{"id":3800,"date":"2021-01-30T12:29:53","date_gmt":"2021-01-30T12:29:53","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3800"},"modified":"2022-03-23T19:34:45","modified_gmt":"2022-03-23T19:34:45","slug":"resultant-force","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/resultant-force\/","title":{"rendered":"\u09b2\u09ac\u09cd\u09a7\u09bf \u09ac\u09b2 (Resultant Force)"},"content":{"rendered":"

\u09ac\u09b2\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u0993 \u0985\u0982\u09b6\u0995 (Resultant of Force and Components)<\/b><\/h2>\n

\u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u0989\u09aa\u09b0 \u098f\u0995\u0987 \u09b8\u09ae\u09af\u09bc\u09c7 \u098f\u0995\u09be\u09a7\u09bf\u0995 \u09ac\u09b2 \u0995\u09be\u09b0\u09cd\u09af\u09b0\u09a4 \u09b9\u09b2\u09c7, \u098f\u09a6\u09c7\u09b0 <\/span>\u09b8\u09ae\u09cd\u09ae\u09bf\u09b2\u09bf\u09a4 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09ab\u09b2 \u09af\u09a6\u09bf \u098f\u0995\u099f\u09bf \u09ae\u09be\u09a4\u09cd\u09b0 \u09ac\u09b2\u09c7\u09b0 \u09ac\u09be \u0995\u09cb\u09a8\u09cb \u098f\u0995\u0995 \u09ac\u09b2\u09c7\u09b0 \u0995\u09cd\u09b0\u09bf\u09df\u09be\u09ab\u09b2\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8 \u09b9\u09af\u09bc, \u09a4\u09ac\u09c7 \u0990 \u098f\u0995\u099f\u09bf\u09ae\u09be\u09a4\u09cd\u09b0 \u09ac\u09b2\u0995\u09c7 \u09ac\u09be \u098f\u0995\u0995 \u09ac\u09b2\u0995\u09c7 \u098f\u0995\u09be\u09a7\u09bf\u0995 \u09ac\u09b2\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u09ac\u09b2\u09c7 \u098f\u09ac\u0982 \u098f\u0995\u09be\u09a7\u09bf\u0995 \u09ac\u09b2\u09c7\u09b0 \u09aa\u09cd\u09b0\u09a4\u09cd\u09af\u09c7\u0995\u099f\u09bf\u0995\u09c7 \u09b2\u09ac\u09cd\u09a7\u09bf \u09ac\u09b2\u09c7\u09b0 \u0985\u0982\u09b6\u0995 \u09ac\u09be \u0989\u09aa\u09be\u0982\u09b6 \u09ac\u09b2\u09c7\u0964 \u099a\u09bf\u09a4\u09cd\u09b0\u09c7 O<\/span>\u00a0\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0\u09a4 <\/span>P<\/span> \u0993 <\/span>Q<\/span> \u09ac\u09b2 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u09b8\u09ae\u09cd\u09ae\u09bf\u09b2\u09bf\u09a4 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09ab\u09b2 \u098f\u0995\u099f\u09bf \u09ae\u09be\u09a4\u09cd\u09b0 <\/span>R<\/span> \u09ac\u09b2\u09c7\u09b0 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09ab\u09b2\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8 \u09b9\u09b2\u09c7, <\/span>R<\/span> \u0995\u09c7 <\/span>P<\/span> \u0993 <\/span>Q<\/span> \u098f\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u098f\u09ac\u0982 <\/span>P<\/span> \u0993 <\/span>Q<\/span> \u098f\u09b0 \u09aa\u09cd\u09b0\u09a4\u09cd\u09af\u09c7\u0995\u0995\u09c7 <\/span>R<\/span> \u098f\u09b0 \u0985\u0982\u09b6\u0995 \u09ac\u09be \u0989\u09aa\u09be\u0982\u09b6 \u09ac\u09b2\u09c7\u0964<\/span>
\n<\/span>\u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09aa\u09cd\u09b0\u09a4\u09c0\u0995\u09c7\u09b0 \u09b8\u09be\u09b9\u09be\u09af\u09cd\u09af\u09c7 \u09b2\u09ac\u09cd\u09a7\u09bf, <\/span>\\bar{R}=\\bar{P}+\\bar{Q}<\/span><\/span>\"Resultant<\/p>\n

\u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09b2\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf (Resultant of two forces)<\/b>\u00a0\u00a0\u00a0<\/b><\/h2>\n

\u098f\u0995\u0987 \u09b8\u09ae\u09af\u09bc \u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u0989\u09aa\u09b0 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09b2 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4 \u09b9\u09b2\u09c7, \u098f\u0987 \u09ac\u09b2\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09b8\u09ae\u09cd\u09ae\u09bf\u09b2\u09bf\u09a4 \u0995\u09cd\u09b0\u09bf\u09df\u09be\u09ab\u09b2 \u09af\u09a6\u09bf \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf\u09b0 \u0989\u09aa\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09a6\u09bf\u0995\u09c7 \u098f\u0995\u099f\u09bf \u09ae\u09be\u09a4\u09cd\u09b0 \u09ac\u09b2\u09c7\u09b0 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09ab\u09b2\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8 \u09b9\u09af\u09bc, \u09a4\u09ac\u09c7 \u0990 \u098f\u0995\u099f\u09bf \u09ae\u09be\u09a4\u09cd\u09b0 \u09ac\u09b2\u0995\u09c7 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4 \u09ac\u09b2 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u09ac\u09b2 \u09ac\u09b2\u09c7\u0964\u00a0<\/span><\/p>\n

