Phys111_lecture07.ppt

Physics 111: Mechanics
Lecture 7
Dale Gary
NJIT Physics Department

4/25/2023
Potential Energy and
Energy Conservation
 Work
 Kinetic Energy
 Work-Kinetic Energy Theorem
 Gravitational Potential Energy
 Elastic Potential Energy
 Work-Energy Theorem
 Conservative and
Non-conservative Forces
 Conservation of Energy

4/25/2023
Definition of Work W
 The work, W, done by a constant force on an
object is defined as the product of the component
of the force along the direction of displacement
and the magnitude of the displacement
 F is the magnitude of the force
 Δ x is the magnitude of the
object’s displacement
 q is the angle between
x
F
W 
 )
cos
( q
and 
F x

4/25/2023
Work Done by Multiple Forces
 If more than one force acts on an object, then
the total work is equal to the algebraic sum of
the work done by the individual forces
 Remember work is a scalar, so
this is the algebraic sum
 
net by individual forces
W W
r
F
W
W
W
W F
N
g
net 



 )
cos
( q

4/25/2023
Kinetic Energy and Work
 Kinetic energy associated with the motion of
an object
 Scalar quantity with the same unit as work
 Work is related to kinetic energy
2
2
1
mv
KE 
2 2
0
1 1
( cos )
2 2
f
i
net
x
x
mv mv F x
d
q
  
 
 F r
   
net f i
W KE KE KE
Units: N-m or J

4/25/2023
Work done by a Gravitational Force
 Gravitational Force
 Magnitude: mg
 Direction: downwards to the
Earth’s center
 Work done by Gravitational
Force
2
0
2
2
1
2
1
mv
mv
Wnet 

cos
W F r q
   
F r
q
cos
r
mg
Wg 


4/25/2023
Potential Energy
 Potential energy is associated with the
position of the object
 Gravitational Potential Energy is the
energy associated with the relative
position of an object in space near the
Earth’s surface
 The gravitational potential energy
 m is the mass of an object
 g is the acceleration of gravity
 y is the vertical position of the mass
relative the surface of the Earth
 SI unit: joule (J)
mgy
PE 

4/25/2023
Reference Levels
 A location where the gravitational potential
energy is zero must be chosen for each
problem
 The choice is arbitrary since the change in the
potential energy is the important quantity
 Choose a convenient location for the zero
reference height
 often the Earth’s surface
 may be some other point suggested by the problem
 Once the position is chosen, it must remain fixed
for the entire problem

4/25/2023
Work and Gravitational
Potential Energy
 PE = mgy

 Units of Potential
Energy are the same
as those of Work and
Kinetic Energy
gravity i f
W KE PE PE PE
     
cos ( )cos180
( )
g f i
f i i f
W F y mg y y
mg y y PE PE
q
   
    

4/25/2023
Extended Work-Energy Theorem
 The work-energy theorem can be extended to include
potential energy:
 If we only have gravitational force, then
 The sum of the kinetic energy and the gravitational
potential energy remains constant at all time and hence
is a conserved quantity
net f i
W KE KE KE
   
f
i
gravity PE
PE
W 

gravity
net W
W 
f
i
i
f PE
PE
KE
KE 


i
i
f
f KE
PE
PE
KE 



4/25/2023
 We denote the total mechanical energy by
 Since
 The total mechanical energy is conserved and remains
the same at all times
PE
KE
E 

i
i
f
f KE
PE
PE
KE 


f
f
i
i mgy
mv
mgy
mv 

 2
2
2
1
2
1

4/25/2023
Problem-Solving Strategy
 Define the system
 Select the location of zero gravitational
potential energy
 Do not change this location while solving the
problem
 Identify two points the object of interest moves
between
 One point should be where information is given
 The other point should be where you want to find
out something

4/25/2023
Platform Diver
 A diver of mass m drops
from a board 10.0 m above
the water’s surface. Neglect
air resistance.
 (a) Find is speed 5.0 m
above the water surface
 (b) Find his speed as he hits
the water

4/25/2023
Platform Diver
 (a) Find his speed 5.0 m above the water
surface
 (b) Find his speed as he hits the water
f
f
i
i mgy
mv
mgy
mv 

 2
2
2
1
2
1
f
f
i mgy
v
gy 

 2
2
1
0
s
m
gy
v i
f /
14
2 

0
2
1
0 2


 f
i mv
mgy
s
m
m
m
s
m
y
y
g
v f
i
f
/
9
.
9
)
5
10
)(
/
8
.
9
(
2
)
(
2
2






4/25/2023
Spring Force
 Involves the spring constant, k
 Hooke’s Law gives the force
 F is in the opposite direction of
displacement d, always back
towards the equilibrium point.
 k depends on how the spring
was formed, the material it is
made from, thickness of the
wire, etc. Unit: N/m.
d
k
F





