### Worksheet 6.5. Work and Energy

```259
Worksheet 6.5.
Work and Energy
1. Find the work in joules to compress a spring 4 cm past equilibrium, assuming that the
spring constant is k = 150 kg/s2
.
3. Calculate the work in joules required to pump all the water out of the conical tank
pictured. Assume the lengths shown are measured in meters.
Water exits here
5
10
260
Solutions to Worksheet 6.5
1. Find the work in joules to compress a spring 4 cm past equilibrium, assuming that the
spring constant is k = 150 kg/s2 .
The work required to compress the spring 4 cm past equilibrium is
Z −.04
−.04
150x dx = 75x2 0 = 0.12 J
0
3. Calculate the work in joules required to pump all the water out of the conical tank
pictured. Assume the lengths shown are measured in meters.
Water exits here
5
10
Place the origin at the vertex of the inverted cone, and let the positive y-axis point
upward. Consider a layer of water at a height of y meters. From similar triangles, the
area of the layer is
y 2
π
m2
2
so the volume is
y 2
π
∆y m3
2
Thus the weight of one layer is
y 2
∆y N
9800π
2
261
The layer must be lifted 10 − y meters, so the total work needed to empty the tank is
Z 10
y 2
(10 − y) dy ≈ 6.414 × 107 J
9800π
2
0
```