\u099a\u09bf\u09a4\u09cd\u09b0\u09c7 <\/span>O<\/span> \u098f\u0995\u099f\u09bf \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be \u098f\u09ac\u0982 <\/span>O<\/span> \u09a4\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0\u09a4 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09b2 <\/span>P<\/span> \u0993 <\/span>Q<\/span> \u098f\u09b0 \u09b8\u09ae\u09cd\u09ae\u09bf\u09b2\u09bf\u09a4 \u0995\u09cd\u09b0\u09bf\u09df\u09be\u09ab\u09b2 \u0985\u09aa\u09b0 \u09ac\u09b2 <\/span>R<\/span> \u098f\u09b0 \u09b8\u09ae\u09be\u09a8 \u09b9\u09b2\u09c7, <\/span>R<\/span> \u09ac\u09b2\u0995\u09c7 <\/span>P<\/span> \u0993 <\/span>Q<\/span> \u09ac\u09b2 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u09ac\u09b2 \u09ac\u09b2\u09c7\u0964 \u098f\u0996\u09be\u09a8\u09c7 <\/span>R=P+Q<\/span><\/span><\/p>\n

\"Resultant<\/p>\n

\u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09aa\u09b0\u09bf\u099a\u09df (Identity)<\/b><\/h3>\n

\u09a6\u09c1\u09b0\u09cd\u0998\u099f\u09a8\u09be\u09ac\u09b6\u09a4: \u098f\u0995\u099f\u09bf \u09b0\u09c7\u09b2\u0997\u09be\u09a1\u09bc\u09c0 \u099f\u09cd\u09b0\u09c7\u09a8 \u09b2\u09be\u0987\u09a8\u099a\u09cd\u09af\u09c1\u09a4 \u09b9\u09af\u09bc\u09c7 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u09aa\u09a1\u09bc\u09c7 \u0986\u099b\u09c7\u0964 \u098f\u0987 \u0997\u09be\u09a1\u09bc\u09c0\u0996\u09be\u09a8\u09be \u09b2\u09be\u0987\u09a8\u09c7\u09b0 \u0989\u09aa\u09b0\u09c7 \u0989\u09a0\u09be\u09a8\u09cb\u09b0 \u099c\u09a8\u09cd\u09af \u0995\u09cb\u09a8\u09cb \u09b0\u09bf\u09b2\u09bf\u09ab \u099f\u09cd\u09b0\u09c7\u09a8\u09c7\u09b0 \u09a6\u09c1\u0987\u099f\u09bf \u0995\u09cd\u09b0\u09c7\u09a8 \u098f\u0995\u09a4\u09cd\u09b0\u09c7 \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09a4\u09c7 \u09b9\u09af\u09bc\u0964 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \u0985\u09a8\u09cd\u09af \u0986\u09b0 \u098f\u0995\u099f\u09bf \u09b0\u09bf\u09b2\u09bf\u09ab \u099f\u09cd\u09b0\u09c7\u09a8 \u0986\u099b\u09c7 \u09af\u09be\u09b0 \u098f\u0995\u099f\u09bf \u0995\u09cd\u09b0\u09c7\u09a8 \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09c7\u0987 \u0990 \u09b0\u09c7\u09b2\u0997\u09be\u09dc\u09c0\u099f\u09bf \u09b2\u09be\u0987\u09a8\u09c7\u09b0 \u0989\u09aa\u09b0\u09c7 \u0989\u09a0\u09be\u09a8\u09cb \u09af\u09be\u09af\u09bc\u0964 \u098f\u0996\u09be\u09a8\u09c7 \u09aa\u09c2\u09b0\u09cd\u09ac\u09cb\u0995\u09cd\u09a4 \u0995\u09cd\u09b0\u09c7\u09a8 \u09a6\u09c1\u099f\u09bf\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u09ac\u09b2 \u09b9\u09b2\u09cb \u09aa\u09b0\u09ac\u09b0\u09cd\u09a4\u09c0 \u098f\u0995\u099f\u09bf \u0995\u09cd\u09b0\u09c7\u09a8\u09c7\u09b0 \u09ac\u09b2\u0964<\/span><\/p>\n

\u0995\u09be\u099c (Work): <\/b>\u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09b2\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u09ac\u09b2\u09c7\u09b0 \u09ae\u09be\u09a8 \u0995\u0996\u09a8 \u09b6\u09c2\u09a8\u09cd\u09af \u09b9\u09af\u09bc, \u098f\u09b0\u0995\u09ae \u0995\u09af\u09bc\u09c7\u0995\u099f\u09bf \u0989\u09a6\u09be\u09b9\u09b0\u09a3 \u09a6\u09be\u0993\u0964<\/span><\/p>\n

\u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09b2\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 (Magnitude and direction of resultant of two forces)<\/b><\/h2>\n

\u098f\u0995\u0987 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09df \u098f\u0995\u0987 \u09a6\u09bf\u0995\u09c7 \u0995\u09cd\u09b0\u09bf\u09df\u09be\u09b6\u09c0\u09b2 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09b2\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8 \u09b9\u09ac\u09c7 \u09ac\u09b2\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09b8\u09ae\u09b7\u09cd\u099f\u09bf\u09b0 \u09b8\u09ae\u09be\u09a8 \u098f\u09ac\u0982 \u09a6\u09bf\u0995 \u09b9\u09ac\u09c7 \u09ac\u09b2\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09a6\u09bf\u0995 \u09ac\u09b0\u09be\u09ac\u09b0\u0964\u00a0<\/span><\/p>\n

\u0986\u09ac\u09be\u09b0 \u098f\u0995\u0987 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09af\u09bc \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a6\u09bf\u0995\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b6\u09c0\u09b2 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09b2\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8 \u09b9\u09ac\u09c7 \u09ac\u09b2\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8 \u098f\u09ac\u0982 \u09a6\u09bf\u0995 \u09b9\u09ac\u09c7 \u09ac\u09c3\u09b9\u09a4\u09cd\u09a4\u09b0 \u09ae\u09be\u09a8\u09c7\u09b0 \u09a6\u09bf\u0995 \u09ac\u09b0\u09be\u09ac\u09b0\u0964<\/span><\/p>\n

