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Page 1 Edurev123 STATICS 1. Fundamental ideas, Statics deals with forces (on bodies) which produce equilibrium. If a single force ?? acting indepencently produces the same effect as that due to a number of forces ?? 1 ,?? 2 ,?? 3 , etc., acting simultaneously; then ??? is called the resustant of ??? 1 ,??? 2 ,??? 3 , etc. in other words ???? is ??? 1 +??? 2 +??? 3 +?, The forces ??? 1 ,??? 2 ,??? 3 , etc., are called components of ???? . The resultant of two forces :??? and ???. along ???? and ???? such that angle ?? ?? ?? is ?? . is of magnitude v?? 2 +?? 2 +2???? cos ?? . This is obtained by applying parallelogram law for addition of vectors. If two forces are in equilibrium, then they must be of equal magnitude and act in opposite directions along the same line. If three forces acting at a point are in equilibrium, they can be represented in magnitude and direction by the sides of a triangle taken in order. This is known as law of triangle of forces. The converse of this law is also true. From the law of triangle of forces we get Lami's, theorem. If three forces acting at a point, are in equilibrium, then each force is proportional to the sine of the angle between the other two and conversely. The magnitude of the resultant ?? of two like parallel forces ???? and ???? acting at ?? and ?? is sum of their magnitudes, that is ?? i. ?? , and the line of action of ???? (which is parallel to that of ?? and ?? ) divides ???? internally inversely in the ratio of the forces ?? and ?? . Page 2 Edurev123 STATICS 1. Fundamental ideas, Statics deals with forces (on bodies) which produce equilibrium. If a single force ?? acting indepencently produces the same effect as that due to a number of forces ?? 1 ,?? 2 ,?? 3 , etc., acting simultaneously; then ??? is called the resustant of ??? 1 ,??? 2 ,??? 3 , etc. in other words ???? is ??? 1 +??? 2 +??? 3 +?, The forces ??? 1 ,??? 2 ,??? 3 , etc., are called components of ???? . The resultant of two forces :??? and ???. along ???? and ???? such that angle ?? ?? ?? is ?? . is of magnitude v?? 2 +?? 2 +2???? cos ?? . This is obtained by applying parallelogram law for addition of vectors. If two forces are in equilibrium, then they must be of equal magnitude and act in opposite directions along the same line. If three forces acting at a point are in equilibrium, they can be represented in magnitude and direction by the sides of a triangle taken in order. This is known as law of triangle of forces. The converse of this law is also true. From the law of triangle of forces we get Lami's, theorem. If three forces acting at a point, are in equilibrium, then each force is proportional to the sine of the angle between the other two and conversely. The magnitude of the resultant ?? of two like parallel forces ???? and ???? acting at ?? and ?? is sum of their magnitudes, that is ?? i. ?? , and the line of action of ???? (which is parallel to that of ?? and ?? ) divides ???? internally inversely in the ratio of the forces ?? and ?? . If two unlike parallel forces ???? and ???? act at ?? and ?? , the magnitude of their "resultant ???? is ?? Q and the tine of action of ???? divides ???? externally inversely in the ratio of the forces. The moment about a point ?? of a force ??? whose line of action passes through a point ?? is ???? ¯¯¯¯ ×??? =???×??? If a number of forces act at a point, the algebraic sum of their (rectangular) components along any direction is equal to the component of their resultant in the same direction; the vector sum of their moments about any point is equal to the moment of their resultant about the same point. It follows that if a number of forces acting at a point are in equilibrium the algebraic sum of their components along each of three mutually perpendicular directions must be zero or the vector sum of their components about each of three non-collinear points must be zero. Fic If three forces are in equilibrium, then they must be co-planar and they must be either concurrent or all of them must be parallel. The resultant of two forces acting at a point O along ???? and ???? and represented in magnitude by ?? : ?? A and ?? ,?? 