Applied Mechanics PDF

Applied Mechanics 783 21 APPLIED MECHANICS 1. Basic Fundamental 2. Velocity & Projectile Motion 3. Friction 4. Force 5. Work, Power & Energy 784 Civil Booster (Civil Ki Goli Publication 9255624029)...

Applied Mechanics 783 21 APPLIED MECHANICS 1. Basic Fundamental 2. Velocity & Projectile Motion 3. Friction 4. Force 5. Work, Power & Energy 784 Civil Booster (Civil Ki Goli Publication 9255624029) Basic Fundamental 1  Units and Measurements  Vector and Scalar Quantity  Newton’s Law of Motion  Friction  Rectilinear Motion  Projectile Motion  Circular Motion  Simple Harmonic Motion. Fundamental and derived quantities.  Large number of physical quantities and every quantity needs a unit.  Fundamental quantities are only seven in number. Quantity SI Unit Symbol 1. Length meter m 2. Mass Kilogram kg 3. Time Second s 4. Electric Current Ampere A 5. Thermodynamic temperature Kelvin K 6. Amount of Substance Mole mol. 7. Luminous intensity Candela Cd. Special points: Two supplimentary unit. (i) Plane angle – radian rad. (ii) Solid angle – Staradian St.  Applied Mechanics 785 CGS System: Units of length, mass and time are centimetre (cm), gram (g) and second(s) respectively. Force – dyne Work – erg. FPS system: Units of length, mass and time are foot, pound and second. Force – poundal. SI Prefix: Power of 10 Prefix Symbol 6 mega M 3 Kilo K –2 Centi C –3 Mili m –6 Micro μ –9 nano n – 12 pico p Vector and scalar quantities  Any physical quantity is either a scalar or a vector. Scalar  Scalar quantity completely can be described by its magnitude only. e.g. Mass, Volume, density etc. Vector  A physical quantity in addition to magnitude it has a specified direction.  Obeys the law of parallelogram of addition eg: Displacement, velocity, acceleration etc. Note: Electric current in a wire shown by a direction but it is not a vector quantity. Reason: It does not obey the law of parallelogram of vector addition. 786 Civil Booster (Civil Ki Goli Publication 9255624029) Representation of Vector A  1. The parallelogram Law:    R = Resultant of A and B B    A Bsin  tan = A  Bcos  A sin  tan   B  A cos  According to parallelogram law  Magnitude of R is given by:  R = | R | = A 2  B2  2ABcos  if = 0°; R = A + B = Max. = 180°; R = A – B = Min. = 90°; R = A 2  B2  Applied Mechanics 787 2. The Triangle Law:  “If the tail of one vector be placed at the head of other, their sum R is drawn from the tail end of first to the head of other.    R AB Sine Rule: R B A P Q R    sin  sin  sin  P    R Q Special Points:  The law of parallelogram of the forces cannot be proved theoretically  The Triangle Law and the polygon Law of Forces are not fundamental laws. they are only derived laws. 788 Civil Booster (Civil Ki Goli Publication 9255624029) Velocity & Projectile Motion 2 Rectilinear Motion  Motion in a straight path. Distance and displacement: Distance:When a particle is moving its successive position in general may lie on a curve, the curve is then called as the path of the particle the total length of the path followed by the body is called the distance travelled by the body.  Scalar quantity. Displacement:  The directional distance between final and initial position of the particles.  It is a vector quantity. Speed  Speed is the rate at which a moving body describes its path. Path may be curve or straight line. s Average Speed, V = t If the interval of time t is infinitisimally small approaching to zero. This ratio is called instantenous speed. s ds Instantenous speed = lt  t 0 t dt  Speed is scalar quantity. Velocity:  Rate of change of position.  r Average velocity,  V t  Applied Mechanics 789     r dr Also, instantenous velocity, V  Lt  t 0 t dt  It is a vector quantity. Acceleration:  Rate of change of velocity.  The change in either magnitude or direction or both of them.   v a Average acceleration avg.  t Note: If an object moves along straight line without change in direction, in a given time interval. (i) It’s displacement and distance travelled are equal. (ii) It’s average speed and velocity are equal. Distance  |displacement| Speed  | Velocity| Uniform motion  Motion is uniform if, (i) Velocity is non zero. (ii) Acceleration is zero. (iii) Direction and velocity do not change.  An object moves along a straight line. Half the time with V1 and rest with V2. V1  V2 Average velocity = 2  While moving half distance with V1 and rest with V2 2V1V2 Average velocity = V  V 1 2 Relative Motion 1. A B Moving in opposite direction, relative speed = VA + VB  790 Civil Booster (Civil Ki Goli Publication 9255624029) 2. Moving along same direction, VA VB Relative Speed = |VA – VB| Resultant motion V2 V1 VR = V1 + V2 Resultant velocity of man wrt ground. V2 V1 VR = V2 – V1 Equation of Motion: Straight line with uniform acceleration. 