PICTURE MISSING<\/strong><\/h1>\n

\u09e7\u09ae \u099a\u09bf\u09a4\u09cd\u09b0\u09be\u09a8\u09c1\u09b8\u09be\u09b0\u09c7, <\/span>P<\/span> \u0993 <\/span>Q<\/span> \u09ac\u09b2\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf <\/span>R<\/span> \u09b9\u09b2\u09c7 R = P + Q<\/span> <\/span>\u098f\u09ac\u0982 \u09a6\u09bf\u0995 \u09b9\u09ac\u09c7 \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 <\/span>P<\/span> \u0993 <\/span>Q<\/span> \u098f\u09b0 \u09a6\u09bf\u0995\u09c7\u0964<\/span><\/p>\n

\u09e8\u09df \u099a\u09bf\u09a4\u09cd\u09b0\u09be\u09a8\u09c1\u09b8\u09be\u09b0\u09c7, <\/span>P<\/span> \u0993 <\/span>Q<\/span> \u09ac\u09b2\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf <\/span>R<\/span> \u098f\u09ac\u0982 <\/span>P>Q<\/span> <\/span>\u09b9\u09b2\u09c7, <\/span>R=P-Q<\/span> <\/span>\u098f\u09ac\u0982 <\/span>R \u098f\u09b0<\/span> \u09a6\u09bf\u0995 \u09b9\u09ac\u09c7 <\/span>P<\/span> \u098f\u09b0 \u09a6\u09bf\u0995\u09c7\u0964<\/span><\/p>\n

\u0986\u09ac\u09be\u09b0, <\/span>\\mathrm{Q}>\\mathrm{P}<\/span> <\/span>\u09b9\u09b2\u09c7 <\/span>R=Q-P<\/span> <\/span>\u098f\u09ac\u0982 <\/span>R \u098f\u09b0<\/span> \u09a6\u09bf\u0995 \u09b9\u09ac\u09c7 <\/span>Q<\/span> \u098f\u09b0 \u09a6\u09bf\u0995\u09c7\u0964<\/span><\/p>\n

\u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u098f\u0995\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09b2 \u098f\u0995\u0987 \u09b8\u09ae\u09df\u09c7 \u09ad\u09bf\u09a8\u09cd\u09a8 \u09ad\u09bf\u09a8\u09cd\u09a8 \u09a6\u09bf\u0995\u09c7 \u0995\u09cd\u09b0\u09bf\u09df\u09be\u09b6\u09c0\u09b2 \u09b9\u09b2\u09c7, \u09a4\u09be\u09a6\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u201c\u09ac\u09b2\u09c7\u09b0 <\/span>\u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995 \u09b8\u09c2\u09a4\u09cd\u09b0\u09c7\u09b0\u201d \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u09be \u09b9\u09df\u0964<\/span><\/p>\n

\u09ac\u09b2\u09c7\u09b0 \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995 \u09b8\u09c2\u09a4\u09cd\u09b0 (Parallelogram law of forces)<\/b>\u00a0\u00a0\u00a0\u00a0<\/b><\/h2>\n

\u09ac\u09b0\u09cd\u09a3\u09a8\u09be (Statement): <\/b>\u09af\u09a6\u09bf \u0995\u09cb\u09a8 \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995\u09c7\u09b0 \u09a6\u09c1\u0987\u099f\u09bf \u09b8\u09a8\u09cd\u09a8\u09bf\u09b9\u09bf\u09a4 \u09ac\u09be\u09b9\u09c1 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0995\u09cb\u09a8\u09cb \u0995\u09a3\u09be\u09b0 \u0989\u09aa\u09b0 \u098f\u0995\u0987 \u09b8\u09ae\u09df\u09c7 \u0995\u09cd\u09b0\u09bf\u09df\u09be\u09b0\u09a4 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09b2\u09c7\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u09b8\u09c2\u099a\u09bf\u09a4 \u09b9\u09df \u09a4\u09ac\u09c7 \u09a4\u09be\u09a6\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995\u09c7\u09b0 \u0989\u0995\u09cd\u09a4 \u09ac\u09be\u09b9\u09c1\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u0995\u09b0\u09cd\u09a3 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c2\u099a\u09bf\u09a4 \u09b9\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n

\u09ac\u09cd\u09af\u09be\u0996\u09cd\u09af\u09be (Explanation):<\/b> \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <\/span>OABC<\/span> \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995\u09c7\u09b0 <\/span>O<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0\u09a4 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09b2 <\/span>P<\/span> \u0993 <\/span>Q<\/span> \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 \u09b8\u09a8\u09cd\u09a8\u09bf\u09b9\u09bf\u09a4 \u09ac\u09be\u09b9\u09c1 <\/span>OA<\/span> \u0993 <\/span>OC<\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c2\u099a\u09bf\u09a4\u0964\u00a0<\/span><\/p>\n

\"Parallelogram<\/p>\n

\u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09b8\u09c2\u099a\u0995\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09b2\u09c7 <\/span>\\overrightarrow{O A}=P<\/span><\/span> \u098f\u09ac\u0982 <\/span>\\overrightarrow{O C}=Q<\/span>\u0964<\/span> \u098f\u0996\u09be\u09a8\u09c7 <\/span>P<\/span> \u0993 <\/span>Q<\/span> \u0989\u09ad\u09af\u09bc\u09c7\u0987 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09b0\u09be\u09b6\u09bf\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09af\u09cb\u099c\u09a8\u09c7\u09b0 \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995 \u09ac\u09bf\u09a7\u09bf \u0985\u09a8\u09c1\u09b8\u09be\u09b0\u09c7 \u09a4\u09be\u09a6\u09c7\u09b0 \u09af\u09cb\u0997\u09ab\u09b2 \u09ac\u09be \u09b2\u09ac\u09cd\u09a7\u09bf \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995 <\/span>OABC<\/span> \u098f\u09b0 \u0995\u09b0\u09cd\u09a3 <\/span>OB<\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c2\u099a\u09bf\u09a4 \u09b9\u09ac\u09c7\u0964 \u09a7\u09b0\u09bf, <\/span>P<\/span> \u0993 <\/span>Q<\/span> \u098f\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf <\/span>R<\/span>; \u09a4\u09be\u09b9\u09b2\u09c7 <\/span>R<\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u0995\u09b0\u09cd\u09a3 <\/span>OB<\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c2\u099a\u09bf\u09a4 \u09b9\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n

