QExample - 4: HC Verma: Concepts of Physics - Volume 1 Physics and Mathematics Example - 4 Solution: The work done by a force F "> during a displacement r "> is…
When the force and displacement point in opposite directions, the angle between them is $180^circ$, so $cos(180^circ)=-1$. Always include this sign in the dot product $W=|mathbf F||mathbf r|cos heta$; otherwise you’ll incorrectly treat the work as positive.
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during a displacement r is given by F⋅r. Suppose a force of 12 N acts on a particle in a vertically upward direction, and the particle is displaced through 2.0 m in a vertically downward direction. Find the work done by the force during this displacement.
Step-by-Step Verified Solution
Solution:
Given:
- Force (F) = 12 N (vertically upward)
- Displacement (r) = 2.0 m (vertically downward)
To Find:
- Work done (W) = F⋅r
Concept:
The work done is the dot product of the force and displacement vectors. Since the force and displacement are in opposite directions, the angle between them is 180°.
Formula:
W = F⋅r = |F| |r| cos(θ)
where θ is the angle between F and r.
Calculation:
- Substitute the given values into the formula:
- W = (12 N) (2.0 m) cos(180°)
- We know that cos(180°) = -1. Substitute this value:
- W = (12 N) (2.0 m) (-1)
- Calculate the result:
- W = -24 N⋅m
- Since 1 N⋅m = 1 Joule (J), we have:
- W = -24 J
Answer:
The work done by the force during this displacement is -24 J. The negative sign indicates that the work done by the force is against the displacement.