Calculus Volume 2 · 免费预览
物理应用
功是对力沿距离的积分，流体静压力是对压力沿深度的积分，抽水问题则将薄层提升经过计算的距离，这些都出自同一个分段近似模板。
1. Work as force through distance
Work as force through distance
2. Hydrostatic force
Hydrostatic force
3. Pumping problems
Pumping problems
4. The template's generality
The template's generality
5. Spring work
A spring follows Hooke's law F(x) = kx. What work stretches it from x = 0 to x = s?
6. Pressure with depth
At what depth d below the surface does water pressure equal rho g d?
7. Force on a dam panel
A rectangular dam panel of width W extends from the surface to depth h. What total force does the water exert?
8. Pumping layer work
In a pumping problem, a thin water layer at height y with cross-sectional area A(y) must be lifted a distance D(y). What is the layer's work element?
9. Constant force recovery
The work formula W = integral of F dx must recover W = F d for a constant force. Substituting F constant, what does the integral give?
10. Spring work number
A spring with k = 10 newtons per meter is stretched 2 meters. The work is one half k s squared. What is it in joules? Answer with a number.
11. Centroid formulation of dam force
The total force on a submerged vertical plate equals the pressure at its centroid times the plate's area. For a rectangle of width W and height h with top at the surface, where is the centroid?
12. Hydrostatic constant
In SI units, water's weight density rho g is approximately 9800 newtons per cubic meter. As a number, what is it? Answer with a number.
13. Pumping cylinder integral
Pumping the cylindrical tank integrates rho g pi r^2 (2H - y) dy from 0 to H. The integral of (2H - y) dy over that range is what multiple of H^2? Answer like 3/2.
14. Work units
Force in newtons integrated over distance in meters gives work in what SI unit, one word? Answer in lowercase.