3 is represented by (?? +?? )???? where ?? is a point on ???? such that ?? CA=?? ?? CB A useful trigonometrical result ?? is a point on the side ?? B of a triangle ?????? such that ???? :???? =?? :?? Then (?? +?? )cot?? =?? cot a-?? cot ?? (?? +?? )cot ?? =?? cos ?? -?? cot ?? Page 3 Edurev123 STATICS 1. Fundamental ideas, Statics deals with forces (on bodies) which produce equilibrium. If a single force ?? acting indepencently produces the same effect as that due to a number of forces ?? 1 ,?? 2 ,?? 3 , etc., acting simultaneously; then ??? is called the resustant of ??? 1 ,??? 2 ,??? 3 , etc. in other words ???? is ??? 1 +??? 2 +??? 3 +?, The forces ??? 1 ,??? 2 ,??? 3 , etc., are called components of ???? . The resultant of two forces :??? and ???. along ???? and ???? such that angle ?? ?? ?? is ?? . is of magnitude v?? 2 +?? 2 +2???? cos ?? . This is obtained by applying parallelogram law for addition of vectors. If two forces are in equilibrium, then they must be of equal magnitude and act in opposite directions along the same line. If three forces acting at a point are in equilibrium, they can be represented in magnitude and direction by the sides of a triangle taken in order. This is known as law of triangle of forces. The converse of this law is also true. From the law of triangle of forces we get Lami's, theorem. If three forces acting at a point, are in equilibrium, then each force is proportional to the sine of the angle between the other two and conversely. The magnitude of the resultant ?? of two like parallel forces ???? and ???? acting at ?? and ?? is sum of their magnitudes, that is ?? i. ?? , and the line of action of ???? (which is parallel to that of ?? and ?? ) divides ???? internally inversely in the ratio of the forces ?? and ?? . If two unlike parallel forces ???? and ???? act at ?? and ?? , the magnitude of their "resultant ???? is ?? Q and the tine of action of ???? divides ???? externally inversely in the ratio of the forces. The moment about a point ?? of a force ??? whose line of action passes through a point ?? is ???? ¯¯¯¯ ×??? =???×??? If a number of forces act at a point, the algebraic sum of their (rectangular) components along any direction is equal to the component of their resultant in the same direction; the vector sum of their moments about any point is equal to the moment of their resultant about the same point. It follows that if a number of forces acting at a point are in equilibrium the algebraic sum of their components along each of three mutually perpendicular directions must be zero or the vector sum of their components about each of three non-collinear points must be zero. Fic If three forces are in equilibrium, then they must be co-planar and they must be either concurrent or all of them must be parallel. The resultant of two forces acting at a point O along ???? and ???? and represented in magnitude by ?? : ?? A and ?? ,?? 3 is represented by (?? +?? )???? where ?? is a point on ???? such that ?? CA=?? ?? CB A useful trigonometrical result ?? is a point on the side ?? B of a triangle ?????? such that ???? :???? =?? :?? Then (?? +?? )cot?? =?? cot a-?? cot ?? (?? +?? )cot ?? =?? cos ?? -?? cot ?? Example 1 Forces acting in at a point I inside a quadrilateral ???????? represented by ???? ,???? ,???? and ?? D in magnitude and direction are in equilibrium. Find the position of ?? . Let ?? and ?? be the midpoint of ???? and ???? respectively. Then PB ¯¯¯¯ ·PC ¯¯¯¯ =2PN ¯¯¯¯ PD ¯¯¯¯ ·PA ¯¯¯¯ =2PM ¯¯¯¯ We get 2(??? |??? +???? ¯¯¯¯ )=???? ¯¯¯¯ +??? +???? ¯¯¯¯ +???? ¯¯¯¯ =0 (given) Therefore ?? is the midpoint of ?? ?? . It is also the midpoint of the line joining the midpoints of ???? and ???? . Example 2 A uniform circular plate of weight ?? is supported at three points in its edge whose distances apart are a, b, c. Find the load carried by each support. Page 4 Edurev123 STATICS 1. Fundamental ideas, Statics deals with forces (on bodies) which produce equilibrium. If a single force ?? acting indepencently produces the same effect as that due to a number of forces ?? 