\u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09b8\u09c2\u099a\u0995\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09b2\u09c7 \u09aa\u09be\u0987, <\/span>\\overrightarrow{O A}+\\overrightarrow{O C}=\\overrightarrow{O B}<\/span><\/span> \u0985\u09b0\u09cd\u09a5\u09be\u09ce <\/span>P + Q = R<\/span><\/p>\n

\u099c\u09c7\u09a8\u09c7 \u09b0\u09be\u0996\u09cb<\/b><\/p>\n

\u09ac\u09bf\u0996\u09cd\u09af\u09be\u09a4 \u09ac\u09c3\u099f\u09bf\u09b6 \u09ac\u09bf\u099c\u09cd\u099e\u09be\u09a8\u09c0 \u09b8\u09cd\u09af\u09be\u09b0 \u0986\u0987\u099c\u09be\u0995 \u09a8\u09bf\u0989\u099f\u09a8 \u09e7\u09ec\u09ee\u09ed \u09b8\u09be\u09b2\u09c7 \u09ac\u09b2\u09c7\u09b0 \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995 \u09b8\u09c2\u09a4\u09cd\u09b0\u099f\u09bf \u09ac\u09b0\u09cd\u09a4\u09ae\u09be\u09a8 \u0986\u0995\u09be\u09b0\u09c7 \u09b2\u09bf\u09aa\u09bf\u09ac\u09a6\u09cd\u09a7 \u0995\u09b0\u09c7\u09a8\u0964<\/span><\/p>\n

\u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \\alpha<\/span> <\/b> \u0995\u09cb\u09a3\u09c7 \u0995\u09cd\u09b0\u09bf\u09df\u09be\u09b6\u09c0\u09b2 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09b2\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df<\/b>\u00a0\u00a0\u00a0\u00a0<\/b><\/h2>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <\/span>O<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u098f\u0995\u099f\u09bf \u0995\u09a3\u09be\u09b0 \u0989\u09aa\u09b0 \u098f\u0995\u0987 \u09b8\u09ae\u09df\u09c7 \\alpha<\/span> <\/span> \u0995\u09cb\u09a3\u09c7 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09b2 <\/span>P<\/span> \u0993 <\/span>Q<\/span> \u0995\u09cd\u09b0\u09bf\u09df\u09be\u09b6\u09c0\u09b2\u0964 <\/span>OA<\/span> \u098f\u09ac\u0982 <\/span>OB<\/span> \u09b0\u09c7\u0996\u09be\u0982\u09b6 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <\/span>P<\/span> \u0993 <\/span>Q<\/span> \u09ac\u09b2\u09c7\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u09b8\u09c2\u099a\u09bf\u09a4 \u0995\u09b0\u09be \u09b9\u09b2\u09cb\u0964 \u098f\u0996\u09be\u09a8\u09c7 <\/span>\\angle A O B=\\alpha, O A C B<\/span><\/span> \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995 \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09bf\u0964<\/span><\/p>\n

\u09a7\u09b0\u09bf <\/span>P<\/span> \u0993 <\/span>Q<\/span> \u09ac\u09b2 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u09ac\u09b2 <\/span>R<\/span> \u098f\u09ac\u0982 \u098f\u0987 \u09ac\u09b2\u099f\u09bf <\/span>P<\/span> \u09ac\u09b2\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \\theta<\/span>\u00a0<\/span> \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09c7\u0964 \u09a4\u09be\u09b9\u09b2\u09c7 \u09ac\u09b2\u09c7\u09b0 \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995 \u09b8\u09c2\u09a4\u09cd\u09b0 \u0985\u09a8\u09c1\u09b8\u09be\u09b0\u09c7 <\/span>OC<\/span> \u0995\u09b0\u09cd\u09a3\u099f\u09bf <\/span>R<\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7\u0964\u00a0<\/span><\/p>\n

\"resultant \"Resultant<\/p>\n

\u09e7\u09ae \u099a\u09bf\u09a4\u09cd\u09b0\u09be\u09a8\u09c1\u09b8\u09be\u09b0\u09c7, <\/span>OA<\/span> \u098f\u09b0 \u09ac\u09b0\u09cd\u09a7\u09bf\u09a4\u09be\u0982\u09b6 \u098f\u09b0 \u0989\u09aa\u09b0 <\/span>CD<\/span> \u09b2\u09ae\u09cd\u09ac \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09bf,<\/span><\/p>\n

\u09a4\u09be\u09b9\u09b2\u09c7 <\/span>\\triangle A D C<\/span><\/span> \u09b9\u09a4\u09c7, <\/span>\\cos C A D=\\frac{A D}{A C}<\/span><\/span> \u09ac\u09be, <\/span>\\cos \\alpha=\\frac{A D}{A C}<\/span><\/span> \u00a0 [\\because A C \\| O B]<\/span><\/span><\/p>\n

\u09ac\u09be, <\/span>A D=Q \\cos \\alpha \\quad[\\because Q=O B=A C]<\/span><\/span><\/p>\n

\\sin \\alpha=\\frac{C D}{A C}<\/span><\/span> \u09ac\u09be, <\/span>C D=Q \\sin \\alpha<\/span><\/span> \u098f\u09ac\u0982 <\/span>O D=O A+A D=P+Q \\cos \\alpha<\/span><\/span><\/p>\n

\u09e8\u09af\u09bc \u099a\u09bf\u09a4\u09cd\u09b0\u09be\u09a8\u09c1\u09b8\u09be\u09b0\u09c7, <\/span>OA<\/span> \u098f\u09b0 \u0989\u09aa\u09b0 <\/span>CD<\/span> \u09b2\u09ae\u09cd\u09ac \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09bf\u0964<\/span><\/p>\n

