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					        Lecture VII


Work & Energy Principles
                               Introduction
   In previous lecture, Newton’s 2nd law (SF = ma) was applied to various
    problems of particle motion to establish the instantaneous relationship
    between the net force acting on a particle and the resulting acceleration
    of this particle. To get the velocity and displacement, the appropriate
    kinematics equations may be applied.
   There are two general classes of problems in which the cumulative
    effects of the unbalanced forces acting on a particle are of interest:
      1) Integration of the forces w.r.t. the displacement of the particle.
        This leads to the equations of work and energy.
      2) Integration of the forces w.r.t. the time. This leads to the
        equations of impulse and momentum.
   Incorporation of the results of these integrations directly into the
    governing equations of motion makes it unnecessary to solve directly for
    the acceleration.
              First: Work & Kinetic Energy
                                        Work
   Work done by the force F during the displacement
    dr is defined by:
                  dU = F . dr
   The magnitude of this dot product is:
                  dU = F ds cosa
    where, a= the angle between the applied force
    and the displacement, ds = the magnitude of the
    displacement dr, F cosa = the tangential
    component of the force, Ft. Thus,
                     dU  Ft  ds
                                                        Note: Work is a
   Work is defined by the displacement multiplied by
    the force component in the direction of that        scalar not a vector.
    displacement.
             First: Work & Kinetic Energy –
                         Cont.
                                      Work – Cont.
   Work is positive if Ft is in the direction of the
    displacement; and work is negative if Ft is in
    opposite direction to the displacement.
   Forces which do work are termed active forces,
    while constraint force which do no work are
    termed reactive forces.
   Work SI unit is Joule (J); where 1 J = 1 N.m
   During a finite movement of the point of
    application of a force, the force does an amount
    work equal to:
     U   Ft ds      or     U   Fx dx  Fy dy              x2
                                                        U12   Fdx   
                                                                x1        x1
                                                                              x2
                                                                                   kxdx   1k x2 2  x12 
                                                                                               2
                  First: Work & Kinetic Energy –
                              Cont.
                               Work – Kinetic Energy
     The kinetic energy T of the particle is defined as:                 Note: T is a scalar;
                                1                                         and it is always +ve
                             T  mv 2                                     regardless of the
                                2
                                                                          direction of v.
     It is the total work that must be done on the
      particle to bring it from a state of rest to a
      velocity v.
     Work and Kinetic Energy relation:
             2          2              2
    U1 2   F  d r   ma  d r   ma t ds
            1          1              1
           dv dv ds    dv
But at            v     at ds  vdv
           dt ds dt    ds
             v2
    U12   mv dv 
             v1
                            1
                            2
                                
                              m v2  v1
                                  2     2
                                                  U12  T2  T1  T      or    T1  U12  T2
            First: Work & Kinetic Energy –
                        Cont.
                                       Power
   Power is a measure of machine capacity; it is the time rate of doing work;
    i.e.                       dU        dr
                           P        F          P  F v
                                dt       dt
   Power is scalar and its SI unit is watt (W), where
     1 W = 1 J/s
     1 hp = 550 ft-lb/sec = 33,000 ft-lb/min
     1 hp = 746 W = 0.746 kW
   Mechanical Efficiency (em)
                                    em  Poutput Pinput         em < 1

   Other sources of energy loss cause an overall efficiency of,

                                 e  em ee et   Where, ee and et is the electrical and thermal
                                                efficiencies, respectively.
                Second: Potential Energy
           Gravitational Potential Energy Vg

Vg  mgh        or    Vg  Vg 2  Vg1  mg h2  h1   mg h


 Vg is positive, but Vg may be +ve or –ve.



                     Elastic Potential Energy Ve

           0
            x      1 2
     Ve   kx dx  kx
                   2
                                 or
                                                               1
                                                               2
                                                                  
                                              Ve  Ve2  Ve1  k x2  x1
                                                                    2     2
                                                                              
 Ve is positive, but Ve may be +ve or –ve.
            General Work and Energy
                    Equation
                                                                 Note: U’1-2 is
       U12  T  Vg  Ve   T  Vg  Ve   E            the work of all
                                                                 external forces
                                                                 other than
                                  or                             gravitational
                            
          T1  Vg1  Ve1  U12  T2  Vg 2  Ve2                and spring
                                                                 forces.

For problems where the only forces are the gravitational, elastic, and
nonworking constraint forces, the U’1-2-term is zero, and the energy equation
becomes:



                       E  0          or   E  Constant
Work and Energy Principles
        Exercises
                                Exercise # 1
3/103: The spring is unstretched when x = 0. If the body
   moves from the initial position x1 = 100 mm to the
   final position x2 = 200 mm, (a) determine the work
   done by the spring on the body and (b) determine the
   work done on the body by its weight.
                                    Exercise # 2
3/105: The 30-kg crate slides down the curved path in the
   vertical plane. If the crate has a velocity of 1.2 m/s down
   the incline at A and a velocity of 8 m/s at B, compute the
   work Uf done on the crate by friction during the motion
   from A to B.
                                    Exercise # 3
3/109: In the design of a spring bumper for a 1500-kg car, it
is desired to bring the car to a stop from a speed of 8 km/h in
a distance equal to 150 mm of spring deformation. Specify
the required stiffness k for each of the two springs behind the
bumper. The springs are undeformed at the start of impact.
                                   Exercise # 4
3/122: A department-store escalator handles a steady load
of 30 people per minute in elevating them from the first to the
second floor through a vertical rise of 7 m. The average person
has a mass of 65 kg. If the motor which drives the unit
delivers 3 kW, calculate the mechanical efficiency em of the
system .
                                         Exercise # 5
3/130: Each of the sliders A and B has a mass of 2 kg and moves with
   negligible friction in its respective guide, with y being in the vertical
   direction. A 20-N horizontal force is applied to the midpoint of the
   connecting link of negligible mass, and the assembly is released from
   rest with q = 0. Calculate the velocity vA with which A strikes the
   horizontal guide when q = 90°.
                                 Exercise # 6
3/131: The ball is released from position A with a velocity of
3 m/s and swings in a vertical plane. At the bottom position,
the cord strikes the fixed bar at B, and the ball continues to
swing in the dashed arc. Calculate the velocity vC of the ball
as it passes position C.
                                     Exercise # 7
3/143: The 0.9-kg collar is released from rest at A and slides
freely up the inclined rod, striking the stop at B with a velocity
v. The spring of stiffness k = 24 N/m has an unstretched length
of 375 mm. Calculate v.
                                    Exercise # 8
3/144: The 4-kg slider is released from rest at A and slides
with negligible friction down the circular rod in the vertical
plane. Determine (a) the velocity v of the slider as it reaches
the bottom at B and (b) the maximum deformation x of the
spring.
                                    Exercise # 9
3/158: If the system is released from rest, determine the
speeds of both masses after B has moved 1 m. Neglect
friction and the masses of the pulleys.

				
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