How much work is done by a force of magnitude 200 newtons and direction 30 degrees from the horizontal, which slides a crate 20 meters horizontally along a dock? Give your answer to two decimal places.
We begin by recalling that we can calculate work done by multiplying force by distance. In vector form, work is equal to the dot product of 𝐅 and 𝐝. When using the formula, it is important to note that, for both force and distance, the direction matters. This means that the work can be positive, negative, or zero. The standard units for force are newtons, and the standard units for distance are meters. This means that we measure work in newton meters, which are also commonly known as joules.
We are told in this question that we have a force of magnitude 200 newtons and direction 30 degrees from the horizontal acting on a crate. This force slides the crate a distance of 20 meters horizontally. When we find work done, we calculate 𝐅 multiplied by 𝐝, where 𝐝 is the distance traveled in the direction of the force. Since we know the horizontal distance traveled, it makes sense to work out the horizontal component of the force.
We can do this using our knowledge of right angle trigonometry. Since we know the hypotenuse and are trying to calculate the adjacent, we will use the cosine ratio, which states that cos 𝜃 is equal to the adjacent over the hypotenuse. Substituting in our values, we have cos of 30 degrees is equal to 𝐹 over 200. We know that the cos of 30 degrees is equal to root three over two. We can then multiply both sides of the equation by 200 such that 𝐹 is equal to 100 root three. The horizontal component of our force is 100 root three newtons.
We can now calculate the work by multiplying this by 20 meters. This is equal to 2000 root three. Typing this into the calculator gives us 3464.1016 and so on. And we are asked to give our answer to two decimal places. The work done by the force is 3464.10 newton meters to two decimal places.