\\triangle A C D<\/span><\/span> \u09b9\u09a4\u09c7, <\/span>\\cos C A D=\\frac{A D}{A C}<\/span><\/span> \u09ac\u09be, <\/span>\\cos (\\pi-\\alpha)=\\frac{A D}{A C}<\/span><\/span> \u09ac\u09be, <\/span>A D=-Q \\cos \\alpha<\/span><\/span><\/p>\n

\\sin C A D=\\frac{C D}{A C}<\/span><\/span> \u09ac\u09be, <\/span>C D=Q \\sin \\alpha<\/span><\/span> \u098f\u09ac\u0982 <\/span>O D=O A-A D=P+Q \\cos \\alpha<\/span><\/span><\/p>\n

\u098f\u0996\u09a8 \u0989\u09ad\u09af\u09bc \u099a\u09bf\u09a4\u09cd\u09b0\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u0987, <\/span>\\triangle O C D<\/span><\/span> \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, <\/span> O C^{2}=O D^{2}+C D^{2}<\/span><\/span><\/p>\n

\u09ac\u09be, <\/span>R^{2}=(P+Q \\cos \\alpha)^{2}+(Q \\sin \\alpha)^{2}=P^{2}+2 P Q \\cos \\alpha+Q^{2} \\cos ^{2} \\alpha+Q^{2} \\sin ^{2} \\alpha<\/span><\/span><\/p>\n

=P^{2}+Q^{2}+2 P Q \\cos \\alpha<\/span><\/span>\u00a0
\n<\/span><\/p>\n

\\therefore R=\\sqrt{P^{2}+Q^{2}+2 P Q \\cos \\alpha}<\/span><\/span> , \u09af\u09be \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8\u0964\u00a0<\/span><\/p>\n

\u0986\u09ac\u09be\u09b0, <\/span>P<\/span> \u0993 <\/span>R<\/span> \u09ac\u09b2\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u0995\u09cb\u09a3 <\/span> \\theta<\/span> \u0985\u09b0\u09cd\u09a5\u09be\u09ce <\/span>\\angle C O D=\\theta<\/span><\/span><\/p>\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982 <\/span>\\triangle O C D<\/span><\/span> \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, <\/span>\\tan \\theta=\\frac{C D}{O D}=\\frac{Q \\sin \\alpha}{P+Q \\cos \\alpha}<\/span><\/span>
\n<\/span><\/p>\n

\\therefore \\theta=\\tan ^{-1} \\frac{Q \\sin \\alpha}{P+Q \\cos \\alpha}<\/span> \u09af\u09be \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09a6\u09bf\u0995 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7\u0964\u00a0<\/span><\/p>\n

\u09ac\u09bf\u0995\u09b2\u09cd\u09aa \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf (\u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf) Alternative method (vector method):<\/b><\/h3>\n

PICTURE MISSING<\/strong><\/h1>\n

\u098f\u0996\u09be\u09a8\u09c7 <\/span>P,Q<\/span> \u0993 <\/span>R<\/span> \u09ac\u09b2 \u09a4\u09bf\u09a8\u099f\u09bf\u09b0 \u09aa\u09cd\u09b0\u09a4\u09cd\u09af\u09c7\u0995\u09c7\u0987 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u098f\u09ac\u0982 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <\/span>\\overrightarrow{O A}, \\overrightarrow{O B}<\/span><\/span> \u0993<\/span> \\overrightarrow{O C}<\/span> <\/span>\u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c2\u099a\u09bf\u09a4\u0964\u00a0<\/span><\/p>\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982 <\/span>\\triangle O A C<\/span><\/span> \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, <\/span>\\overrightarrow{O A}+\\overrightarrow{A C}=\\overrightarrow{O C}<\/span><\/span> \u09ac\u09be, <\/span>\\overrightarrow{O C}=\\overrightarrow{O A}+\\overrightarrow{O B}<\/span><\/span><\/p>\n

\\therefore R=P+Q \\quad \\ldots \\ldots(i)<\/span><\/span><\/p>\n

\u09ac\u09be, <\/span>R \\cdot R=(P+Q) \\cdot(P+Q)=P \\cdot P+2 P \\cdot Q+Q \\cdot Q<\/span><\/span><\/p>\n

\u09ac\u09be, R^{2}=P^{2}+Q^{2}+2 P Q \\cos \\alpha<\/span><\/span>
\n<\/span><\/p>\n

\u2234<\/span> \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8 <\/span>R=\\sqrt{P^{2}+Q^{2}+2 P Q \\cos \\alpha}<\/span><\/span> \u00a0<\/span><\/p>\n

\u0986\u09ac\u09be\u09b0, <\/span>\\text { P. } R=P \\cdot(P+Q)=P \\cdot P+P \\cdot Q=P^{2}+P Q \\cos \\alpha<\/span><\/span> \u00a0 [<\/span>(<\/span>i<\/span>) <\/span>\u09a8\u0982 \u09a6\u09cd\u09ac\u09be\u09b0\u09be]<\/span><\/p>\n

\u09ac\u09be, P R \\cos \\theta=P^{2}+P Q \\cos \\alpha<\/span><\/span>\u00a0
\n<\/span><\/p>\n

\u09ac\u09be, R \\cos \\theta=P+Q \\cos \\alpha \\quad \\ldots . . .(i i)<\/span><\/span><\/p>\n

\u098f\u09ac\u0982 <\/span>P \\times R=P \\times(P+Q)=P \\times P+P \\times Q=0+P \\times Q<\/span><\/span><\/p>\n

\u09ac\u09be, <\/span>P R \\sin \\theta=P Q \\sin \\alpha \\quad[\\because P \\times P=0]<\/span><\/span><\/p>\n

\u09ac\u09be, <\/span>R \\sin \\theta=Q \\sin \\alpha<\/span><\/span> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u2026 \u2026 <\/span>(<\/span>iii<\/span>)<\/span><\/p>\n

