Question

A force does positive work on a particle that has a displacement pointing in +ve X-direction. The same force does negative work on a particle that has a displacement pointing in the +ve Y-direction. In which quadrant does force lie? Explain your answer.

Answer 1

Step 1: Given data:

1. Positive work done by force
2. Displacement is along the positive X direction

Step 2: To find the direction of force:

1. By definition, work $W=F.S=FS\cos \left(\theta \right)$; F is the force , S is the displacement and $\theta$is the angle between force and displacement.
2. Work can be positive or negative.
3. Work is positive when the force has a component in the same direction as the direction of the displacement, or equivalently when the angle between the force ‘F’ and the displacement ‘S’ is less than 90°.
4. $$\begin{gathered} W=FS\cos \left(\theta \right)=+ve \\ \Rightarrow \cos \left(\theta \right)=+ve \\ \Rightarrow 0^o<\theta <{90}^o \end{gathered}$$
5. The work is negative when the force has a component in the direction opposite to the displacement, that is when is greater than 90°.
6. $$\begin{gathered} W=FS\cos \left(\theta \right)=-+ve \\ \Rightarrow \cos \left(\theta \right)=-ve \\ \Rightarrow \theta >{90}^o \end{gathered}$$
7. The work done is positive when the body moves on the x-axis, so the angle between force and displacement must be less than 90°.
8. Similarly when the body makes displacement on the positive Y-axis the work done is negative.
9. So, the angle between force and displacement is more than 90°.
10. From this data, we can conclude that the force lies in the Fourth Quadrant.

Final Answer: The force lies in the fourth quadrant.