Transcript Lecture_1 - Department of Mathematics

USSC2001 Energy Lecture 1
Potential and Kinetic Energy
Wayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Email [email protected]
Tel (65) 6874-2749
QUESTIONS
What falls faster – heavy or light objects ?
hypotheses and supporting arguments
experimental method
Measurement
what to measure
how to measure
HEIGHT
Why is height a relative concept ?
How does it depend on a reference position ?
How does it depend on a reference length ?
Does the distance that an object fall depend on
the reference position ? Does it depend of the
reference length ? Does A or B report a longer
distance dropped ?
A’s reference length = one stick
B’s reference length = one stick
LENGTH CONVERSION
Question: If A’s stick has length = c times the
length of B’s stick, how to convert from length
in A-sticks into length in B-sticks ?
Consider the equation
length L = x (A-sticks) = y (B-sticks)
Substitute the equation
1 (A-sticks) = c (B-sticks)
to obtain x c (B-sticks) = y (B-sticks)
Question: How is y related to x ? Why ?
Question: How is x related to y ? Why ?
HEIGHT CONVERSION
Question: How are A’s and B’s height measurements
related if they both use the same reference point (ie a
point that has height zero) ?
Question: What happens if A’s reference point
is d (B-sticks) above B’s reference point ?
Question: Let a point have height u (A-sticks) in
A’s world and v (B-sticks) in B’s world. Derive
an equation that expresses u in terms of v and
another equation that expresses v in terms of u
(both equations will also involve both c and d)
Question: Graph u as a function of v and v as a
function of u. What is the ‘geometry’ of c and d ?
TUTORIAL 1
1. Two observers A and B measure the heights of a
point x and a point y to be
H_A(x) = 312, H_A(y) = 512
H_B(x) = 125, H_B(y) = 290
Compute constants (real numbers) r and c such that
for any point z
H_A(z) = r H_B(z) + c
CLOCKS AND INVARIANCE
How can we compare two time durations that start at
different times?
Our duration-length analogy might provide a clue!
Clue: Find a repeatable physical process whose
start and finish are points in time.
Examples: Sandclock,
Pendulum, Spring
TUTORIAL 1
2. Three clocks A, B, C run at different (uniform) rates
and have simultaneous readings, expressed in seconds,
shown by the figure below? Express the readings of B
and of C as functions of the reading of A.
Hint: consider the previous tutorial problem
312
25
125
512
200
A
290
B
92
142
C
THE MOTION OF A FALLING OBJECT
Empirical measurements
show that an object in a vacuum and starting with
zero velocity falls a distance
L = L(T) = gT^2 / 2
in a time interval having length T. Here g is a
constant equal to 9.8 meters per squared second.
Question: according to this formula does L
depend on the position in time when or the
position in space where the object starts to fall ?
does it depend of the mass of the object ?
Question: express height H = H(t) as a function of
time t given H(t_0) and graph it
THE SUPERPOSITION PRINCIPLE
Empirical measurements
show that the distance that an object in a vacuum
drops satisfies the following superposition
principle: the distance it drops if it starts with a
nonzero velocity (positive for upwards and negative
for downwards) is the sum of the distance that it
would fall if it moved with constant velocity and
the distance that it would fall if it’s initial velocity
were equal to zero
Question: express height H = H(t) as a function of
time t given H(t_0) and V(t_0) and graph it
TUTORIAL 1
3. A gun is fired directly at an object thrown directly
upward when its height is maximum height. Show
that the bullet will strikes the target unless it hits the
ground before the object hits the ground (ignore air
friction and assume that the ground is flat). Hint: the
horizontal motion is superimposed on the vertical
motion of the bullet fired from the gun. Also ignore
the effects of wind and friction and Earth’s rotation.
GRAPHICAL REPRESENTATION
Consider a particle thrown upward from the ground
h(t)
t
h(t 4 )
t2
t4
t
t3
t1
We can use analytic geometry to construct a graph
of this function using orthonormal planar coordinates
AVERAGE AND INSTANTANEOUS
VELOCITIES
The average velocity over
h
(
t
)
the interval [ t 3 , t 4 ] is the
slope of the dotted line
h(t4 )  h(t3 )
Vav 
t t
4
Tangent
Secant
3
The instantaneous velocity
at t  t 4is the slope of the
solid line and equals the
derivative dh/dt evaluated
at t  t
4
t2
t1
t4
t3
t
COMPUTING INSTANTANEOUS VELOCITIES
v( t )
t2
t1
t4
t
t3
The instantaneous velocity v(t) is computed using
differential calculus
h ( t 4  )  h ( t 4 )
dh
v( t 4 ) 
| t  t  lim
4
dt

0
2
1
If h ( t )  v ( 0 ) t  2 gt the graph of v is shown above
TUTORIAL 1
4. Compute the velocity V(t) for a falling object
given H(t_0) and V(t_0). Show that the quantity
M V(t)^2 / 2 + M gH(t) does not change with time.
The first term is called kinetic energy and the second
is called (gravitational) potential energy.
SCALES AND MASS MEASUREMENTS
Masses of objects can be compared using a scale.
A
B
Question: in which direction will the right side move ?
Answer: up, down iff mass B < , > mass A
TUTORIAL 1
5. Discuss how the sum of the potential energy of
two masses at opposite sides of a balanced board of a
fulcrum changes as the board is very slowly moved.
COMPUTING DISTANCE FROM VELOCITY

v( t )
t

t
The fundamental theorem of calculus says that h
equals the signed area under the graph of v
h(t) 
s  t dh ( u )
s  0
du
| u  s ds 
st
s  0
v ( s ) ds
SYSTEMS OF UNITS
SI (International System) of seven base units include:
meter (m)– length or distance traveled by light in a
vacuum during 1/299792458 of a second
second (s) – time taken by 9192631770 oscillations
of the light (of a specified frequency) emitted by a
cesium-133 atom
kilogram (kg) – mass of a certain platinum-iridium
cylinder
These base units can be used to define derived units
2
newton (N) – force unit = (1 kg )  (1 m / s )
joule (J) - energy unit = (1 N) x (1 m)