Work EX 3

Select the y-axis pointing downward. Then, draw the forces acting on the mass when it has moved a distance $y$. The gravitational force $F_{G}$ and the force of the spring $F_{sp}$. The y-components of these forces are: ${F_{G}}_{y}=mg$ and ${F_{sp}}_{y}=-ky$. The first graph shows the y-component of the gravitational force ${F_{G}}_{y}$ as a function of the position $y$ of the falling mass. The second graph shows the y-component of the force of the spring ${F_{sp}}_{y}$ as a function of the position $y$ of the falling mass.

(a) Compute the area of the graph from y = 1 cm to y = 5 cm.

	Displacement	The area under the $F_{y}-y$ (measured in $N\times m$)	Work
$W_{G}$	1 to 5 cm	$10\times (5-1)\times 10^{-2}$	0.4 J
$W_{sp}$	1 to 5 m	$(0.5\times (-10)\times 5\times 10^{-2})-(0.5\times (-2)\times 1\times 10^{-2})$	-0.24 J

(b) The total work is $W_{G}+W_{sp}=0.16J$

(c) The total work done is zero when the mass falls from 0 to some point y:

$W_{G}+W_{sp}=0$

$mgy-{\frac {1}{2}}ky^{2}=0$

$y={2mg \over k}={\frac {20}{200}}=0.1m=10cm$