4/25/2023
Potential Energy in a Spring
 Elastic Potential Energy:
 SI unit: Joule (J)
 related to the work required to
compress a spring from its
equilibrium position to some final,
arbitrary, position x
 Work done by the spring
2
2
2
1
2
1
)
( f
i
x
x
s kx
kx
dx
kx
W
f
i



 
2
2
1
kx
PEs 
sf
si
s PE
PE
W 


4/25/2023
 The work-energy theorem can be extended to include
potential energy:
 If we include gravitational force and spring force, then
net f i
W KE KE KE
   
f
i
gravity PE
PE
W 

s
gravity
net W
W
W 

0
)
(
)
(
)
( 




 si
sf
i
f
i
f PE
PE
PE
PE
KE
KE
si
i
i
sf
f
f KE
KE
PE
PE
PE
KE 




sf
si
s PE
PE
W 


4/25/2023
 We denote the total mechanical energy by
 Since
 The total mechanical energy is conserved and remains
the same at all times
s
PE
PE
KE
E 


i
s
f
s PE
PE
KE
PE
PE
KE )
(
)
( 




2
2
2
2
2
1
2
1
2
1
2
1
f
f
f
i
i
i kx
mgy
mv
kx
mgy
mv 





4/25/2023
A block projected up a incline
 A 0.5-kg block rests on a horizontal, frictionless surface.
The block is pressed back against a spring having a
constant of k = 625 N/m, compressing the spring by
10.0 cm to point A. Then the block is released.
 (a) Find the maximum distance d the block travels up
the frictionless incline if θ = 30°.
 (b) How fast is the block going when halfway to its
maximum height?

4/25/2023
 Point A (initial state):
 Point B (final state):
m
cm
x
y
v i
i
i 1
.
0
10
,
0
,
0 





m
s
m
kg
m
m
N
mg
kx
d i
28
.
1
30
sin
)
/
8
.
9
)(
5
.
0
(
)
1
.
0
)(
/
625
(
5
.
0
sin
2
2
2
2
1





q
2
2
2
2
2
1
2
1
2
1
2
1
f
f
f
i
i
i kx
mgy
mv
kx
mgy
mv 




0
,
sin
,
0 


 f
f
f x
d
h
y
v q
q
sin
2
1 2
mgd
mgy
kx f
i 


4/25/2023
 Point A (initial state):
 Point B (final state):
m
cm
x
y
v i
i
i 1
.
0
10
,
0
,
0 





s
m
gh
x
m
k
v i
f
/
5
.
2
......
2




2
2
2
2
2
1
2
1
2
1
2
1
f
f
f
i
i
i kx
mgy
mv
kx
mgy
mv 




0
,
2
/
sin
2
/
?, 


 f
f
f x
d
h
y
v q
)
2
(
2
1
2
1 2
2 h
mg
mv
kx f
i 
 gh
v
x
m
k
f
i 
 2
2
m
m
d
h 64
.
0
30
sin
)
28
.
1
(
sin 

 
q

4/25/2023
Types of Forces
 Conservative forces
 Work and energy associated
with the force can be recovered
 Examples: Gravity, Spring Force,
EM forces
 Nonconservative forces
 The forces are generally
dissipative and work done
against it cannot easily be
recovered
 Examples: Kinetic friction, air
drag forces, normal forces,
tension forces, applied forces …

4/25/2023
Conservative Forces
 A force is conservative if the work it does on an
object moving between two points is
independent of the path the objects take
between the points
 The work depends only upon the initial and final
positions of the object
 Any conservative force can have a potential energy
function associated with it
 Work done by gravity
 Work done by spring force
f
i
f
i
g mgy
mgy
PE
PE
W 



2
2
2
1
2
1
f
i
sf
si
s kx
kx
PE
PE
W 




4/25/2023
Nonconservative Forces
 A force is nonconservative if the work it does
on an object depends on the path taken by the
object between its final and starting points.
 The work depends upon the movement path
 For a non-conservative force, potential energy can
NOT be defined
 Work done by a nonconservative force
 It is generally dissipative. The dispersal
of energy takes the form of heat or sound

 



 s
otherforce
k
nc W
d
f
d
F
W



4/25/2023
 The work-energy theorem can be written as:
 Wnc represents the work done by nonconservative forces
 Wc represents the work done by conservative forces
 Any work done by conservative forces can be accounted
for by changes in potential energy
 Gravity work
 Spring force work
net f i
W KE KE KE
   
c
nc
net W
W
W 

2
2
2
1
2
1
f
i
f
i
s kx
kx
PE
PE
W 



f
i
f
i
g mgy
mgy
PE
PE
W 



f
i
c PE
PE
W 


4/25/2023
 Any work done by conservative forces can be accounted
for by changes in potential energy
 Mechanical energy includes kinetic and potential energy
2
2
2
1
2
1
kx
mgy
mv
PE
PE
KE
PE
KE
E s
g 







