Calculate the work done by a force of five newtons acting on a body that moved 10 meters to the north if the force was acting 30 degrees south of west. State your answer in joules.
The force of five newtons we can represent with a letter capital 𝐹. And the distance of 10 meters that the force causes a body to move we can call 𝑑. We want to solve for the work done by this force on the body. We’ll call this work capital 𝑊.
We’re told both the direction that the body moved, to the north, and the direction the force was acting, 30 degrees south of west. Let’s draw this motion on a diagram. If we put all of our motion on a compass grid with directions of north, south, east, and west, then we have a displacement of our body 𝑑, which is in a northerly direction. That displacement occurs when a force 𝐹 acting 30 degrees south of west is acting on the body.
To solve for the work that this force 𝐹 does on the body that is displaced at distance 𝑑, we can recall that the work done on an object is equal to the dot product of the force on that object and its displacement 𝑑.
In a two-dimensional plane, we can simplify this expression by writing that work is equal to the magnitude of the force multiplied by the magnitude of the body’s displacement times the cosine of the angle between those two vectors. In our scenario, the angle between 𝐹 and 𝑑 is 30 plus 90, or 120 degrees. So we can write 𝑊, the work done on the body, is equal to 𝐹 times 𝑑 times the cos of 120 degrees.
Plugging in for our values of force 𝐹 and displacement 𝑑, when we calculate 𝑊, we find it’s equal to negative 25 joules. This is the work that the force 𝐹 does on the body as it’s displaced the distance 𝑑.