Animated Solution for Physics - Work, Energy, and Power: A porter lifts a heavy suitcase of mass $80\text{ kg}$ and at the destination lowers it down by a distance of $80\text{ cm}$ with a constant velocity. Calculate the work done by the porter in lowering the suitcase. [Take, $g=9.8{\text{ ms}}^{-2}$]

Imagine you are a porter at a busy railway station, tasked with carefully lowering a heavy $80\text{ kg}$ suitcase to the ground. If you simply let go, the suitcase would plummet, accelerating under the force of gravity. To prevent this, you must exert an upward force. Even though the suitcase is moving downwards, your muscles are straining upwards. This physical reality is the heart of understanding negative work.

In physics problems, certain phrases act as massive neon signs pointing to the solution. Here, the phrase is "constant velocity".

According to Newton's First Law of Motion, if an object is moving at a constant velocity, its acceleration is exactly zero. If the acceleration is zero, the net force acting on the object must also be zero.

For our suitcase, there are two primary forces at play: 1. The downward pull of gravity: $F_g=mg$ 2. The upward pull from the porter: $F_p$

To maintain a net force of zero, the porter's upward force must perfectly balance gravity:

Work is defined mathematically as the dot product of the force vector and the displacement vector:

Here is where the magic happens. The porter is applying a force upwards, but the suitcase is being displaced downwards. The angle $\theta$ between these two vectors is exactly $180^{\circ }$.

Let's plug in our values. The mass $m=80\text{ kg}$, the acceleration due to gravity $g=9.8{\text{ m/s}}^2$, and the displacement $d=80\text{ cm}=0.8\text{ m}$.

The negative sign is not just a mathematical artifact; it has profound physical meaning. It tells us that the porter is doing work against the direction of motion, effectively removing kinetic energy from the suitcase that gravity is trying to add. Without this negative work, the suitcase would crash into the ground!