)
(
)
( i
i
f
f
nc PE
KE
PE
KE
W 



)
(
)
( i
f
i
f
nc PE
PE
KE
KE
PE
KE
W 







PE
PE
PE
PE
PE
W i
f
f
i
c 






 )
(
i
f
nc E
E
W 


4/25/2023
Problem-Solving Strategy
 Define the system to see if it includes non-conservative
forces (especially friction, drag force …)
 Without non-conservative forces
 With non-conservative forces
 Select the location of zero potential energy
 Do not change this location while solving the problem
 Identify two points the object of interest moves between
 One point should be where information is given
 The other point should be where you want to find out something
2
2
2
2
2
1
2
1
2
1
2
1
i
i
i
f
f
f kx
mgy
mv
kx
mgy
mv 




)
(
)
( i
i
f
f
nc PE
KE
PE
KE
W 



)
2
1
2
1
(
)
2
1
2
1
( 2
2
2
2
i
i
i
f
f
f
s
otherforce kx
mgy
mv
kx
mgy
mv
W
fd 






 

4/25/2023
 A block of mass m = 0.40 kg slides across a horizontal
frictionless counter with a speed of v = 0.50 m/s. It runs into
and compresses a spring of spring constant k = 750 N/m.
When the block is momentarily stopped by the spring, by
what distance d is the spring compressed?
Conservation of Mechanical Energy
)
(
)
( i
i
f
f
nc PE
KE
PE
KE
W 



2
2
2
2
2
1
2
1
2
1
2
1
i
i
i
f
f
f kx
mgy
mv
kx
mgy
mv 




0
0
2
1
2
1
0
0 2
2




 mv
kd
cm
v
k
m
d 15
.
1
2


0
0
2
1
2
1
0
0 2
2




 mv
kd

4/25/2023
Changes in Mechanical Energy for conservative forces
 A 3-kg crate slides down a ramp. The ramp is 1 m in length and
inclined at an angle of 30° as shown. The crate starts from rest at the
top. The surface friction can be negligible. Use energy methods to
determine the speed of the crate at the bottom of the ramp.
)
2
1
2
1
(
)
2
1
2
1
( 2
2
2
2
i
i
i
f
f
f kx
mgy
mv
kx
mgy
mv 




)
0
0
(
)
0
0
2
1
( 2




 i
f mgy
mv
0
,
5
.
0
30
sin
,
1 


 i
i v
m
d
y
m
d 
s
m
gy
v i
f /
1
.
3
2 

?
,
0 
 f
f v
y
)
2
1
2
1
(
)
2
1
2
1
( 2
2
2
2
i
i
i
f
f
f
s
otherforce kx
mgy
mv
kx
mgy
mv
W
fd 






 

4/25/2023
Changes in Mechanical Energy for Non-conservative forces
top. The surface in contact have a coefficient of kinetic friction of 0.15.
Use energy methods to determine the speed of the crate at the bottom
of the ramp.
N
fk
)
2
1
2
1
(
)
2
1
2
1
( 2
2
2
2
i
i
i
f
f
f
s
otherforce kx
mgy
mv
kx
mgy
mv
W
fd 






 
)
0
0
(
)
0
0
2
1
(
0 2







 i
f
k mgy
mv
Nd

?
,
5
.
0
30
sin
,
1
,
15
.
0 



 N
m
d
y
m
d i
k


0
cos 
 q
mg
N
i
f
k mgy
mv
dmg 

 2
2
1
cosq

s
m
d
y
g
v k
i
f /
7
.
2
)
cos
(
2 

 q


4/25/2023
Changes in Mechanical Energy for Non-conservative forces
top. The surface in contact have a coefficient of kinetic friction of 0.15.
How far does the crate slide on the horizontal floor if it continues to
experience a friction force.
)
2
1
2
1
(
)
2
1
2
1
( 2
2
2
2
i
i
i
f
f
f
s
otherforce kx
mgy
mv
kx
mgy
mv
W
fd 






 
)
0
0
2
1
(
)
0
0
0
(
0 2







 i
k mv
Nx

?
,
/
7
.
2
,
15
.
0 

 N
s
m
vi
k

0

 mg
N
2
2
1
i
k mv
mgx 

 
m
g
v
x
k
i
5
.
2
2
2




4/25/2023
Block-Spring Collision
 A block having a mass of 0.8 kg is given an initial velocity vA = 1.2
m/s to the right and collides with a spring whose mass is negligible
and whose force constant is k = 50 N/m as shown in figure. Assuming
the surface to be frictionless, calculate the maximum compression of
the spring after the collision.
m
s
m
m
N
kg
v
k
m
x A 15
.
0
)
/
2
.
1
(
/
50
8
.
0
max 


