第三版流体力学习题答案（部分）.doc

<p> 第三章　流体静力学
【3-2】　图3-35所示为一直煤气管，为求管中静止煤气的密度，在高度差H=20m的两个截面装U形管测压计，内装水。已知管外空气的密度ρa=1.28kg/m3，测压计读数h1=100mm，h2=115mm。与水相比，U形管中气柱的影响可以忽略。求管内煤气的密度。

图3-35　习题3-2示意图
【解】　 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp;

【3-10】　试按复式水银测压计（图3-43）的读数算出锅炉中水面上蒸汽的绝对压强p。已知：H=3m，h1=1.4m，h2=2.5m，h3=1.2m，h4=2.3m，水银的密度ρHg=13600kg/m3。

图3-43　习题3-10示意图
【解】　 &nbsp; &nbsp; &nbsp;
&nbsp;

【3-15】　图3-48所示为一等加速向下运动的盛水容器，水深h=2m，加速度a=4.9m/s2。试确定：（1）容器底部的流体绝对静压强；（2）加速度为何值时容器底部所受压强为大气压强？（3）加速度为何值时容器底部的绝对静压强等于零？

图3-48　习题3-15示意图
【解】　，，
压强差公式 &nbsp;

（1） &nbsp; &nbsp;
（2） &nbsp; &nbsp;
（3） &nbsp; &nbsp;
【3-16】　图3-49所示为一圆柱形容器，直径d=300mm，高H=500mm，容器内装水，水深h1=300mm，使容器绕铅直轴作等角速旋转。（1）试确定水正好不溢出时的转速n1；（2）求刚好露出容器底面时的转速n2；这时容器停止旋转，水静止后的深度h2等于多少？

图3-49　习题3-16示意图
【解】　初始状态圆筒中没有水的那部分空间体积的大小为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (1)
圆筒以转速n1旋转后，将形成如图所示的旋转抛物面的等压面。令h为抛物面顶点到容器边缘的高度。空体积旋转后形成的旋转抛物体的体积等于具有相同底面等高的圆柱体的体积的一半
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(2)
由(1)(2)，得
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(3)
即
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (4)
等角速度旋转容器中液体相对平衡时等压面的方程为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (5)
对于自由液面，C=0。圆筒以转速n1旋转时，自由液面上，边缘处，，，则
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(6)
得
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(7)
由于
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(8)
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (9)
（1）水正好不溢出时，由式(4)(9)，得
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (10)
即

（2）求刚好露出容器底面时，h=H，则

（3）旋转时，旋转抛物体的体积等于圆柱形容器体积的一半
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (11)
这时容器停止旋转，水静止后的深度h2，无水部分的体积为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(12)
由(11)(12)，得
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(13)
得

【3-21】　图3-54所示为绕铰链O转动的倾斜角α=60°的自动开启式水闸，当水闸一侧的水位H=2m，另一侧的水位h=0.4m时，闸门自动开启，试求铰链至水闸下端的距离x。

图3-54　习题3-21示意图
【解】　设水闸宽度为b，水闸左侧水淹没的闸门长度为l1，水闸右侧水淹没的闸门长度为l2。
作用在水闸左侧压力为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (1)
其中
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
则
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(2)
作用在水闸右侧压力为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (3)
其中
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
则
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(4)
由于矩形平面的压力中心的坐标为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(5)
所以，水闸左侧在闸门面上压力中心与水面距离为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(6)
水闸右侧在闸门面上压力中心与水面距离为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(7)
对通过O点垂直于图面的轴取矩，设水闸左侧的力臂为d1，则
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (8)
得
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (9)
设水闸右侧的力臂为d2，则
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(10)
得
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(11)
当满足闸门自动开启条件时，对于通过O点垂直于图面的轴的合力矩应为零，因此
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(12)
则
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(13)

【3-29】　如图3-62所示，直径d=1m，高H=1.5m的圆柱形容器内充满密度ρ=900kg/m3的液体，顶盖中心开孔通大气。若容器绕中心轴以n=50r/min的转速旋转，求容器的上盖、底面和侧面所受的液体总压力。

图3-62　习题3-29示意图
【解】　由题设所述的容器内的压强分布与等角速度旋转运动容器中液体的静压强分布相同，为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (1)
则计示压强为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(2)
（1）作用在上盖的计示压强为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(3)
设圆柱体的底面积为Ad，则作用在上盖的总压力为
(4)
由于
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (5)
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (6)

（2）作用在底面的计示压强为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (7)
则作用在底面的总压力为
(8)

（3）由式(3)可知，作用在上盖边缘的计示压强为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(9)
则作用在侧面的计示压强为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (10)
设圆柱体的侧面积为Ac，则作用在侧面的总压力为
&nbsp; &nbsp; (11)
将式(5)代入，得
&nbsp; &nbsp; &nbsp; &nbsp; (12)

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
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