As you lift a chair to store is upside down on a desk, gravity exerts a force, mg, downward and you exert a force of the same magnitude upwards. The two forces are equal in magnitude and opposite in direction. From this is would appear that no work was done, but you know you did work. Explain how you did work.

To understand how work is done in this scenario, we need to clarify the concept of work in physics. In physics, work is defined as the product of the force applied to an object and the displacement of the object in the direction of the force.

In this case, when you lift a chair to store it upside down on a desk, you are applying an upward force on the chair, opposite to the force of gravity pulling it downward. This upward force counteracts the downward force of gravity, preventing the chair from falling.

Even though the force you exert and the force of gravity on the chair are equal in magnitude and opposite in direction, work is still being done. This is because work depends not only on the force being applied but also on the displacement that occurs due to that force.

In this situation, as you lift the chair, you are exerting a force on it over a distance against the force of gravity. As a result, you are displacing the chair upwards, and this upward displacement is what defines the work being done.

According to the work formula W = F*d*cos(θ), where F is the force applied, d is the displacement, and θ is the angle between the force and the displacement vectors, the cosine of the angle between the force and displacement vectors is 1 in this case because they are in the same direction. Therefore, the work done is simply W = F*d.

So, when you lift the chair against gravity's force and displace it upward, you are indeed doing work because you are applying a force over a certain distance, resulting in the upward displacement of the chair.