\u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/span>(<\/span>ii<\/span>)<\/span> \u0993 <\/span>(<\/span>iii<\/span>)<\/span> \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, <\/span>\\tan \\theta=\\frac{Q \\sin \\alpha}{P+Q \\cos \\alpha}<\/span><\/span>
\n<\/span><\/p>\n

\\therefore \\theta=\\tan ^{-1} \\frac{Q \\sin \\alpha}{P+Q \\cos \\alpha}<\/span><\/span> \u09af\u09be \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09a6\u09bf\u0995 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7\u0964\u00a0<\/span><\/p>\n

\u09a6\u09cd\u09b0\u09b7\u09cd\u099f\u09ac\u09cd\u09af: <\/span>\\tan \\theta=\\frac{Q \\sin \\alpha}{P+Q \\cos \\alpha}<\/span> <\/span>\u09b8\u09c2\u09a4\u09cd\u09b0\u099f\u09bf \u0995\u09c7\u09ac\u09b2\u09ae\u09be\u09a4\u09cd\u09b0 <\/span>P+Q \\cos \\alpha \\neq 0<\/span><\/span> \u098f\u09b0 \u099c\u09a8\u09cd\u09af \u09aa\u09cd\u09b0\u09af\u09cb\u099c\u09cd\u09af\u0964\u00a0\u00a0\u00a0<\/span><\/p>\n

\u09e9\u09df \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf (3<\/b>rd<\/b> Method) :<\/b><\/h3>\n

PICTURE MISSING<\/strong><\/h1>\n

O<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \\alpha<\/span><\/span> \u0995\u09cb\u09a3\u09c7 \u098f\u0995\u0987 \u09b8\u09ae\u09df\u09c7 \u0995\u09cd\u09b0\u09bf\u09df\u09be\u09b6\u09c0\u09b2 <\/span>P<\/span> \u0993 <\/span>Q<\/span> \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09b2 \u09ae\u09be\u09a8\u09c7 \u0993 \u09a6\u09bf\u0995\u09c7 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <\/span>OA<\/span> \u0993 <\/span>OB<\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c2\u099a\u09bf\u09a4\u0964 <\/span>OACB<\/span> \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995\u099f\u09bf \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09c7 <\/span>O,\u00a0C\u00a0<\/span>\u09af\u09cb\u0997 \u0995\u09b0\u09bf\u0964 \u09a4\u09be\u09b9\u09b2\u09c7 \u09ac\u09b2\u09c7\u09b0 \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995\u09c7\u09b0 \u09b8\u09c2\u09a4\u09cd\u09b0\u09be\u09a8\u09c1\u09b8\u09be\u09b0\u09c7 <\/span>OC<\/span> \u0995\u09b0\u09cd\u09a3\u099f\u09bf \u09ac\u09b2 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u09b8\u09c2\u099a\u09bf\u09a4 \u0995\u09b0\u09ac\u09c7\u0964 \u09a7\u09b0\u09bf, \u09ac\u09b2 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8 <\/span>R<\/span>, \u09af\u09be <\/span>OA<\/span> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 \\theta<\/span><\/span> \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09c7, \u0985\u09b0\u09cd\u09a5\u09be\u09ce <\/span>\\angle A O C=\\theta<\/span><\/span><\/p>\n

OB<\/span> \u098f\u09b0 \u09b8\u09ae\u09be\u09a8 \u0993 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09ac\u09b2\u09c7, <\/span>AC<\/span> \u098f\u0995\u0987 \u09ac\u09b2 <\/span>Q<\/span> \u0995\u09c7 \u09b8\u09c2\u099a\u09bf\u09a4 \u0995\u09b0\u09c7\u0964 \u098f\u0996\u09be\u09a8\u09c7 <\/span>\\angle O A C=\\pi-\\alpha<\/span><\/span> \u098f\u09ac\u0982 <\/span>\\angle A C O=\\alpha-\\theta.<\/span><\/span><\/p>\n

0<\u03b1<\u03c0<\/span> \u09b8\u09c0\u09ae\u09be\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 <\/span> \\alpha<\/span> \u098f\u09b0 \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09ae\u09be\u09a8\u09c7\u09b0 \u099c\u09a8\u09cd\u09af, <\/span>OAC<\/span> \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7 \u0995\u09cb\u09b8\u09be\u0987\u09a8 \u09b8\u09c2\u09a4\u09cd\u09b0 \u09aa\u09cd\u09b0\u09df\u09cb\u0997 \u0995\u09b0\u09c7 \u09aa\u09be\u0987,\u00a0<\/span><\/p>\n

O C^{2}=O A^{2}+A C^{2}-2 . O A \\cdot A C \\cos (\\pi-\\alpha)<\/span><\/span>\u00a0
\n<\/span><\/p>\n

\\Rightarrow R^{2}=P^{2}+Q^{2}+2 P Q \\cos \\alpha<\/span><\/span>
\n<\/span><\/p>\n

\\therefore R=\\sqrt{P^{2}+Q^{2}+2 P Q \\cos \\alpha}<\/span><\/span> , \u09af\u09be \u09ac\u09b2 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8\u0964\u00a0<\/span><\/p>\n

O A C<\/span><\/span> \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7 \u09b8\u09be\u0987\u09a8 \u09b8\u09c2\u09a4\u09cd\u09b0 \u09aa\u09cd\u09b0\u09df\u09cb\u0997 \u0995\u09b0\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\n\\frac{O A}{\\sin O C A}=\\frac{A C}{\\sin A O C} \\Rightarrow \\frac{P}{\\sin (\\alpha-\\theta)}=\\frac{Q}{\\sin \\theta}<\/span>\n

\\Rightarrow P \\sin \\theta=Q(\\sin \\alpha \\cos \\theta-\\sin \\theta \\cos \\alpha) \\Rightarrow(P+Q \\cos \\alpha) \\sin \\theta=Q \\sin \\alpha \\cos \\theta<\/span><\/span> \u00a0\u00a0 <\/span>
\n<\/span><\/p>\n