1. V = u + at 1 2 2. S = ut  at 2  Applied Mechanics 791 3. V 2  u 2  2as  in a given interval of time initial velocity + final velocity V 2  Displacement = (Average velocity) × time Important point: 1. Average velocity during first t seconds– 1 = u  at 2 2. For a body starting from rest (u = 0) with uniform acceleration, the ratio of distances convered in ts, 2ts, 3ts etc is 1 : 4 : 9 etc. 3. A body starting from rest with uniform acceleration covers distances in the ratio 1 : 3 : 5 (odd numbers) in consecutive equal internal of time. 4. Distance traversed by the particle in the nth second of its motion: 1 Sn = u  a(2n  1) 2 Vertical motion under gravity. 1. Body released from rest– A point object is released from rest from a point at height h. 2h – Time taken to reach ground = g  Velocity with which it strikes ground = 2gh. 2. Body thrown vertically upward: A point object is thrown vertically upward with velocity u. u time of ascend = g u time of descend = g 2u time of flight = g 792 Civil Booster (Civil Ki Goli Publication 9255624029) V2 = u2 – 2gh, V = 0 at highest point u2 = 2ghmax u2 hmax = 2g Projectile Motion  It is two dimentional motion with constant acceleration. u sin  u ax = 0 u cos H = + Horizontal ay = –g  motion u cos  Vertical motion R (Ux)t = u cos (Vy)t = u sin– gt 1. Time of flight: The displacement along vertical direction is zero for the complete flight. 1 2  S = ut  gt 2 1  0  (u sin )t  gt 2 2 2u sin   t= g 2. Horizontal Range (R): R = ux.t 2u sin   R = u cos . g u 2 sin 2  R= g 3. Max. Height (H): At highest point velocity component equal to zero.  Applied Mechanics 793 V2 = u2 + 2as 0 = u 2 sin 2   2gH u 2 sin 2   H= 2g 4. Resultant velocity:  ˆ ˆ ˆ ˆ V = v x i  v y j  u cos i  (u sin   gt) j  |V| = u 2 cos 2   (u sin   gt) 2 and tan = Vy/Vx Important Points  For max. Range = 45° u2 Rmax = g  We get the same range for  and (90 – ) but in both cases, max. heights attained by the particles are different.  Equation of Trajactory: gx 2 y = x tan   2u 2 cos 2  Projectile throw Parallel to the horizontal from some height: u ux  uy v 2h 1. Time of flight (T): T = g 2h 2. Range (R): R  u g 3. Velocity at general point P(x, y): V= u 2x  v 2y ux = u , uy = gt (downward). 794 Civil Booster (Civil Ki Goli Publication 9255624029) Circular Motion  When a particle moves in plane such that its distance from a fixed point remains constant, motion with respect to that fixed point is called as circular motion. 1. Angular velocity (W): Wav = Average Angular velocity Angular displacement 2  1  Wav =   Total time taken t 2  t1 t  Angular displacement is scalar but angular velocity is Pseudo vector.  SI unit rad/s, dimention [T–1] 2  wav = (T-Time period) T  wav = 2f (f-frequency) 2n  wav = (n-rotations) t 2. Angular Acceleration:    w 2  w1 w av   t 2  t1 t If = 0; Motion is called uniform  Unit rad/s2  Dimention = [T–2].  Speed and angular velocity Relation    V  w r Position w.r.t. centre of circular motion. 3. Centripetal Force (Fc): At constant speed, the net force acting on the body is along the inside normal to the path of the body, called centripetal force. mv 2 Fc = ma c   mw 2 r r  Applied Mechanics 795 Ft Tangential force = max Fc C 4. Centrifugal Force:  Magnitude equal to centripetal force.  Always directed radialy outward. mv 2  r w mv2 r mg  When a body is rotating in circular path and the centripetal force vanishes, body would leave the circular path. Special Point: Lame’s constant E E  = 2(1  ) ,   (1  )(1  2)  Constants which relate stress to strain in an isotropic, elastic material.  Depends upon temperature and material. 796 Civil Booster (Civil Ki Goli Publication 9255624029) Friction 3  Comes into play between two surfaces whenever there is relative motion or a tendency of relative motion between them.  It opposes the relative motion between two surfaces in contact. Types of Friction Static Kinetic It acts between Surface in It acts between surface in contact but not in relative contact which are in relative motion, opposes the tendency motion. of relative motion It opposes the relative motion between surfaces. F  Dynamic friction (kinetic friction) is the friction between surfaces which acts between the body moves relative to one another.  In Static friction body tends to move when a force is applied on it (the bodies are not moving relative to each other).  