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JEE Revision Notes

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  • Work:-  Work done W is defined as the dot product of force F and displacement s.
     Notes | EduRev

    Here θ is the angle between Notes | EduRevand Notes | EduRev.
    Work done by the force is positive if the angle between force and displacement is acute (0°<θ<90°) as cos θ is positive. This signifies, when the force and displacement are in same direction, work done is positive. This work is said to be done upon the body.
  • When the force acts in a direction at right angle to the direction of displacement (cos90° = 0), no work is done (zero work).
     Notes | EduRev
  • Work done by the force is negative if the angle between force and displacement is obtuse (90°<θ<180°)  as cosθ is negative. This signifies, when the force and displacement are in opposite direction, work done is negative. This work is said to be done by the body.
  • Work done by a variable force:-
    If applied force F is not a constant force, then work done by this force in moving the body from position A to B will be,
     Notes | EduRev
    Here ds is the small displacement.
  • Units:  The unit of work done in S.I is joule (J) and in C.G.S system is erg.
    1J = 1 N.m , 1 erg = 1 dyn.cm
  • Relation between Joule and erg:- 1 J = 107 erg
  • Power:-The rate at which work is done is called power and is defined as,
    P = W/t = F.s/v = F.v
    Here s is the distance and v is the speed.
  • Instantaneous power in terms of mechanical energy:- P = dE/dt
  • Units: The unit of power in S.I system is J/s (watt) and in C.G.S system is erg/s.
  • Energy:-
    1) Energy is the ability of the body to do some work. The unit of energy is same as that of work.
    2) Kinetic Energy (K):- It is defined as,
    K= ½ mv2
    Here m is the mass of the body and v is the speed of the body.
  • Potential Energy (U):- Potential energy of a body is defined as, U = mgh
    Here, m is the mass of the body, g is the free fall acceleration (acceleration due to gravity) and h is the height.
  • Gravitational Potential Energy:- An object’s gravitational potential energy U is its mass m times the acceleration due to gravity g times its height h above a zero level.
    In symbol’s,
    U = mgh
  • Relation between Kinetic Energy (K) and momentum  (p):-
    K = p2/2m
  • If two bodies of different masses have same momentum, body with a greater mass shall have lesser kinetic energy.
  • If two bodies of different mass have same kinetic energy, body with a greater mass shall have greater momentum.
  • For two bodies having same mass, the body having greater momentum shall have greater kinetic energy.
  • Work energy Theorem:- It states that work done on the body or by the body is equal to the net change in its kinetic energy .
  • For constant force,
    W = ½ mv– ½ mu2
    = Final K.E – Initial K.E
  • For variable force,
     Notes | EduRev
  • Law of conservation of energy:- It states that, “Energy can neither be created nor destroyed. It can be converted from one form to another. The sum of total energy, in this universe, is always same”.
  • The sum of the kinetic and potential energies of an object is called mechanical energy. So, E = K+U
  • In accordance to law of conservation of energy, the total mechanical energy of the system always remains constant.
    So, mgh + ½ mv2 = constant
    In an isolated system, the total energy Etotal of the system is constant.
    So, E = U+K = constant
    Or, Ui+Ki = Uf+Kf
    Or, ?U = -?K
    Speed of particle v in a central force field:
    v = √2/m [E-U(x)]
  • Conservation of linear momentum:-
    In an isolated system (no external force ( Fext = 0)), the total momentum of the system before collision would be equal to total momentum of the system after collision.
     Notes | EduRevSo, p= pi
  • Coefficient of restitution (e):- It is defined as the ratio between magnitude of impulse during period of restitution to that during period of deformation.
    e = relative velocity after collision / relative velocity before collision
    = v– v1/u1 – u2
    Case (i) For perfectly elastic collision, e = 1. Thus, v2 – v1 = u– u2. This signifies the relative velocities of two bodies before and after collision are same.
    Case (ii) For inelastic collision, e<1. Thus, v2 – v1 < u1 – u2. This signifies, the value of e shall depend upon the extent of loss of kinetic energy during collision.
    Case (iii) For perfectly inelastic collision, e = 0. Thus, v2 – v=0, or v2 = v1. This signifies the two bodies shall move together with same velocity. Therefore, there shall be no separation between them.
  • Elastic collision:- In an elastic collision, both the momentum and kinetic energy conserved.
  • One dimensional elastic collision:-
     Notes | EduRev
    After collision, the velocity of two body will be,
    v1 = (m1-m2/ m1+m2)u1 + (2m2/ m1+m2)u2
    and
    v2 = (m2-m1/ m1+m2)u+ (2m1/ m1+m2)u1
    Case:I
    When both the colliding bodies are of the same mass, i.e., m1 = m2, then,
    v1 = u2 and v2 = u1
    Case:II
    When the body B of mass m2 is initially at rest, i.e., u2 = 0, then,
    v1 = (m1-m2/ m1+m2)u1 and v= (2m1/ m1+m2)u1
    (a) When  m2<<m1, then, v1 = u1 and v2 = 2u1
    (b) When  m2=m1, then, v1 =0  and v2 = u1
    (c) When  m2>>m1, then, v= -u1 and v2 will be very small.
  • Inelastic collision:- In an inelastic collision, only the quantity momentum is conserved but not kinetic energy.
    v = (m1u1+m2u2) /(m1+m2)
    and
    loss in kinetic energy, E = ½ m1u12+ ½ m2u22 - ½ (m1+ m2)v2
    or,
    E= ½ (m1u12 + m2u22) – ½ [(m1u1+ m2u2)/( m1+ m2)]2
    = m1 m2 (u1-u2)/ 2( m1 + m2)
  • Points to be Notice:-
    (i) The maximum transfer energy occurs if m1= m2
    (ii) If Kis the initial kinetic energy and Kf is the final kinetic energy of mass m1, the fractional decrease in kinetic energy is given by,
    K– Kf / Ki = 1- v12/u21
    Further, if m= nm1 and u2 = 0, then,
    Ki – Kf / K= 4n/(1+n)2
  • Conservation Equation:
    (i)  Momentum – m1u1+m2u2 = m1v1+m2v2
    (ii) Energy – ½ m1u12+ ½ m2u22 = ½ m1v12+ ½ m2v22
  • Conservative force (F):- Conservative force is equal to the negative gradient of potential V of the field of that force. This force is also called central force.
    So,  F = - (dV/dr)
  • The line integral of a conservative force around a closed path is always zero.
    So,
     Notes | EduRev
  • Spring potential energy (Es):- It is defined as,
     Notes | EduRev
    E= ½ kx2
    Here k is the spring constant and x is the elongation.
  • Equilibrium Conditions:
    (a) Condition for equilibrium, dU/dx = 0
    (b) For stable equilibrium,
    U(x) = minimum,
    dU/dx = 0,
    d2U/dx2 = +ve
    (c) For unstable equilibrium,
    U(x) = maximum
    dU/dx = 0
    d2U/dx2 = -ve
    (d) For neutral equilibrium,
    U(x) = constant
    dU/dx = 0
    d2U/dx= 0

