Saved Bookmarks
| 1. |
Bernaullis theorem derivation\xa0 |
|
Answer» Ans\xa0.\xa0Proof\xa0:\xa0Let\xa0the\xa0velocity,\xa0pressure\xa0and\xa0area\xa0of\xa0a\xa0fluid\xa0column\xa0at\xa0a\xa0point\xa0X\xa0be\xa0v1,\xa0p1and\xa0A1\xa0and\xa0at\xa0another\xa0point\xa0Y\xa0be\xa0v2,\xa0p2\xa0and\xa0A2.\xa0Let\xa0the\xa0volume\xa0that\xa0is\xa0bounded\xa0by\xa0X\xa0and\xa0Y\xa0be\xa0moved\xa0to\xa0M\xa0and\xa0N.\xa0let\xa0XM\xa0=\xa0L1\xa0and\xa0YN\xa0=\xa0L2.\xa0Now\xa0if\xa0we\xa0can\xa0compress\xa0the\xa0fluid\xa0then\xa0we\xa0have,\xa0\xa0A1\xa0×\xa0L1=\xa0A2\xa0×\xa0L2We\xa0know\xa0that\xa0that\xa0the\xa0work\xa0done\xa0by\xa0the\xa0pressure\xa0difference\xa0per\xa0volume\xa0of\xa0the\xa0unit\xa0is\xa0equal\xa0to\xa0the\xa0sum\xa0of\xa0the\xa0gain\xa0in\xa0kinetic\xa0energy\xa0and\xa0gain\xa0in\xa0potential\xa0energy\xa0per\xa0volume\xa0of\xa0the\xa0unit.This\xa0implies\xa0Work\xa0done\xa0=\xa0force\xa0×\xa0distance\xa0⇒\xa0Work\xa0done\xa0=\xa0\xa0p\xa0×\xa0volumeTherefore,\xa0net\xa0work\xa0done\xa0per\xa0volume\xa0=\xa0p1\xa0–\xa0p2\xa0\xa0Also,\xa0kinetic\xa0energy\xa0per\xa0unit\xa0volume\xa0=\xa012\xa0mv2\xa0=\xa012\xa0ρv2Therefore,\xa0we\xa0have,Kinetic\xa0energy\xa0gained\xa0per\xa0volume\xa0of\xa0unit\xa0=\xa012\xa0ρv22\xa0-\xa0v12And\xa0potential\xa0energy\xa0gained\xa0per\xa0volume\xa0of\xa0unit\xa0=\xa0pg\xa0(h2\xa0–\xa0h1)Here,\xa0h1\xa0and\xa0h2\xa0are\xa0heights\xa0of\xa0X\xa0and\xa0Y\xa0above\xa0the\xa0reference\xa0level\xa0taken\xa0in\xa0common.Finally\xa0we\xa0have\xa0\xa0p1\xa0–\xa0p2\xa0=\xa012ρ\xa0v22-v12\xa0+\xa0ρg\xa0(h2\xa0–\xa0h1)⇒\xa0\xa0p1\xa0+\xa012\xa0ρ(v1)2\xa0+\xa0ρgh1\xa0=\xa0p2\xa0+\xa012ρ\xa0(v2)2\xa0+\xa0ρgh2⇒\xa0\xa0p\xa0+\xa012ρv2\xa0+\xa0ρgh\xa0is\xa0a\xa0constant\xa0\xa0When\xa0we\xa0have\xa0h1\xa0=\xa0h2\xa0\xa0Then\xa0we\xa0have,\xa0p\xa0+\xa012\xa0ρv2is\xa0a\xa0constant.\xa0\xa0This\xa0proves\xa0the\xa0Bernoulli’s\xa0Theorem Ans.\xa0According\xa0to\xa0Bernoulli’s\xa0theorem\xa0in\xa0physics,\xa0whenever\xa0there\xa0is\xa0an\xa0increase\xa0in\xa0the\xa0speed\xa0of\xa0the\xa0liquid,\xa0there\xa0is\xa0a\xa0simultaneous\xa0decrease\xa0in\xa0the\xa0potential\xa0energy\xa0of\xa0the\xa0fluid\xa0or\xa0we\xa0can\xa0say\xa0that\xa0there\xa0is\xa0a\xa0decrease\xa0in\xa0the\xa0pressure\xa0of\xa0the\xa0fluid.\xa0Basically,\xa0it\xa0is\xa0a\xa0principle\xa0of\xa0conservation\xa0of\xa0energy\xa0in\xa0the\xa0case\xa0of\xa0ideal\xa0fluids.\xa0If\xa0the\xa0fluid\xa0flows\xa0horizontally\xa0such\xa0that\xa0there\xa0is\xa0no\xa0change\xa0in\xa0the\xa0gravitational\xa0potential\xa0energy\xa0of\xa0the\xa0fluid\xa0then\xa0increase\xa0in\xa0velocity\xa0of\xa0the\xa0fluid\xa0results\xa0in\xa0a\xa0decrease\xa0in\xa0pressure\xa0of\xa0the\xa0fluid\xa0and\xa0vice\xa0versa. |
|