1 ,?? 2 ,?? 3 , etc., acting simultaneously; then ??? is called the resustant of ??? 1 ,??? 2 ,??? 3 , etc. in other words ???? is ??? 1 +??? 2 +??? 3 +?, The forces ??? 1 ,??? 2 ,??? 3 , etc., are called components of ???? . The resultant of two forces :??? and ???. along ???? and ???? such that angle ?? ?? ?? is ?? . is of magnitude v?? 2 +?? 2 +2???? cos ?? . This is obtained by applying parallelogram law for addition of vectors. If two forces are in equilibrium, then they must be of equal magnitude and act in opposite directions along the same line. If three forces acting at a point are in equilibrium, they can be represented in magnitude and direction by the sides of a triangle taken in order. This is known as law of triangle of forces. The converse of this law is also true. From the law of triangle of forces we get Lami's, theorem. If three forces acting at a point, are in equilibrium, then each force is proportional to the sine of the angle between the other two and conversely. The magnitude of the resultant ?? of two like parallel forces ???? and ???? acting at ?? and ?? is sum of their magnitudes, that is ?? i. ?? , and the line of action of ???? (which is parallel to that of ?? and ?? ) divides ???? internally inversely in the ratio of the forces ?? and ?? . If two unlike parallel forces ???? and ???? act at ?? and ?? , the magnitude of their "resultant ???? is ?? Q and the tine of action of ???? divides ???? externally inversely in the ratio of the forces. The moment about a point ?? of a force ??? whose line of action passes through a point ?? is ???? ¯¯¯¯ ×??? =???×??? If a number of forces act at a point, the algebraic sum of their (rectangular) components along any direction is equal to the component of their resultant in the same direction; the vector sum of their moments about any point is equal to the moment of their resultant about the same point. It follows that if a number of forces acting at a point are in equilibrium the algebraic sum of their components along each of three mutually perpendicular directions must be zero or the vector sum of their components about each of three non-collinear points must be zero. Fic If three forces are in equilibrium, then they must be co-planar and they must be either concurrent or all of them must be parallel. The resultant of two forces acting at a point O along ???? and ???? and represented in magnitude by ?? : ?? A and ?? ,?? 3 is represented by (?? +?? )???? where ?? is a point on ???? such that ?? CA=?? ?? CB A useful trigonometrical result ?? is a point on the side ?? B of a triangle ?????? such that ???? :???? =?? :?? Then (?? +?? )cot?? =?? cot a-?? cot ?? (?? +?? )cot ?? =?? cos ?? -?? cot ?? Example 1 Forces acting in at a point I inside a quadrilateral ???????? represented by ???? ,???? ,???? and ?? D in magnitude and direction are in equilibrium. Find the position of ?? . Let ?? and ?? be the midpoint of ???? and ???? respectively. Then PB ¯¯¯¯ ·PC ¯¯¯¯ =2PN ¯¯¯¯ PD ¯¯¯¯ ·PA ¯¯¯¯ =2PM ¯¯¯¯ We get 2(??? |??? +???? ¯¯¯¯ )=???? ¯¯¯¯ +??? +???? ¯¯¯¯ +???? ¯¯¯¯ =0 (given) Therefore ?? is the midpoint of ?? ?? . It is also the midpoint of the line joining the midpoints of ???? and ???? . Example 2 A uniform circular plate of weight ?? is supported at three points in its edge whose distances apart are a, b, c. Find the load carried by each support. S (cicumcentre of, BC ) is the centre of the circuar plate. The weight ?? acts at ?? vertically downward. Let ?? 