Mechanics
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Syllabus

#### Mechanics

MATHEMATICAL TOOLS

1. Variables, functions, angles, units of angles (Degree & radion) , conversion of units, Trigonometric ratios/ functions, values of trigonometric ratios, values of trigonometric ratios for angles grater than 90°,
2. unit circle method, CAST rule. Trigonometric formula, sine rule & cosine rule , Logrithem , exponential and inverse functions.
3. Coordinate geometry , slope of a line , equation of straight line, parabola , ellipse, circle and rtectangular hyperbola.
4. Differentiation, geometrical meaning of differentiation, slope of a line, formulae for differentiation, rules of differentiation- addition/subtraction rule, product rule, quotient rule, constant multiple rule, chain rule.
5. Higher order Differentiation , implicit functions , important problems .
6. Differentiation  as rate measurement, maxima & minima.
7. Integration, geometrical meaning of integration, formulae of integration,
8. Definite integration,  rules of integration, addition/ subtraction rule, method of substitution. Integration by parts, Integration as area under curve, indefinite integration , area under curve.
9. Introduction to vectors, null vector, unit vector, negative of a vector, graphical representation and mathematical representation of a vector, angle b/w two given vectors,
10.  Resolution of vector.Addition of vectors, triangle method and parallelogram method, substraction of vectors.
11. Dot product and its uses.
12. Cross product and its uses ,  right hand screw rule

RECTILINEAR MOTION

1. Rest & motion, distance & displacement, speed, average speed ,  time average and space average,  instantaneous speed, Uniform speed and non uniform speed,
2. velocity, average and instantaneous velocity, acceleration, average and instantaneous acceleration.
3. Equations of kinematics with constant acceleration, steps used to solve the problems based on equation of kinematics, motion under gravity.
4. graphical analysis, some important graphs, conversion of graphs, information collected from graphs.
5. Variable acceleration, steps used to solve the problems based on variable acceleration, when acceleration is dependent on time, distance and velocity.

PROJECTILE MOTION

1. Ground to ground projectile, time of flight, net velocity, trajectory equation, maximum height,
2.  horizontal range.Projection at complementary angles from ground, some important  relations and problems.
3. Problems based on ground to ground projectile.
4. (a) Projectile from tower projected horizontally, , time of flight, net velocity, trajectory equation, horizontal range

(b)  Projectile from tower projected above horizontal, time of flight, net velocity, trajectory equation, maximum height.  horizontal range

(c) Projectile from tower projected below horizontal. time of flight, net velocity, trajectory equation, horizontal range

1. Problem on projectiles from tower
2.       (a)Projectile from inclined plane, projected up the incline plane , time of flight, net

velocity, trajectory equation, maximum height. range

1.  Projectile from inclined plane, projected down the incline plane , time of flight, net velocity, trajectory equation, maximum height. Range
1. Problems based on projectile on incline plane.
2. Projectiles from moving platform, Collision of a projectile with vertical wall, some miscellaneous examples.

RELATIVE MOTION

1. Introduction to relative motion, one dimensional relative motion and two dimensional relative motion . Uses of equations of kinematics in 1D relative motion.
2. uses of equations of kinematics in 2D relative motion , Velocity of approach and velocity of separation in 1D, Velocity of approach and velocity of separation in 2D, condition for two particles to collide, minimum separation b/w two moving particle, time taken to come at minimum separation miscellaneous problems .
3. miscellaneous problems
4. River boat problem in one dimension.
5. River boat problem in two dimensions, direct crossing, minimum time taken to cross the river, minimum drift , minimum velocity
6. Wind-aeroplane problem. Rain man problem, some illustrations.

Newton’s laws of motion (NLM)

1. Force, fundamental forces, normal force, tension force, Newton’s lst law, 2nd law , and 3rd law, equation of motion, Inertia.
2. free body diagram ,Equilibrium, types of equilibrium, steps to solve the problems based on equilibrium, problems
3.  steps used to solve the problems of accelerated motion, problems , atwood machine
4.  Constrained motion, string constraint, displacement method, tension method, differentiation method, two block one pulley system,
5. constrained motion when string is inclined, wedge constraint.
6. Weighing machine, motion inside lift, apparent weight, weightlessness, spring balance , spring and spring force.
7. Reference frame, inertial frame and non-inertial frame, pseudo force, illustrations
8. Newton’s laws for system , problems

FRICTION

1. Introduction to friction, properties of friction. Kinetic friction,coefficient of kinetic friction.
2. Static friction, coefficient of static friction, self adjustable nature of static friction, driving force,    graph relating friction with driving force.
3. Contact force, angle of friction, minimum force required to slide a block , why pulling is easier than pushing?
4. Angle of repose, minimum and maximum force on the inclined plane so that block does not   move , graph
5. System of two blocks, steps used to check the slipping b/w two blocks, problems
6. System of three blocks and miscellaneous examples.

