Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com Mechanics Force System When a member of forces simultaneously acting on the body, it is known as force system. A force system is a collection of forces acting at specified locations. Thus, the set of forces can be shown on any free body diagram makes-up a force system. Truss It is a rigid structure composed of number of straight members pin jointed to each other. It can sustain static or dynamic load without any relative motion to each other. Types of Truss. Plane Truss It is defined as a truss in which members are essentially lies in a single plane.. igid Truss igid means there is no deformation take place due to internal strain in members. 3. Simple Truss This type of trusses built a basic triangle by adding different members are known as simple truss. Classification of a Truss (based on joints) Truss can be classified on the basis of joints (j) and members (m) in the structure. It can be easily understood with the help of following hierarchical approach. D:\Book Final\G5 Handbook Mechanical Engineering\Mechanical Engineering\Chap_0\Chap_.vp Thursday, October 4, 03 4:49:35 AM Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com
Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com Handbook Mechanical Engineering Truss Perfect Truss m = j 3 Imperfect Truss (unstable truss) n j 3 where m number of members and j number of joints Key Points edundant Truss m > = j 3 Analysis of a Framed Structure (Section Method). This method is used when the forces in the few members of a truss is required to found out in a truss structure. To find out force in AC, BC and BD First cut a section which passes through AC, BC and BD members. Find out reaction at point A and B Deficient Truss m < = j 3 m = 6, j = 4 m = 4, j = 4 6 > 4 3 4 < 4 3 6 > 5 4 < 5 Classification of truss A C F B D Framed structure using section method Find out forces FCA, FCB and F DB in membersca, CBand DB respectively by taking moment about A and B. E D:\Book Final\G5 Handbook Mechanical Engineering\Mechanical Engineering\Chap_0\Chap_.vp Thursday, October 4, 03 4:49:35 AM Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com
Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com Mechanics 3. Analytical Method In this method, the free body diagram of each joint is separately analysed to find magnitude of stresses in the truss members. Lami s Theorem Friction Force It is resistant force which acts in opposite direction at the surface in body which tend to move or its move. Normal force mg If mg F, the body will not move. mg F the body will tend to move. mg F the body will move. Angle of Friction Q It is defined as the angle between normal reaction and resultant reaction when the body is in condition of just sliding. esultant mg tan reaction φ mg (Q mg mg ) F F µmg P tan coefficient of friction γ β Three concurrent forces P, Q and α µ F mg Friction force on a body mg Angle of friction due to resultant reaction D:\Book Final\G5 Handbook Mechanical Engineering\Mechanical Engineering\Chap_0\Chap_.vp Thursday, October 4, 03 4:49:35 AM 3 Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 3
Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 4 Handbook Mechanical Engineering Angle of epose ( ) It is defined as angle of inclined plane with horizontal at which body is in condition of just sliding. Angle of friction is equal to angle of repose. Plane Motion When all parts of the body move in a parallel planes then a rigid body said to perform plane motion. Key Points Straight Line Motion It defines the three equations with the relationship between velocity, acceleration, time and distance travelled by the body. In straight line motion, acceleration is constant. v u at s ut at v u as where, u initial velocity v final velocity a acceleration of body t time s distance travelled by body Distance travelled in nth second sn u a ( n ) w sin α α α w cos α w Angle of epose ( ) µ D:\Book Final\G5 Handbook Mechanical Engineering\Mechanical Engineering\Chap_0\Chap_.vp Thursday, October 4, 03 4:49:35 AM 4 Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 4
Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com Mechanics 5 Projectile Motion Projectile motion defines that motion in which velocity has two components, one in horizontal direction and other one in vertical direction. Horizontal component of velocity is constant during the flight of the body as no acceleration in horizontal direction. Let the block of mass is projected at angle from horizontal direction Maximum height h where, u initial velocity Key Points max u sin g u Time of flight T sin g ange u sin g otational Motion with Uniform Acceleration Uniform acceleration occurs when the speed of an object changes at a constant rate. The acceleration is the same over time. So, the rotation motion with uniform acceleration can be defined as the motion of a body with the same acceleration θ over time. Let the rod of block rotated about a point in horizontal plane with angular otational motion velocity. Angular velocity d (change in angular displacement per unit time) dt Angular acceleration d dt where angle between displacement. d dt θ u h max range Projectile motion D:\Book Final\G5 Handbook Mechanical Engineering\Mechanical Engineering\Chap_0\Chap_.vp Thursday, October 4, 03 4:49:35 AM 5 Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 5
Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 6 Handbook Mechanical Engineering In case of angular velocity, the various equations with the relationships between velocity, displacement and acceleration are as follows. t 0 0 t 0t t where 0 initial angular velocity final angular velocity angular acceleration angular displacement Angular displacement in nth second n 0 ( n ) 0 elation between Linear and Angular Quantities There are following relations between linear and angular quantities in rotational motion. e r e t e r and e t are radial and tangential unit vector. Linear velocity v r e t Linear acceleration (Net) Tangential acceleration a a r e r dv dt e t t Centripetal acceleration ar r v r Net acceleration a a a where a r centripetal acceleration a t tangential acceleration dv (rate of change of speed) dt r t v d v r dt Y O e t θ e r Position of radial and tangential vectors X ( Q v r ) D:\Book Final\G5 Handbook Mechanical Engineering\Mechanical Engineering\Chap_0\Chap_.vp Thursday, October 4, 03 4:49:35 AM 6 Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 6
Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com Mechanics 7 Centre of Mass of Continuous Body Centre of mass of continuous body can be defined as Centre of mass about x, x Centre of mass about y, y Centre of mass about z, z CM CM CM xdm dm y dm dm z dm dm x dm M y dm M z dm CM of uniform rectangular, square or circular plate lies at its centre. CM of semicircular ring CM of semicircular disc CM of hemispherical shell CM of solid hemisphere O O O Law of Conservation of Linear Momentum CM CM CM The product of mass and velocity of a particle is defined as its linear momentum (p). p mv p Km M π 4 3π 3 8 D:\Book Final\G5 Handbook Mechanical Engineering\Mechanical Engineering\Chap_0\Chap_.vp Thursday, October 4, 03 4:49:35 AM 7 Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 7
Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 8 Handbook Mechanical Engineering p F d dt where, K kinetic energy of the particle F net external force applied to body P momentum ocket Propulsion Let m 0 be the mass of the rocket at time t 0, m its mass at any time t and v its velocity at that moment. Initially, let us suppose that the velocity of the rocket is u. u u At t = 0 v =u m=m 0 Thrust force on the rocket F v t r dm dt dm where, rate at which mass is ejecting dt v r relative velocity of ejecting mass (exhaust velocity) Weight of the rocket w mg Net force on the rocket Fnet Ft w v Net acceleration of the rocket F a m d v vr dm g dt m dt where, m 0 mass of rocket at time t 0 m mass of rocket at time t m0 v u gt vr ln m r At t= t m =m v =v exhaust velocity =v r ocket propulsion dm mg dt D:\Book Final\G5 Handbook Mechanical Engineering\Mechanical Engineering\Chap_0\Chap_.vp Thursday, October 4, 03 4:49:35 AM 8 Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 8
Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com Mechanics 9 Impulse The product of constant force F and time t for which it acts is called the impulse (J) of the force and this is equal to the change in linear momentum which it produces. Impulse J F t p pf pi where, F constant force P linear momentum Instantaneous Impulse e.g., bat and ball contact Key Points Collision J F dt p p p A Collision is an isolated event in which two or more moving bodies exert forces on each other for a relatively short time. Collision between two bodies may be classified in two ways Head-on collision Oblique collision. Head-on Collision Let the two balls of masses m and m collide directly with each other with velocities v and v in direction as shown in figure. After collision the velocity become v and v along the same line. m v m v Before collision m em v m m v m em m m v f m m v v After collision i D:\Book Final\G5 Handbook Mechanical Engineering\Mechanical Engineering\Chap_0\Chap_.vp Thursday, October 4, 03 4:49:35 AM 9 Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 9
Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 0 Handbook Mechanical Engineering m em v m m v m em m m v where, m mass of body m mass of body v velocity of body v velocity of body v velocity of body after collision v velocity of body after collision where e coefficient restitution Separation speed e Approach speed v v e v v In case of head-on elastic collision e In case of head-on inelastic collision 0 e In case of head-on perfectly inelastic collision e 0 If e is coefficient of restitution between ball and ground, then after nth collision with the floor, the speed of ball will n remain e v and it will go upto a height e h. n 0 v e v e gh n n n h e h Oblique Collision 0 n 0 In case of oblique collision linear momentum of individual particle do change along the common normal direction. No component of impulse act along common tangent direction. So, linear momentum or linear velocity remains unchanged along tangential direction. Net momentum of both the particle remain conserved before and after collision in any direction. v h O u = 0 O u = gh 0 Collision of a ball with floor Oblique collision y x D:\Book Final\G5 Handbook Mechanical Engineering\Mechanical Engineering\Chap_0\Chap_.vp Thursday, October 4, 03 4:49:35 AM 0 Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 0
Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com Mechanics Moment of Inertia Momentum of inertia can be defined as r distance of the body of mass, m from centre of axis. Very thin circular loop (ring) I m r I i i i r dm I M where, M mass of the body radius of the ring I moment of inertia Uniform circular loop I M M Uniform solid cylinder I A Uniform circular loop A Uniform solid cylinder A Thin circular ring D:\Book Final\G5 Handbook Mechanical Engineering\Mechanical Engineering\Chap_0\Chap_.vp Thursday, October 4, 03 4:49:35 AM Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com
Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com Handbook Mechanical Engineering Uniform solid sphere Uniform thin rod I M 5 (AA ) moment of inertia about the centre and perpendicular axis to the rod moment of inertia about the one corner point and perpendicular (BB ) axis to the rod. Ml I Very thin spherical shell Thin circular sheet M I 4 A A Uniform solid sphere l I Ml 3 A Uniform thin rod Thin sperical shell I M 3 Thin circular sheet D:\Book Final\G5 Handbook Mechanical Engineering\Mechanical Engineering\Chap_0\Chap_.vp Thursday, October 4, 03 4:49:35 AM Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com
Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com Mechanics 3 Thin rectangular sheet Uniform right cone Uniform cone as a disc I M a b I 3 M 0 Suppose the given section is th part of the disc, then mass of disc will n be nm. O Inertia of the disc, Inertia of the section, I Idisc ( nm) section a A Thin rectangular sheet A M Uniform right cone A part of uniform cone as a disc b n I disc M D:\Book Final\G5 Handbook Mechanical Engineering\Mechanical Engineering\Chap_0\Chap_.vp Thursday, October 4, 03 4:49:36 AM 3 Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 3
Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 4 Handbook Mechanical Engineering Torque and Angular Acceleration of a igid Body For a rigid body, net torque acting I where, angular acceleration of rigid body I moment of inertia about axis of rotation Kinetic energy of a rigid body rotating about fixed axis KE I Angular moment of a particle about same point L r p L m( r v) where L angular displacement ( angular velocity) Angular moment of a rigid body rotating about a fixed axis. L I A B Angular moment of a rigid body Angular moment of a rigid body in combined rotation and translation L LCM M ( r0 v0) Conservation of angular momentum d L dt dl r F v p dt r 0 O CM O ω Combined rotation and translation in a rigid body ω r P V 0 θ r p = mv Angular moment D:\Book Final\G5 Handbook Mechanical Engineering\Mechanical Engineering\Chap_0\Chap_.vp Thursday, October 4, 03 4:49:36 AM 4 Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 4
Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com Mechanics 5 Kinetic energy of rigid body in combined translational and rotational motion Uniform Pure olling K mvcm I CM Pure rolling means no relative motion or no slipping at point of contact between two bodies. If v v P Q v if v v p Q v if v v P CM Q v no slipping forward slipping backward slipping Accelerated Pure olling ω v ω v s = π Pure olling No slipping s Forward slipping s Backward slipping s A pure rolling is equivalent to pure translation and pure rotation. It follows a uniform rolling and accelerated pure rolling can be defined as ω F f Ma ( F f ) I P Q Uniform Pure olling v D:\Book Final\G5 Handbook Mechanical Engineering\Mechanical Engineering\Chap_0\Chap_.vp Thursday, October 4, 03 4:49:36 AM 5 Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 5
Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 6 Handbook Mechanical Engineering α F C a F force acting on a body, f friction on that body Angular Impulse The angular impulse of a torque in a given time interval is defined as t t t t dt dt L L where, L and L are the angular momentum at time t and t respectively. Key Points f Accelerated pure rolling D:\Book Final\G5 Handbook Mechanical Engineering\Mechanical Engineering\Chap_0\Chap_.vp Thursday, October 4, 03 4:49:36 AM 6 Get Discount Coupons for your Coaching institute and FEE Study Material at www.pickmycoaching.com 6