1 ,?? 2 ,?? 3 be the forces acting vertically upward on the plate at ?? ,?? , C respectively. The forces acting on the plate are in equilibrium. Taking moments of the forces about the line [???? , we get ?? 1 ???? =?? ·?? ?? ' ??? 1 :?? · ?? ?? ' ???? =?? · 2cos ?? 2sin ?? .(?? is circum radius =SA ) (? ?? sin ?? =2?? and ?? 2 =?? 2 +?? 2 -2???? ·cos ?? ) =?? ?? 2 2cos ?? ???? =?? ?? 2 ?? 2 ?? 2 (?? 2 +?? 2 -?? 2 ) = ?? ?? 2 ?? 2 ?? 2 ?? 2 [?? 2 (?? 2 +?? 2 -?? 2 )] Therefore Page 5 Edurev123 STATICS 1. Fundamental ideas, Statics deals with forces (on bodies) which produce equilibrium. If a single force ?? acting indepencently produces the same effect as that due to a number of forces ?? 1 ,?? 2 ,?? 3 , etc., acting simultaneously; then ??? is called the resustant of ??? 1 ,??? 2 ,??? 3 , etc. in other words ???? is ??? 1 +??? 2 +??? 3 +?, The forces ??? 1 ,??? 2 ,??? 3 , etc., are called components of ???? . The resultant of two forces :??? and ???. along ???? and ???? such that angle ?? ?? ?? is ?? . is of magnitude v?? 2 +?? 2 +2???? cos ?? . This is obtained by applying parallelogram law for addition of vectors. If two forces are in equilibrium, then they must be of equal magnitude and act in opposite directions along the same line. If three forces acting at a point are in equilibrium, they can be represented in magnitude and direction by the sides of a triangle taken in order. This is known as law of triangle of forces. The converse of this law is also true. From the law of triangle of forces we get Lami's, theorem. If three forces acting at a point, are in equilibrium, then each force is proportional to the sine of the angle between the other two and conversely. The magnitude of the resultant ?? of two like parallel forces ???? and ???? acting at ?? and ?? is sum of their magnitudes, that is ?? i. ?? , and the line of action of ???? (which is parallel to that of ?? and ?? ) divides ???? internally inversely in the ratio of the forces ?? and ?? . If two unlike parallel forces ???? and ???? act at ?? and ?? , the magnitude of their "resultant ???? is ?? Q and the tine of action of ???? divides ???? externally inversely in the ratio of the forces. The moment about a point ?? of a force ??? whose line of action passes through a point ?? is ???? ¯¯¯¯ ×??? =???×??? If a number of forces act at a point, the algebraic sum of their (rectangular) components along any direction is equal to the component of their resultant in the same direction; the vector sum of their moments about any point is equal to the moment of their resultant about the same point. It follows that if a number of forces acting at a point are in equilibrium the algebraic sum of their components along each of three mutually perpendicular directions must be zero or the vector sum of their components about each of three non-collinear points must be zero. Fic If three forces are in equilibrium, then they must be co-planar and they must be either concurrent or all of them must be parallel. The resultant of two forces acting at a point O along ???? and ???? and represented in magnitude by ?? : ?? A and ?? ,?? 3 is represented by (?? +?? )???? where ?? is a point on ???? such that ?? CA=?? ?? CB A useful trigonometrical result ?? is a point on the side ?? B of a triangle ?????? such that ???? :???? =?? :?? Then (?? +?? )cot?? =?? cot a-?? cot ?? (?? +?? )cot ?? =?? cos ?? -?? cot ?? Example 1 Forces acting in at a point I inside a quadrilateral ???????? represented by ???? ,???? ,???? and ?? D in magnitude and direction are in equilibrium. Find the position of ?? . Let ?? and ?? be the midpoint of ???? and ???? respectively. Then PB ¯¯¯¯ ·PC ¯¯¯¯ =2PN ¯¯¯¯ PD ¯¯¯¯ ·PA ¯¯¯¯ =2PM ¯¯¯¯ We get 2(??? |??? +???? ¯¯¯¯ )=???? ¯¯¯¯ +??? +???? ¯¯¯¯ +???? ¯¯¯¯ =0 (given) Therefore ?? is the midpoint of ?? ?? . It is also the midpoint of the line joining the midpoints of ???? and ???? . Example 2 A uniform circular plate of weight ?? is supported at three points in its edge whose distances apart are a, b, c. Find the load carried by each support. S (cicumcentre of, BC ) is the centre of the circuar plate. The weight ?? acts at ?? vertically downward. Let ?? 