Confused about answer key's solution to a work/energy problem

[physics801]

I'm studying for my college's physics placement and I'm frustrated because I don't get the solution to problem 1d.

Homework Statement

The problem is here: http://ocw.edu.ht/courses/physics/8-01-physics-i-classical-mechanics-fall-1999/exams/exam_2.pdf

Solution manual is here: http://ocw.edu.ht/courses/physics/8-01-physics-i-classical-mechanics-fall-1999/exams/quiz2_sol.pdf

The Attempt at a Solution

I drew a mspaint diagram of what I have: http://cl.ly/image/3b1o082l1k3a ($x_{max}$ is defined in 1a, which is in the soln manual)

Basically since gravity is opposing the direction of travel (up the ramp), I think the solution to 1d(i) should be $work=-mg(h-h_{max})=-mgsin\theta(x-x_{max}).$ However the solution manual has the same thing except $x_{max}-x$ in the parentheses. Why is this? Shouldn't it be $final - initial$, which means $x-x_{max}$ in the parentheses?

edit: I also have the same question for 1d(ii) and (iii) but hopefully it will be answered by explaining (i)

 

[haruspex]

You have to decide which way is positive and stick to it.
If the force due to gravity is being taken as negative along the slope then the positive direction is up the slope. But x is measured down the slope, so it's (-x) - (-x_max)
It is always a good idea to sanity-check the sign of a result. It is clear that the work done by gravity here is negative.

 

[/AH]

Look at the direction of x, what does x mean in terms of the spring?

 

[physics801]

You have to decide which way is positive and stick to it.
If the force due to gravity is being taken as negative along the slope then the positive direction is up the slope. But x is measured down the slope, so it's (-x) - (-x_max)
It is always a good idea to sanity-check the sign of a result. It is clear that the work done by gravity here is negative.

Oh whoops, I completely missed that. Thanks for the help!

 

[HalfLostAndSearching]

first

As a scientist, it is important to carefully analyze and understand the solution to a problem before coming to any conclusions or making assumptions. In this case, it may be helpful to review the problem and solution manual multiple times to ensure a full understanding of the problem and the given solution.

It is also important to remember that there may be multiple ways to approach and solve a problem, and there may be slight variations in the approach or notation used in different solutions. It is important to carefully read and understand the solution manual and any given equations or variables, as well as to check for any typos or errors that may have been made.

In this specific case, it seems that the solution manual has used a slightly different notation for the final and initial positions, but the overall approach and solution is the same. It may be helpful to compare your solution to the given solution step by step to identify any differences and understand why they may have occurred. Additionally, reaching out to a tutor or a professor for clarification may also be beneficial in understanding the solution to this problem.