero.

Solution

Physics sir · 3 steps

Zero work means the force is perpendicular to the displacement, so set their dot product to zero.

1. Work as dot product. Work done by a constant force is W = F⃗ · s⃗. Multiply matching components and add: î with î, ĵ with ĵ, k̂ with k̂.

   W = (2)(1) + (b)(-2) + (1)(-1) = 1 - 2b

2. Set it to zero. 1 - 2b = 0 gives b = 1/2. Check: the force becomes 2î + 1/2ĵ + k̂, and 2 - 1 - 1 = 0.
3. So. b = 1/2, which is option (b).

The idea to keep. Whenever work is zero with nonzero force and displacement, the two vectors are perpendicular. Set F⃗ · s⃗ = 0 and watch the signs of each component.

Answer from the NTA final answer key. Solution by PrepWiser.