WORK POWER AND ENERGY

1. Introduction to work, definition of work, point of application of force. Calculation of work done when force is constant
2.  Sign of work done . work done by  variable force,
3. work done from force-displacement graph, work done by friction, normal and gravity
4. work done by  spring force.Work done by variable force  along given path, conservative and non-conservative forces
5. methods to identify conservative forces , Del-operator, curl, Potential energy, its definition, external agent,
6. relation b/w conservative force and potential energy, how to find P.E. if conservative force is given and vise-versa. Refrence line ,  gravitational Potential energy and spring potential energy
7. Equilibrium, types of equilibrium, stable, unstable and neutral equilibrium.
8. Kinetic energy , Work energy theorem, some examples.
9. Problems based on work energy theorem
10. Energy conservation, some examples, power, instantaneous power and average power.

CIRCULAR MOTION

1. Similarities b/w translational and rotational motion, angular displacement and its direction .
2. angular velocity and angular acceleration, equations of circular kinematics.
3. Relation b/w linear and rotational quantities, tangential acceleration centripetal/redial/normal acceleration. Total acceleration.
4. Time period , frequency , angular frequency , Problems
5. Radius of curvature of path, radius of curvature in projectile motion.
6. Types of circular motion, horizontal circular motion. Some important examples. Steps used to solve the problems based on circular dynamics. Vertical circular motion, some important examples.
7. Vertical circular motion of a ball attached with string , vertical circular motion of a ball attached with light rod.
8. Problems , Banking of roads with  and without friction.
9. Centrifugal force, its direction and magnitude. Some examples.

CENTER OF MASS

1. Center of Mas, definitions, Type of  mass distribution, discrete and continuous mass distribution, linear mass density, surface mass density, volume mass density. Calculation of com for system of particles. Com of system of two particles.
2. Calculation of com for continuous mass distribution, com of rod, semi-circular ring, semi-circular disc, solid hemi-sphere, hollow hemi-sphere, solid cone.
3. Com of a body with hole, problems
4. Motion of com, velocity of com, acceleration of com, impulsive force, impulse, impulse-momentum equation, important examples.Conservation of momentum, some important conclusions and examples.
5. Miscellaneous  problems
6. Some important points related to center of mass and miscellaneous problems.
7. Spring mass system, steps to solve  the problems based on spring-mass-system. Problems , Collision, line of impact, coefficient of restitution,
8. classification of collision, head-on-inelastic collision, head on elastic collision, head on-perfectly in elastic collision. Problems on collision.
9.  collision with heavy mass.   Oblique collision, problems
10. oblique collision with wall , problems
11. Variable mass, thrust force, rocket propulsion.

ROTATIONAL MOTION

1. Introduction, similarities b/w rotational and translational motion. Rigid body, types of motion of rigid body.
2. Moment of inertia definitions, calculation of MOI of a point mass, MOI of system of particles, MOI of a rod,
3. MOI of ring, MOI of disc, MOI of solid sphere, MOI of hollow sphere, MOI of cone, MOI of solid cylinder, MOI of hollow cylinder
4.  perpendicular axes theorem, parallel axes theorem. MOI of a body with hole,
5.  Radius of Gyration.Torque, Calculation of torque,
6. Force couple, point of application.
7. Rotational and translational equilibrium.
8. Rotational equation of motion, accelerated rotational motion. Some important examples.
9. Combined motion, rolling motion, slipping, skidding, perfect rolling,
10. Some important problems , trajectory of a point on wheel performing perfect rolling and radius of curvature of trajectory.
11.  instantaneous axis of rotation,  rotational K.E. , conversion of imperfect rolling to perfect rolling
12. Direction of friction in perfect rolling , Angular momentum, calculation  of angular momentum,
13. calculation  of angular momentum,
14. conservation of angular momentum in pure rotational motion , in pure translational motion  and in combined motion , angular impulse momentum equation.
15. Collision of a particle with rigid body
16. Toppling and sliding.