How much work would it take to push a # 3 kg # weight up a # 8 m # plane that is at an incline of # pi / 3 #?
We know that the technical, calculus-based definition of work is defined as:
Assuming a constant force, this expression simplifies to the following (which you are taught if you're not doing a calculus based physics course):
We know how far the block travels, but we do not know how much force is acting on the block. Hence, we'll need to set up a free body diagram and use Newton's 2nd Law to figure that out:
Before we start talking about work, let's draw a free body diagram here to analyze the forces acting on the crate as we push it up:
I've assumed that the ramp is frictionless, since taking friction into account would need a bit more information that you don't have.
Hence, the only forces we care about here are those acting along the axis of the ramp. In this case, those are
Recall that in order to get the ramp moving, the minimum net force you need to have acting on the object is 0 (meaning that the object would have zero acceleration). Therefore, we can set up the following:
x signifies the direction up the ramp
Now, we just solve for
That is the minimum force we need to move the block. Now, we can go ahead and set up our work equation:
We can set
Rounded to one significant figure
And there you are. Note that this is the minimum amount of work you'd have to do. You could apply a larger force on the block, and move it up faster, but you'd be doing more work in those cases.
Hope that helped :)