Transcript Lecture_2 - Department of Mathematics

USSC2001 Energy
Lecture 2 Kinetic Energy in Motion
Wayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Email [email protected]
http://www.math.nus/~matwml
Tel (65) 6874-2749
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MODERN CALCULUS
Although integral calculus (to compute areas and
volumes) had been invented (by Archimedes – and
independently in China and Japan), as well as
differential calculus (to compute tangents to curves),
the relation between integration and differentiation
that constitutes modern calculus was discovered
independently (but not without bitter disputes), by
Sir Isaac Newton Newton (1642-1727) and
Gottfried Wilhelm Leibniz (1646-1716)
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Newton.html
http://www.maths.tcd.ie/pub/HistMath/People/Leibniz/RouseBall/RB_Leibnitz.html
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NEWTON’S FIRST LAW
If no force acts on a body, then the body’s velocity
cannot change; that is, the body cannot accelerate.
Note: force is a vector quantity
– it has both magnitude and direction!
What happens if two or more people
pull on an object? This question leads
to the following more precise statement
If no net force acts on a body, then the body’s velocity
cannot change; that is, the body cannot accelerate.
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NEWTON’S SECOND LAW
The net force on a body is equal to the product of
the body’s mass and the acceleration of the body.


F  ma
Question: what constant horizontal force must
be applied to make the object below (sliding on
a frictionless surface) stop in 2 seconds?
v  6m / s
4
STATICS
Why is this object static (not moving) ?


mg
Hint: What are the forces acting on this object?
What is the net force acting on this object?
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VECTOR ALGEBRA FOR STATICS
The tension forces are

Fl 
 a cos θ 
 a sin θ 
The gravity force is



Fg 

Fr 
 0 
 mg
b cos 
 b sin  
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TUTORIAL 2
1. Compute the magnitudes a and b of the tensile
forces on vufoil 6.
2. Compute the mass of the object on the side of
the block below that has length 2m so that the
system is in equilibrium (there is no movement).
? kg
3kg
1m
2m
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TUTORIAL 2
3. Analyse the forces on an object that slides down
a frictionless inclined plane. What is the net force?
h
θ
Compute the time that it takes for an object with
initial speed zero to slide down the inclined plane.
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WORK-KINETIC ENERGY THEOREM
Consider a net force that is applied to an object
having mass m that is moving along the x-axis
The work done is
xf
W   F ( x ) dx
xi
Newton’s 2nd Law F ( x )
 ma( x)
dv dv dx
dv
Chain Rule a ( x) 

v
dt dx dt
dx
Kinetic Energy T ( x )  1 mv 2 ( x )
2
xf
W   m v d v  T ( x f )  T ( xi )
xi
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TEXAS ENERGY
Question Page 117 in the 2001, 6th edition of Fundamentals of Physics by D. Halliday,
R. Resnick and J. Walker has the following problem: "In 1896 in Waco Texas William
Crush of the 'Katy' railway parked two locomotives at opposite ends of a 6.4 km long
tack, fired them up, tied their throttles open, and allowed them to crash head on in front
of 30,000 spectators. Hundreds of people were hurt by flying debris; several were
killed. Assume that each locomotive weighed 1.2 million Newtons and that its
acceleration along the track was a constant a  0.26 m / s  s , what was the total
kinetic energy of the two locomotives just before the collision?
Answer The crash speeds v satisfied v 2  2a  3200 m and each had mass
1.2 106 N
5
m

1
.
22

10
Kg
2
9.8 m/s
so
E  2  12 mv 2  2 108 J
Question What does this event suggest about the value of force & power in Texas?
Question Can ‘work’ in the nontechnical sense of the word have a negative effect ?
Hint: Can two Americans, both born and educated in Connecticut and who lived in
Texas and then in the Washington DC area, have opposite effects on national security ?
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FALLING BODIES
Consider a particle thrown upward from the ground
h(t)
t
h(t 4 )
t2
t4
t
t3
t1
Question What happens? How can ‘gravitational
potential energy’ V be defined so that if T is
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kinetic energy then T + V is constant?
POTENTIAL ENERGY
Definition V is a potential energy function if
dV ( x)
F ( x)  
dx
and in that case we can also compute the work as
xf
W   F ( x) dx  V ( xi )  V ( x f )
xi
so the total energy
E  T V
is conserved since
E ( x f )  T ( x f )  V ( x f )  T ( xi )  V ( xi )  E ( xi )
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CONSERVATION OF ENERGY
gives a powerful method to solve physical problems.
The total energy of an object on a spring is
E  mx  kx
1
2
(where
2
1
2
x  dx / dt )
2
x0  x
therefore since dE/dt = 0
0  m x x  k x x  x   x
 x(t )  a cos ( t   )
k
m
R
a  2E k
amplitude
 k m
angular frequency T  2 period
phase

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TUTORIAL 2
4. (from Halliday, Resnick and Walker, p. 162)
A 60kg skier starts from rest at a height of 20 m
above the end of ski-jump ramp as shown below. As
the skier leaves the ramp, his velocity makes
an angle of 28 degrees with the horizontal. Ignoring
friction and air resistance, use conservation of energy
to compute the maximum height h of his jump?
20 m
end of ramp

28
h
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TUTORIAL 2
5. (from Halliday, Resnick and Walker, p. 164)
The potential energy of a diatomic molecule is
A B
V (r )  12  6 where r is the separation of the
r
r
two atoms of the molecule and A and B are positive
constants. This potential energy is associated with the
force that binds the two atoms together. (i) Find the
equilibrium separation, that is, the distance between
the atoms at which the force on each atom is zero. Is
the force repulsive or attractive if their separation is
(ii) smaller, (iii) larger than the equilibrium
separation?
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