\\Rightarrow \\tan \\theta=\\frac{Q \\sin \\alpha}{P+Q \\cos \\alpha}<\/span><\/span>
\n<\/span><\/p>\n

\\therefore \\theta=\\tan ^{-1} \\frac{Q \\sin \\alpha}{P+Q \\cos \\alpha}<\/span><\/span> \u09af\u09be \u09ac\u09b2 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09a6\u09bf\u0995\u0964<\/span><\/p>\n

\u09b2\u09ac\u09cd\u09a7\u09bf <\/b>R<\/b> \u09ac\u09c3\u09b9\u09a4\u09cd\u09a4\u09ae \u09b9\u0993\u09df\u09be\u09b0 \u09b6\u09b0\u09cd\u09a4 <\/b>(Condition for maximum Resultant, R)<\/b><\/h3>\n\\Rightarrow R^{2}=P^{2}+Q^{2}+2 P Q \\cos \\alpha=(P+Q)^{2}-2 P Q(1-\\cos \\alpha)=(P+Q)^{2}-4 P Q \\sin ^{2} \\frac{\\alpha}{2}<\/span>\n

\\therefore \\sin \\frac{a}{2}=0 \\Rightarrow \\sin \\frac{\\alpha}{2}=\\sin 0 \\Rightarrow \\alpha=0<\/span> \u09b9\u09b2\u09c7, R \u09ac\u09c3\u09b9\u09a4\u09cd\u09a4\u09ae \u09b9\u09ac\u09c7\u0964<\/p>\n

\u0985\u09a4\u098f\u09ac, R \u09ac\u09c3\u09b9\u09a4\u09cd\u09a4\u09ae \u09b9\u09ac\u09c7 \u09af\u0996\u09a8 \u03b1 = 0 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09af\u0996\u09a8 P, Q \u09ac\u09b2 \u09a6\u09c1\u0987\u099f\u09bf \u098f\u0995\u0987 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09af\u09bc \u098f\u0995\u0987 \u09a6\u09bf\u0995\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be \u0995\u09b0\u09c7, \u098f\u09ac\u0982 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ac\u09c3\u09b9\u09a4\u09cd\u09a4\u09ae \u09ae\u09be\u09a8, R_max=P+Q<\/p>\n

\u09b2\u09ac\u09cd\u09a7\u09bf R \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09ae \u09b9\u0993\u09df\u09be\u09b0 \u09b6\u09b0\u09cd\u09a4 (Condition for minimum Resultant, R)<\/b><\/h3>\n

\\Rightarrow R^{2}=P^{2}+Q^{2}+2 P Q \\cos \\alpha=(P-Q)^{2}+2 P Q(1+\\cos \\alpha)=(P-Q)^{2}+4 P Q \\cos ^{2} \\frac{\\alpha}{2}<\/span><\/span>
\n<\/span><\/p>\n

\\therefore \\cos \\frac{\\alpha}{2}=0 \\Rightarrow \\cos \\frac{a}{2}=\\cos \\frac{\\pi}{2} \\Rightarrow \\alpha=\\pi<\/span><\/span> \u09b9\u09b2\u09c7, R \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09ae \u09b9\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n

\u0985\u09a4\u098f\u09ac, <\/span>R<\/span> \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09ae \u09b9\u09ac\u09c7 \u09af\u0996\u09a8 <\/span>\\alpha=\\pi<\/span><\/span> \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09af\u0996\u09a8 <\/span>P,\u00a0Q\u00a0<\/span>\u09ac\u09b2 \u09a6\u09c1\u0987\u099f\u09bf \u098f\u0995\u0987 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09af\u09bc \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4\u00a0 \u09a6\u09bf\u0995\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be \u0995\u09b0\u09c7, \u098f\u09ac\u0982 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09ae \u09ae\u09be\u09a8, <\/span>R_{\\min }=P-Q,(P>Q)<\/span><\/span><\/p>\n

\u09ac\u09bf.\u09a6\u09cd\u09b0 .: <\/b>\u09af\u09a6\u09bf\u0993 <\/span>\\pi=0<\/span><\/span> \u09ac\u09be <\/span>\\alpha=\\pi<\/span><\/span> \u098f\u09b0 \u099c\u09a8\u09cd\u09af \u0995\u09cb\u09a8\u09cb \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995 \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc \u09a8\u09be, \u09a4\u09a5\u09be\u09aa\u09bf <\/span>\\alpha \\in[0, \\pi]<\/span> <\/span>\u098f\u09b0 <\/span>\u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09ae\u09be\u09a8\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u09ac\u09b2\u09c7\u09b0 \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995 \u09ac\u09bf\u09a7\u09be\u09a8 \u09b9\u09a4\u09c7 \u09aa\u09cd\u09b0\u09be\u09aa\u09cd\u09a4 \u09b8\u09c2\u09a4\u09cd\u09b0\u09b8\u09ae\u09c2\u09b9 \u09b8\u09a4\u09cd\u09af \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n

\u0995\u09df\u09c7\u0995\u099f\u09bf \u09aa\u09cd\u09b0\u09df\u09cb\u099c\u09a8\u09c0\u09df \u0985\u09a8\u09c1\u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4<\/b>\u00a0\u00a0<\/b><\/h3>\n
    \n
  • \u0985\u09a8\u09c1\u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4 \u2013 1: <\/b>\u09af\u0996\u09a8 <\/span>P<\/span> \u0993 <\/span>Q<\/span> \u09ac\u09b2\u09a6\u09cd\u09ac\u09af\u09bc \u09b8\u09ae\u09be\u09a8 \u0993 \u098f\u0995\u0987 \u09b0\u09c7\u0996\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4\u09ae\u09c1\u0996\u09c0, \u098f \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 <\/span>\\pi=180^{\\circ}<\/span><\/span> \u098f\u09ac\u0982<\/span><\/li>\n<\/ul>\n