  • UNITS AND DIMENSIONS OF WORK, POWER AND ENERGY
    Work and Energy are measured in the same units. Power, being the rate at which work is done, is measured in a different unit.
    Quantity and  Units/Dimensions  
    Work (Energy)
    Power
    Dimension
    ML2T-2
    ML2T-3
    Absolute
    MKS
    Joule
    Watt
    FPS
    ft-Poundal
    ft-poundal/sec
    CGS
    erg
    Erg/sec.
    Gravitational
    MKS
    kg-m
    Kg-m/sec
    FPS
    ft-lb
    ft-lb/sec.
    CGS
    gm-cm
    Gm-cm/sec
    Practical
    (Other)
     
    kwh, eV, cal
    HP, kW, MW
  • Conversions between Different Systems of Units
    1 Joule = 1 Newton ´ 1 m = 105 dyne ´ 102 cm = 107 erg
    1 watt = 1 Joule/ sec = 107 erg/sec.
    1 kwh  = 103 watt ´ 1 hr  = 103 watt ´ 3600 sec
    = 3.6 ´ 106 Joule
    1HP = 746 watt.
    1 MW = 106 watt.
    1 cal = 1 calorie = 4.2 Joule
    1eV = "e" Joule  = 1.6 ´ 10-19 Joule
    (e = magnitude of charge on the electron in coulombs)
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