Question

A particle is displaced from P \( (3i + 2j - k) \) to Q \( (2i + 2j + 2k) \) by a force \( \mathbf{F} = i + j + k \). The work done on the particle (in J) is:

• A. 2
• B. 1
• C. 2.5
• D. 3
• E. 5

Hint:

Remember that the work done by a force is the dot product of the force vector and the displacement vector. Make sure to calculate the displacement accurately by subtracting the position vectors.

Answer 1

The Correct Option is A

Work done by a force is given by the formula: \[ W = \mathbf{F} \cdot \mathbf{d} \] where \( \mathbf{F} \) is the force vector and \( \mathbf{d} \) is the displacement vector.
First, we calculate the displacement vector \( \mathbf{d} \) between points P and Q: \[ \mathbf{d} = \mathbf{Q} - \mathbf{P} = (2i + 2j + 2k) - (3i + 2j - k) = (-i + 3k). \] Next, the force vector \( \mathbf{F} \) is given as: \[ \mathbf{F} = i + j + k. \] Now, calculate the work done: \[ W = \mathbf{F} \cdot \mathbf{d} = (i + j + k) \cdot (-i + 3k). \] Using the dot product: \[ W = (1)(-1) + (1)(0) + (1)(3) = -1 + 0 + 3 = 2 \, {J}. \] Thus, the work done on the particle is 2 J, which corresponds to option (A).