0
0
2
1
0
0
2
1 2
2
max 



 A
mv
mv
2
2
2
2
2
1
2
1
2
1
2
1
i
i
i
f
f
f kx
mgy
mv
kx
mgy
mv 





4/25/2023
Block-Spring Collision
 A block having a mass of 0.8 kg is given an initial velocity vA = 1.2
m/s to the right and collides with a spring whose mass is negligible
and whose force constant is k = 50 N/m as shown in figure. Suppose
a constant force of kinetic friction acts between the block and the
surface, with µk = 0.5, what is the maximum compression xc in the
spring.
)
0
0
2
1
(
)
2
1
0
0
(
0 2
2







 A
c
k mv
kx
Nd

)
2
1
2
1
(
)
2
1
2
1
( 2
2
2
2
i
i
i
f
f
f
s
otherforce kx
mgy
mv
kx
mgy
mv
W
fd 






 
c
k
A
c mgx
mv
kx 


 2
2
2
1
2
1
c
x
d
mg
N 
 and
0
58
.
0
9
.
3
25 2


 c
c x
x m
xc 093
.
0


4/25/2023
Energy Review
 Kinetic Energy
 Associated with movement of members of a
system
 Potential Energy
 Determined by the configuration of the system
 Gravitational and Elastic
 Internal Energy
 Related to the temperature of the system

4/25/2023
Conservation of Energy
 Energy is conserved
 This means that energy cannot be created nor
destroyed
 If the total amount of energy in a system
changes, it can only be due to the fact that
energy has crossed the boundary of the
system by some method of energy transfer

4/25/2023
Ways to Transfer Energy
Into or Out of A System
 Work – transfers by applying a force and causing a
displacement of the point of application of the force
 Mechanical Waves – allow a disturbance to propagate
through a medium
 Heat – is driven by a temperature difference between
two regions in space
 Matter Transfer – matter physically crosses the
boundary of the system, carrying energy with it
 Electrical Transmission – transfer is by electric
current
 Electromagnetic Radiation – energy is transferred by
electromagnetic waves

4/25/2023
Connected Blocks in Motion
 Two blocks are connected by a light string that passes over a
frictionless pulley. The block of mass m1 lies on a horizontal surface
and is connected to a spring of force constant k. The system is
released from rest when the spring is unstretched. If the hanging
block of mass m2 falls a distance h before coming to rest, calculate the
coefficient of kinetic friction between the block of mass m1 and the
surface.
2
2
2
1
0 kx
gh
m
Nx
k 



 
PE
KE
W
fd s
otherforce 




 
h
x
mg
N 
 and
)
0
2
1
(
)
0
( 2
2 







 kx
gh
m
PE
PE
PE s
g
2
2
1
2
1
kh
gh
m
gh
m
k 


  g
m
kh
g
m
k
1
2
2
1




4/25/2023
Power
 Work does not depend on time interval
 The rate at which energy is transferred is
important in the design and use of practical
device
 The time rate of energy transfer is called power
 The average power is given by
 when the method of energy transfer is work
W
P
t



4/25/2023
Instantaneous Power
 Power is the time rate of energy transfer. Power
is valid for any means of energy transfer
 Other expression
 A more general definition of instantaneous
power
v
F
t
x
F
t
W
P 





v
F
dt
r
d
F
dt
dW
t
W
P
t












 0
lim
q
cos
Fv
v
F
P 





4/25/2023
Units of Power
 The SI unit of power is called the watt
 1 watt = 1 joule / second = 1 kg . m2 / s3
 A unit of power in the US Customary
system is horsepower
 1 hp = 550 ft . lb/s = 746 W
 Units of power can also be used to
express units of work or energy
 1 kWh = (1000 W)(3600 s) = 3.6 x106 J

4/25/2023
 A 1000-kg elevator carries a maximum load of 800 kg. A
constant frictional force of 4000 N retards its motion upward.
What minimum power must the motor deliver to lift the fully
loaded elevator at a constant speed of 3 m/s?
Power Delivered by an Elevator Motor
y
y
net ma
F 
,
0


 Mg
f
T
N
Mg
f
T 4
10
16
.
2 



W
s
m
N
Fv
P
4
4
10
48
.
6
)
/
3
)(
10
16
.
2
(





hp
kW
P 9
.
86
8
.
64 


Phys111_lecture07.ppt

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