Help with physics question

Question

A system in which only one particle can move has the potential energy shown in the figure. Suppose U1 = 92 J. What is the y -component of the force on the particle at y = 0.5 m ? What is the y -component of the force on the particle at y = 4 m ?

Figure image: https://prnt.sc/vCC2xP-CHjPO

Thanks

Answer 1

The picture shows Work (U) on the y axis and Distance (Y) on the x axis. (The x and y labels might get confusing here. Capital Y will be the distance and lower case y will be the typical y axis)

It makes a triangle shape in straight lines going from (U,Y) = (0,0) to (92 J, 1 m) then back down in a straight line to point (0, 5 m)

Work = F d. so since we are looking for the Force, F = W/d

At Y= 0.5m we need to find the Work(at0.5). I'm stating W(0.5) = 92/2= 46J because the work went from 0 to 92 when Y went from 0 to 1 m in the picture. So in the middle (Y = 0.5m) we are half way to U1=92 J.

So at Y = 0.5m, F = 46J/0.5m = 92 N

Looking at the second part to find W(at 4m) you could get it by looking at the graph in a similar manner as above. Starting at Y = 1m we have U=92J, it goes straight down at Y = 5m where U=0J. Note Y has increased 4m. At half the distance(Y=3) U is half = 46, half this again (at Y=4m) U is half again or 23J.

So at Y=5m, F = 23J/5m = 5.75 N

Bonus: Another way of finding the Work at Y=5m is to get the slope of the triangle and then use the line formula y=mx+b. It is easier to do the formula if you shift the y axis over one so we know where the line crosses the y axis (92) but now the "Y" (x axis here in the chart) values are shifted to 0 1 2 3 4 instead of 1 2 3 4 5 .

slope = change in y/change in x = 0-92/ (4-0) = -23 J/m

so at Y=4m. we are actually looking for x=3 (due to the shift)

y=mx+b

y= -23 (3) + 92 = 23 J

F= w/d= 23J/4m = 5.75 N