Work and Energy

 

Work and Energy Energy exists in various forms: There is mechanical energy, chemical energy, electrical energy,, heat energy energy energy,, nuclea nuclearr energy, a and nd so on. A ttransfor ransformatio mation n from one form to another may take place, but the total amount of energy is conserved or always remains the same.

Work Done by a Constant Force

The word work is commonly used in a variety of ways: We go to work. We work on proects. proec ts. We work on our desk desks s or on computers! W We e work on probl problems. ems. "n #hysic #hysics, s, work rk in invo volv lves es fo forc rce e an and d work wo rk ha has s a very sp spec ecif ific ic me mean anin ing. g. $e $ech chan anic ical ally ly,, wo displa disp lace ceme ment nt,, an and d we us use e th this is wo word rd to de desc scri ribe be qu quan anti tita tati tive vely ly wh what at is accomplished when a force moves an object through a distance !n the simplest simplest case of a constant force" #he work done by a constant force in moving an object is equal to the product of  the magnitudes of the displacement and the component of thte force parallel to the displacement

Work then involves moving moving an obect through a distance. A force may be applied but if  there the re is no mot motion ion %no disp displac laceme ement&, nt&, th then en no wor work k is done done.. 'or a cons constan tantt force ' acting in the same direction as the displacement, the work W is simply W   Fd =

and  F =mg

"n general, work doneiswhen force an obect through a at distance and some component of theisforce along athe linemoves of motion i.e. if the force is an angle ( to the obect)s displacement, then ' x * ' cos(. The more general formula for work is now W   Fd cosθ =

Work is measured in +ewtonmeter %+m& or oule %-&. The vertical component of the force ', which is ' y, does no work since there is no displacement in this direction. Work is a scalar quantity We commonly commonly specify wh what at is doing work on what. 'or example example,, the force of gravity does work on a falling falling obect. When you lif liftt an obect, yo you u do work on the obect. We

 

refer to this as doing work against gravity because the force of gravity is in the direction opposite that of the applied lift force, and opposes it. 'orces are variable variable i.e. they cha change nge with tim time e andor posit position. ion. 'or exampl example, e, a force applied to an obect to overcome the force of static friction may be increased until it exceeds f smax /owever,, the force of stat static ic fricti friction on does no work because ther there e is no smax. /owever motion of displacement. An example of a variable force doing work is stretching a spring. As a spring iis s stretched or compressed farther and farther, the restoring force of  the spring gets gets greater and an increasi increasing ng applied for force ce is re0uired. "n e0uati e0uation on form, this is expressed as  F  kx =

where x represents the compressor or extensor of the spring from its unstretched length len gth.. The k is a con consta stant nt of proport proportion ionali ality ty and is com common monly ly call called ed the spring constant expressed in +ewtons per meter %+m&. The work done by a uniformly varying force of the form ' * kx W 

=

1 2

kx

2

1ample #roblems: 2. A st student udent hol holds ds a #hysic #hysics s book whic which h has a mass of 2. 2.34 34 kg, out of a cl classroom assroom window until his arm is tired, and then releases it. %a& /ow much work is done on the book by the student in simply holding it out the window5 %b& /ow much work will have been done by the force of gravity during the time in which the book falls 6.7 m5 3. "f a garde gardener ner pus pushes hes a lawn lawnmow mower er on the gras grass s fiel field d with a constant constant fo force rce of  87.7 + at an angle of 92 degrees to the horiontal, how much work does he do in pushing it at a horiontal distance of ;.4 m5 6. A 7. 7.24 24 kg mass is susp suspende ended d from a verti vertical cal spr spring ing and des descen cends ds a distan distance ce of  9. 9.< < cm cm,, afte afterr wh whic ich h it han hang g s at res rest. t. An ad addi diti tion onal al 7.4 7.47 7 kg mass is then suspended suspend ed from th the e first. What is tthe he total ex extensi tension on of the spring. spring. +eglec +eglectt the mass of the spring. 9. A cr crane ane lilifts fts a 3. 3.7 7 metr metric ic ton lload oad a ver vertic tical al dis distan tance ce of 24. 24.7 7 m. "' th the e spee speed d of  the load is constant, how much work is done in lifting it5 4. A ttract ractor or exe exerts rts a co const nstant ant fo force rce of 4 4.7 .7 x 276 + on a horiontal chain while moving a load horiontal distance of 67 cm. /ow much work is done by the tractor5

 

ne form of energy that is closely associated associated with work is kinetic kinetic energy. energy. ?onside ?onsiderr an obect at rest on a frictionless frictionless surface. @et a horiontal force act on the the obect and set it in motion. motion. Work iis s done o on n the ob obect, ect, b but ut where does th the e work g go, o, so to speak5 "t goes into setting the obect into motion or changing its kinetic kinetic conditions. =ecause of its motion, we say that the obect has energy  kinetic energy, which gives it the capability to do work. Binetic energy is often called the energy of motion. "t is defined mathematically as one half of the product of the mass and the s0uare of the instant instantaneous aneous speed of a moving obect.  K 

1

=

2

mv

2

and is measured in oules %-&. The workenergy formula, depicted by a

v =

2

2

vo 2 x −

W = Fd = m

(  )  v

2

2

−v o

2 x

W net  ∆ K   K   K o =

=

−

 x  x

1 =

2

mv

1

2 −

2

2

m vo

work&energy work&en ergy theorem The above is called and relates workby done on an obecte0uation to the change in the its kinetic energy i.e. the net work donethe onnet a body an

 

external force is e0ual to the change in the kinetic energy of the body body.. Work and energy both have the units of oules and are both scalar 0uantities. 1ample #roblems: 2. A sh shuffl uffleboard eboard pl player ayer pushe pushes s a 7.34 kg puck, ini initiall tially y at rest, in a way tthat hat causes a constant horiontal force of