1 ,?? 2 ,?? 3 be the forces acting vertically upward on the plate at ?? ,?? , C respectively. The forces acting on the plate are in equilibrium. Taking moments of the forces about the line [???? , we get ?? 1 ???? =?? ·?? ?? ' ??? 1 :?? · ?? ?? ' ???? =?? · 2cos ?? 2sin ?? .(?? is circum radius =SA ) (? ?? sin ?? =2?? and ?? 2 =?? 2 +?? 2 -2???? ·cos ?? ) =?? ?? 2 2cos ?? ???? =?? ?? 2 ?? 2 ?? 2 (?? 2 +?? 2 -?? 2 ) = ?? ?? 2 ?? 2 ?? 2 ?? 2 [?? 2 (?? 2 +?? 2 -?? 2 )] Therefore ?? 1 ?? 2 (?? 2 +?? 2 -?? 2 ) = ?? 2 ?? 2 (?? 2 +?? 2 -?? 2 ) = ?? 3 ?? 2 (?? 2 +?? 2 -?? 2 ) = ?? 1 +?? 2 +?? 3 (?? +?? +?? )(?? +?? -?? ) (?? +?? -?? )(?? +?? -?? ) or ?? 1 = ?? ?? 2 (?? 2 +?? 2 -?? 2 ) (?? +?? +?? )(?? +?? -?? )(?? +?? -?? )(?? +?? -?? ) (??? 1 +?? 2 +?? 3 =?? ) Similar Expressions for ?? 2 and ?? 3 . Example 3 A rod of length (?? +?? ) whose centre of gravity is at a distance a from one end remains in (stable) - equilibrium in the vertical plane inside a fixed smooth isphere. Show that if ?? be the inclination of the rod to the horizontal then tan ?? =( ?? -?? ?? +?? )tan ?? , where 2?? is the angle subtended by thie rod at the centre of the sphere: The rod is acted by three forces, its weight W and the reactions ?? and ?? at its ends and is in equilibrium. Therefore (?? +?? )cot (90-?? )=?? cot ?? -?? cot ?? or (?? +?? )tan ?? =?? cot (90-?? )-?? cot (90-?? ) =(?? -?? )tan ?? or tan ?? =( ?? -?? ?? +?? )tan ?? Example 4 Three forces ?? ,?? ,?? act along the sides BC, CA,AB of a triangle ABC . The resultant ties along the line joining the centre of the circle inscribed in ?????? and the C.G. of theRead More
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FAQs on Statics - Mathematics Optional Notes for UPSC
| 1. What is the importance of studying statics in the UPSC exam? |  |
Ans. Studying statics is important for the UPSC exam as it helps in understanding the principles of equilibrium, forces, and moments which are essential in various subjects like physics, engineering, and economics that are covered in the exam.
| 2. How can statics help in solving problems related to civil services? | |
Ans. Statics provides a foundation for analyzing and solving complex problems that are often encountered in civil services. Understanding statics principles can help in making informed decisions, assessing risks, and designing effective strategies for public administration.
| 3. What are some common topics covered in the statics syllabus for the UPSC exam? | |
Ans. Some common topics covered in the statics syllabus for the UPSC exam include laws of motion, equilibrium of particles and rigid bodies, friction, centroids, moments of inertia, and analysis of trusses and frames.
| 4. How can practicing statics problems enhance performance in the UPSC exam? | |
Ans. Practicing statics problems can help in improving problem-solving skills, enhancing analytical thinking, and increasing familiarity with the types of questions that may appear in the UPSC exam. It also helps in building confidence and reducing exam-related stress.
| 5. Are there any online resources available for studying statics for the UPSC exam? | |
Ans. Yes, there are several online resources such as study materials, video lectures, practice tests, and forums that can help in studying statics for the UPSC exam. These resources provide additional support and guidance to supplement traditional study methods.
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