Showing work is path independent

[hellocello]

1. The problem:
A 5-kg block is moved from the ground to a height of 1 m via two different routes, vertically
and along an inclined plane with angle of 30° to the horizontal. How much work is done on
the block against gravity in each case?

Homework Equations

W=Fcosθd where θ is between the displacement and force, Fg=mg

The Attempt at a Solution

I realize the answer is that the work done is the same because it is path independent, but I don't understand why I can't show this in my calculations. Any help is greatly appreciated.

For the vertical path, I did
W=Fcosθd
W=mgcos(0)(1m)
W=(5kg)(9.8m/s^2)(1)(1m) ≈ 50 J

For the inclined path I did
W=Fcosθd where the force required to move the object along the plane is =mgsin(30°) and the angle between the displacement and force would now be 60° so
W=mgsin(30°)cos(60°)(1m) ≈12.5 J
Where I get only a fourth of the work I found above. What am I doing wrong?

(I assumed there was no friction because the question only asks about the work done against gravity)

 

[TSny]

For the inclined path I did
W=Fcosθd where the force required to move the object along the plane is =mgsin(30°) and the angle between the displacement and force would now be 60° so
W=mgsin(30°)cos(60°)(1m) ≈12.5 J

For the inclined path, the displacement is not 1 m vertically. It's a certain distance along the incline.

 

[hellocello]

Thank you! However wouldn't that distance be 1/tan(30) ? Because then the value I get is around 20...

 

[TSny]

However wouldn't that distance be 1/tan(30) ?

That's not quite right. Make a right triangle with one side horizontal and the other vertical. Let the hypotenuse represent the distance along the incline, and the vertical side represent the height of 1 m.

 

[hellocello]

Oops, it should be 1/sin(30), right? But then I'm still left with a cos(60) that doesn't cancel

 

[hellocello]

Oh it should be cos(0)- I forgot about that when you corrected my displacement. Thanks so much for your help!

 

[TSny]

You're welcome. Good work!