General Physics I (22101) : Chapter 7
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Instructor : Dr. Iyad SAADEDDIN
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Chapter 7: Energy of a system
Systems and Environments
Work Done by a Constant Force
Scalar Vector Product
Work Done by a Variable Force
Kinetic Energy
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A System and and Its Environment Environment
Identify a system One ore more objects of space space A region of
Set a system boundary (not necessarily a physical boundary)
Everything else is the environment
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Work and Energy Transfer
A system s ystem (one or more mo re than th an one o ne obj ect) or surrounding has energy when they are able to produce a significant changes or activities on others things. The act of making these changes or activities is called “Work”
Work done by
Work done by the force F is defined as: The product of the displacement magnitude (?r) and the force component the direction of the displacement.
W F r cos The work done by the environment (force) or on the object ? is the angle angle betwee between n F and and ?r The work work is scale scalerr quanti quantity ty
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6 Ex: A man cl eaning a floor pulls a vacuum cleaner with a force of magnitude F =50 N at an angle angle of 30° with the horizontal if the vacuum cleaner is displaced 3 m to the right right .
Work done by
=F
W F r cos
=50N
30°
If ? = 90° 90°
If ? = 0° 0°
If ? = 180° 180°
F - ?r
F // // ? r (paral (paralle lel) l)
F and ?r are anti-paralel anti-paralelll
W= 0 J
W= F?r F?r (+ve (+ve work work))
W= -F?r (-ve (-ve work work))
No energy transfer
Energy transfer from environment environment (force) to system (object)
Energy transferred out of the system (from system to environment)
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3m
What is the work done by: a) the force force F (the man) man) b) The gravi gravity ty c) the normal normal force force 8 Ex: a 50 kg objec objectt pulled pulled 25m by by force F on on floor floor with with friction, if μk=0.1, find
Ex:
a) W F F r cos 13 0 J 50( 3) cos 30 130
a)
Work Work done done by each each force force
b)
Total Total work done on the the object object
k
W n
g
F r cos 80( 25) cos 30 1730 J
f k r cos 180 f k r k nr k (mg F sin 30) r 0.1(50)(9. 8)(25)
W F 0 J (n and Fg r ) cos 90 0
g
W F W f
b) and c)
W F 0 J (n and Fg r ) cos 90 0
a) W n
W T
W F W f W n W F 1730 1120 0 0 610 J k
g
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More on Work
The Dot Product of Two Vectors
Work is a scalar quantity produced from vectors (F and d ).
A B AB cos
It is the dot product (scalar product) of the force vector F and the displacement vector d.
A B B A
F
A BC
W Fd cos F d
d
ˆ A A x iˆ A y jˆ A z k
ˆ B B x iˆ B y jˆ B z k
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Ex:A particle particle moving in in the xy plane undergoes a displacement displacement Δr as a constant force force F acts on the particle. a)Displacem a)Displacement ent magni magnitude r = r = ? and Force magnitude F = F = ? b) Work done on the object W = ? c) W ork done in the y-direction y-direction =?
r 2.0iˆ 3.0 jˆ m N F 5.0ˆi 2.0 jˆ N
a)
r 2² 3² 13 F 5² 2 ² 29
b ) W F r 5( 2) 2(3) 16 J
c)
W y
2(3) 6 J
A B A C
A B A x B x
A y B y A z Bz A A A x A x A y A y A z A z A2
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Work Done by a Variable Force Consider force and displacement along the x-axis W F x x x f
x f
W lim F x
W lim
x 0
xi
x 0
x
xi
x f
W F x d x
Work is equal to the area under the F(x) vs. x vs. x graph graph
x i
For more than one force acting upon a particle
x f
W W net
F dx x
x i
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Ex: from from the grap graph h Calcul Calculate ate the work work done done to move move a particl particle e from from x=0 x=0 to x=6 x=6
A Spring Spring
W=area W=area under under graph graph
F s
=rectangle area +triangle area
kx
Hooke’s Law Spring Constant
W=4(5)+1/2 (2)(5) =20 + 5 = 25J
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Work Done by a Spring
Ex: A 0.55 kg object is attached to a vertical hanged spring, If the spring is stretched 2 cm, find a) spring constant b) the work done done by spring spring as it stretched m = 0.55 0.55 kg, d = 2 cm (y=-d=-2cm) (y=-d=-2cm)
For arbitrary displacement of spring between xi and xf x f
W s
x f
1
1
a) After After streach, streach, the object object is at equili equilibri brium um
F s dx kxdx W s kx kx x i
2
x i
2 i
2
2 f
When an external force is applied to compress or expand the spring from xi to xf
W F app
x f
x f
x f
xi
xi
xi
F appdx F s dx kx dx
W F app
1
1
2
2
kx f 2 kxi2
F s mg ky mg k ( d ) mg
FBD
k b) W s
mg 0. 55(9.8) d 2 10 2
2.7 102 N / m
1 kyi2 1 ky f 2 2
2
1
1
2
2
0 kd 2 (2.7 102 )(2 102 ) 2 5.4 102 J
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Work-Kinetic Energy Theorem
Work-Kinetic Energy Theorem 1
W 2 mv
Consider a block m oving under the influence of a net force force F ma r xiˆ x f xi iˆ
x f
W
x f
W m xi
1
2
The total work done by the force
K
1 2
mv 2
xi
dv dt
x f
dx m
W 2 mv
2 f
xi
1
dv dx dx dt
mvi 2 2
v f
W K f K i K
dx mvdv vi
The total work done by the force
Work-Kinetic Energy Theorem If there is only a change in particle speed, then net work done on the particle particle =Change in the kinetic energy of a particle particle
or K f K i 19 Ex:A 6 kg block initially at rest is pulled to the right along a horizontal, frictionless frictionless s urface urface by a constant horizontal force of 12 N. Find th e block’s speed after it has m oved 3 m. m = 6 kg, kg, vi=0 F = F = 12 N, x = x = 3 m v f = ?