Generally, the order of friction forces are: Limiting friction Frictional force Kinetic friction Static friction Applied force  Applied Mechanics 797 Limiting friction force > Maximum static friction force > Dynamic friction force Note Coefficient of friction is the ratio of friction force to the normal force which is acting to the normal of frictional force surface Law of Static friction  It is independent of area of contacting surfaces. f s max N f s max  µRN, (N-Normal reaction, µs- Co-efficient of static friction) 0  f s  N  When F exceeds fs block starts moving and frictional force decreases to a constant value fk. fk is kinematic friction. fk = µkN µk-Co-efficient of kinematic friction. Angle of friction  Angle made by the resultant force with the vertical is known as the angle of the friction in OAB, AB AB tan(90  )   cot   OB OB N A  O B f OB  OB   tan      tan 1   AB  AB  798 Civil Booster (Civil Ki Goli Publication 9255624029) Force 4 Newtons Law of Motion Force: A pull or push which changes or tends to change the state of rest or of uniform motion or direction of motion of any object is called force. It is a vector quantity. kg.m unit – (MKS) s2 g  cm dyne and (CGS) s2 1 N = 105 dyne. Effect of resultant force: 1. May change only speed. 2. May change only direction of motion. 3. May change both speed and direction of motion. 4. May change size and shape of body. Contact forces: Tension, Normal reaction, Friction etc. Forces that act between bodies in contact. Field Forces: Weight, electrostatic force etc. Forces that act between bodies separated by a distance without any actual contact. Contact forces: (i) Tension: When string, thread, wire or a spling is held tight, the ends of the string pull on whatever bodies are attached to them in the direction  Applied Mechanics 799 of the string. This force is known as tension. W = mg (ii) Normal:  Normal force is perpendicular to surface.  When two surface are in contact, then the surface exert forces on each other. (iii) Friction:  Force that acts between bodies in contact with each other. Special points: The forces whose lines of action is lie on the same plane are known as coplanar force. If lines of Action of forces in a system of force meet a point. Then these forces are called concurrent forces. Newton’s first Law Each body continues to be in its rest state or of uniform motion in a Straight line unless compelled by some external force to act otherwise.  The Newton’s first law leads to the definition of force Example: (i) A bullet fired on a glass window makes a clean hole through it while a stone breaks the whole of it. Bullet speed – high – large inertia of motion – cut clean hole. (ii) Passenger sitting in a bus gets a jerk when the bus starts or stops suddenly. Second Law of motion  The rate of change of momentum of a body is directly proportional to the applied force.  dp  F or F  ma dt   Where P  mv 800 Civil Booster (Civil Ki Goli Publication 9255624029) important points about second law:  Obviously consistent with the first law as F = 0 implies a = 0.  It is a vector law.  It is strictly applicable to a single point mass. Third law of motion.  Every Action has equal and opposite reaction. Ex: Rowing a boat, Hitting a wall, Walking, Jet Weighing Machine  A weighing machine does not measure the weight but the force exerted by object on its upper surface. Spring balance  It does not measure the weight.  Measures the force exerted by the object at the hook. = mg – T = 0 T = mg. m mg  Applied Mechanics 801 Work, Power & Energy 5 Work: Following two conditions must be fulfilled: (i) A force must be applied. (ii) The applied force must produce a displacement in any direction except perpendicular to the force direction. W=F.S dot product F Scalar quantity Note: The tension in the string of a simple pendulum is always perpendicular to displacement. So, Work done = zero. by tension T S F  S Case 1. If = 0; cos  = 1. So, work done is max. < 90°; cos = +ve; So, work done positive. = 90°; cos = 0; So, work done is zero. is obtuse, cos = – ve, So W is negative. Dimention and Unit: Unit: erg [CGS], Joule or N-m. 802 Civil Booster (Civil Ki Goli Publication 9255624029) 1 erg = 1 dyne × 1 cm = 1g cm s–2 × cm = 1g cm2 s–2 Dimention: – [ML2 T–2] Energy Intinal capacity of doing work. Kinetic energy: Internal capacity of doing work of the object by virtue of its motion. 1 2 KE = mv 2 p2 EK = ; P = momentum 2m EK = Kinetic energy. Work - Energy theorem According to this theorem, the workdone by all the forces on a particle is equal to the change in its kinetic energy. WC + WNC + WPS = K Work done by conservative force Work done by Non-conservative Work done by force psuedo forces Potential energy: Internal capacity to do work by virtue of relative motion. Ex. Gravitation P.E = mgh. Special Points: Centre of gravity of a body is the point through which the resultant gravitational force act Centre of gravity apply to the bodies with mass and weight Centroid is a point in a plane area such that the moment of area about any Axis through that point is zero. the term centroid applied to the plain areas When a rigid body is taken out in spacecraft its weight change