    R^{2}=P^{2}+P^{2}+2 P^{2} \\cos 180^{\\circ}=2 P^{2}-2 P^{2}=0 \\quad[\\because P=Q] \\quad \\therefore R=0<\/span><\/span><\/p>\n

    \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u098f\u0995\u0987 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u098f\u0995\u0987 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a6\u09bf\u0995\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b6\u09c0\u09b2 \u09a6\u09c1\u0987\u099f\u09bf \u09b8\u09ae\u09be\u09a8 \u09ac\u09b2\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u09b6\u09c2\u09a8\u09cd\u09af (<\/span>0<\/span>)\u0964 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09ac\u09b2\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u0995\u09cb\u09a8\u09cb \u09aa\u09cd\u09b0\u09ad\u09be\u09ac \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u0993\u09aa\u09b0 \u09aa\u09a1\u09bc\u09c7 \u09a8\u09be \u098f \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u0995\u09c7 \u09b8\u09be\u09ae\u09cd\u09af\u09be\u09ac\u09b8\u09cd\u09a5\u09be (Equilibrium position) \u09ac\u09b2\u09c7\u0964<\/span><\/p>\n

      \n
    • \u0985\u09a8\u09c1\u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4 \u2013 2: <\/b>\u09af\u0996\u09a8 <\/span>P<\/span> \u0993 <\/span>Q<\/span> \u09ac\u09b2\u09a6\u09cd\u09ac\u09af\u09bc \u098f\u0995\u0987 \u09b0\u09c7\u0996\u09be\u09df \u0995\u09cd\u09b0\u09bf\u09df\u09be\u09b6\u09c0\u09b2 (\u09ae\u09be\u09a8 \u09b8\u09ae\u09be\u09a8 \u09ac\u09be \u0985\u09b8\u09ae\u09be\u09a8 \u0989\u09ad\u09af\u09bc \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u0987 \u09aa\u09cd\u09b0\u09af\u09cb\u099c\u09cd\u09af); \u098f\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09a6\u09c1\u0987 \u09a7\u09b0\u09a8\u09c7\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be \u09b9\u09a4\u09c7 \u09aa\u09be\u09b0\u09c7, \u098f\u0995\u099f\u09bf \u09b9\u09b2 \u09a4\u09be\u09a6\u09c7\u09b0 \u09a6\u09bf\u0995 \u098f\u0995\u0987 \u0985\u09aa\u09b0\u099f\u09bf \u09b9\u09b2 \u09a6\u09bf\u0995 \u09ad\u09bf\u09a8\u09cd\u09a8\u0964<\/span><\/li>\n<\/ul>\n

      \u09aa\u09cd\u09b0\u09a5\u09ae\u09a4:<\/b> P<\/span> \u0993 <\/span>Q<\/span> \u098f\u09b0 \u09a6\u09bf\u0995 \u098f\u0995\u0987 \u09b9\u09b2\u09c7, <\/span>\\alpha=0^{\\circ}<\/span><\/span> \u098f\u09ac\u0982<\/span>\u00a0<\/span>R=\\sqrt{P^{2}+Q^{2}+2 P Q \\cos 0^{0}}=\\sqrt{P^{2}+Q^{2}+2 P Q}=P+Q<\/span><\/span>
      \n<\/span><\/p>\n

      \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u098f\u0995\u0987 \u09b0\u09c7\u0996\u09be\u09af\u09bc \u098f\u0995\u0987 \u09a6\u09bf\u0995\u09c7 \u098f\u0995\u0987 \u09b8\u09ae\u09af\u09bc\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b6\u09c0\u09b2 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09b2\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u0989\u0995\u09cd\u09a4 \u09ac\u09b2\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09b8\u09ae\u09b7\u09cd\u099f\u09bf\u09b0 \u09b8\u09ae\u09be\u09a8 \u098f\u09ac\u0982 \u098f\u099f\u09be\u0987 \u09ac\u09c3\u09b9\u09a4\u09cd\u09a4\u09ae \u09b2\u09ac\u09cd\u09a7\u09bf\u0964<\/span><\/p>\n

      \u09a6\u09cd\u09ac\u09bf\u09a4\u09c0\u09df\u09a4:<\/b> P<\/span> \u0993 <\/span>Q<\/span> \u098f\u09b0 \u09a6\u09bf\u0995 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09b9\u09b2\u09c7, <\/span>\\alpha=180^{\\circ}<\/span><\/span> \u098f\u09ac\u0982<\/span>\u00a0<\/span>R=\\sqrt{P^{2}+Q^{2}+2 P Q \\cos 180^{\\circ}}=\\sqrt{P^{2}+Q^{2}-2 P Q}=(P-Q)<\/span><\/span>; \u09af\u0996\u09a8 <\/span>P>Q<\/span><\/span>
      \n<\/span><\/p>\n

      \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u098f\u0995\u0987 \u09b0\u09c7\u0996\u09be\u09af\u09bc \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a6\u09bf\u0995\u09c7 \u098f\u0995\u0987 \u09b8\u09ae\u09af\u09bc\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b6\u09c0\u09b2 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09b2\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u0989\u0995\u09cd\u09a4 \u09ac\u09b2\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8 \u098f\u09ac\u0982 \u098f\u099f\u09be\u0987 \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09ae \u09b2\u09ac\u09cd\u09a7\u09bf\u0964<\/span><\/p>\n

      P>Q<\/span><\/span> \u09b9\u09b2\u09c7 <\/span>R=P-Q<\/span><\/span> \u098f\u09ac\u0982 <\/span>P<Q<\/span> <\/span>\u09b9\u09b2\u09c7 R=Q-P<\/span><\/span> \u0985\u09b0\u09cd\u09a5\u09be\u09ce <\/span>R<\/span> \u098f\u09b0 \u09a6\u09bf\u0995 \u09b9\u09ac\u09c7 \u09ac\u09dc\u099f\u09bf\u09b0 \u09a6\u09bf\u0995\u09c7\u0964\u00a0<\/span><\/p>\n