W K K
f
F x F x
v f
1
mvi 2
The Kinetic Energy of the particle
F dx madx
xi
x f
2 f
2 F x m
2(12)(3) 6
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Potential Energy of a system (U)
Potential energy is Energy associated with the position of the object
Think of it as stored energy in a system that can do work or change the system’s kinetic energy.
K i
1 1 mv f 2 mvi2 2 2 1 2 mv f 0 2
3. 5m / s
W
Gravitational P.E. (Ug) Elastic P.E. (Us) Chemical, electrical, etc.
When work gets done on a system, its potential and/or kinetic energy increases.
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22 Ex:A 7kg bowli bowling ng ball slips from the bowler’s bowler’s hands hands (0.5m (0.5m from from ground) ground) and drops drops on the bowler’s bowler’s toe (3cm from from ground) ground).. Find the change change in gravitational gravitational potenti potential al energy energy
Gravitational Potential Energy To raise an obje object ct through through a displace displacement ment ?r by an an applie applied d force, force, you have to push against against Fg (mg) W F app
Fapp r
mg y
mg jˆ yb ya jˆ
b
U g U f U i mgy f mgyi mg ( y f yi ) 7(9.8)(0.03 0.5) 32.24 J
ya
W mgyb mgya
U b U a W U g U f U i
Hence, for an object raised a vertical distance distance y from some reference reference in gravity force field, its P.E. is
U g mgy Gravitational potential energy only depends on height (y) and not on distance (x).
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24 Ex: A 0.08 kg ball on spring compressed 5 cm, assume the spring spring force is 50N when pressed 3.5 cm, find the potential energy energy store d in the spring
Elastic Potential Energy The work done by the applied force force
W F app
1
1
2
2
U s kx f 2 kxi2
U s
1 2
kx
x = 5cm = 0.05m
2
k= ?? Fro From hook’ hook’s s law, law, we find find k
If xi =0 and xf =x The potential energy stored in t he spring
U s
1 2
kx 2
F s
kx F s kx
k
F s x
1
1
2
2
50 3.5 10 2
1430 N / m
U s kx 2 (1430)(5 1 0 2 )² 1.79 J
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Relationship between a conservative conservative force and potential energy
Conservative Conservative Forces
Conservative Forces:
W F
app app
The work done by conservative forces is independent of path; it depends on the displacement. The work done over a closed path is zero. We can always associate a potential energy with them; by pushing against them. Gravity force Fg (mg) and spring force (Fs) are examples.
Work done by a conservative force: W C U i
x f
W C F x dx U (U f U i ) xi
x f
U F x dx xi
U f U
If U f U and U i
0 then
x f
A forc e is nonconservative if it does does not not satisfy satisfy proper propertie ties s of conservative forces mechan mechanica icall energy energy is not not conserved part part of energy energy conver converts ts into interna internall energy energy ? Eint in form of thermal, thermal, sound, sound, etc. 27
U W C W C U
F x
U ( x ) F x dx
xi
For spring, U s
1 2 kx 2
F s
dU s dx
d ( kx ²) 2 dx
Review Work done by a constant force
W Fd cos F d x f
1
Work done by a variable force
W F x dx xi
kx
If you you draw U s versus versus x you will have have x=0 is called called stable stable equil equilibrium ibrium (object (object tends to go to x=0)
dx
Hence Hence from from the potenti potential al energy energy we can have the the force force 28
Relationship between a conservative conservative force and potential energy
dU
Hooke’s law Kinetic energy Work – kinetic energy energy theorem theorem
F s
kx
K
1 2
mv
2
W K
f
If you you U(x) U(x) graph graph is is as shown shown here here
Gravitational potential energy
U g
x=0 is called called unstable unstable equilibr equilibrium ium (obje (object ct tends tends to go away away from x=0) x=0)
Elastic potential energy
U s
K i K
mgy 1 2
kx 2
