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GRADE TWELVE PHYSICS
SPH 4U(G) MR. MOORS
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General booklet
Physics is the original science that studies the physical world. It used to be known as natural philosophy. Today it is mostly concerned with the behaviour and structure of matter.
The study of physics often leads to mathematical relationships. These relation-ships are useful because they describe the nature of matter and they allow the accurate prediction of future events.
PHYSICS IS MATHEMATICAL.
PHYSICS CAN BE ABSTRACT, YOU MUST THINK.
PHYSICS DEMANDS HARD WORK.
SAY THE F-WORD OR ABOVE??????
(I decide what is above or below, e.g. the s-word is below)
THREE CHOICES
1. DO TEN PUSH-UPS
2. DO TWENTY CRUNCHES
3. SING I’M A LITTLE TEAPOT (with proper actions) IN FRONT OF CLASS
(this could also be foisted upon you in the halls!!)
PROPORTIONALITY STATEMENTS
The expression of how one quantity varies in relation to another is called a proportionality statement. In science and many other areas of thought people are looking for patterns that exist between different types of quantities. If there is a pattern to the data it is important to be able to deduce this relationshipLet’s start by graphing some given data. Graph this data on one quarter of a standard piece of graph paper. Place distance on the y-axis and time on the x-axis. Label your graph properly.
time (s) 1 2 3 4 5
distance (m) 0.2 1.6 5.4 12.8 25.0
The goal of this exercise is to find the pattern that exists (if one exists at all) between distance and time. The general proportionality statement is, y xn, while for this set of data it is d tn. In the first graph d vs. t1 was plotted. How does one decide if distance is proportional to time? Two quantities are proportional to each other if they produce a linear relationship when graphed. A linear relationship means that the ratio of any two pairs of data are equal or proportional. After graphing this data you should see that distance is not proportional to time.Plot d vs.t2 on another quarter of your graph paper. Is distance proportional to the square of time? Plot d vs. t3 on another quarter of your graph paper. Is distance proportional to the cube of time?
Since this graph is linear the proportionality statement is d t3. A couple of items to note, firstly we will only be studying data that includes the origin as a data point even though it will not be listed. Secondly, if we had graphed time on the y-axis and distance on the x-axis we would have obtained t d1/3 , after a graphical analysis.A proportionality statement can be analyzed more completely by finding the constant of proportionality, k.
y xn y = kxn (proportionality equation)
Therefore for our example d = kt3
We can determine the value of k by calculating the slope of the line in our final graph. This means the proportionality equation for the data given is d = 0.2t3 .
It is important to see the visual relationship in each of the graphs but there is a faster way to determine the
proportionality equation without graphing the data.
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
loglog
)(
x
xn
y
y
x
x
y
y
x
x
y
y
kx
kx
y
y
n
n
n
n
n
Any two sets of data points could be used.
n = 3
Express all exponents in integer or rational form.
Use any set of data to calculate k.
k = 0.232.0 td
This equation can be rearranged to another form of the equation.
dt
td
5
53
3
Write n values as fractions or whole numbers at all times in this unit. Write k values as decimals at all times.
Up to this point we have dealt with artificial data where the proportionality expression is perfectly upheld by all of the data points. In experiments that will be performed later the data obtained will not behave the same way and we will return to our graphing techniques to discover the proportionality expression. Graphing will determine the exponent while calculation of the slope of the best-fit line will provide the proportionality constant.
3
1
710.1 dt
Of course there are many mathematical relationships between variables that have already been discovered such as the following:circumference = 2r surface area of sphere = 4r2 F = ma F = Gm1m2/r2
If one was to plot circumference vs. radius one would expect to find the slope to be 2Plotting SA vs. r2 should produce a straight line with slope 4Plotting F vs. a should produce a straight line with a slope of whatever the constant mass is. Plotting F vs. 1/r2 should produce a straight line with a slope of Gm1m2 (these are held constant).Homework: Booklet Proportion 1-4
don’t do graphing for any
Using Proportionality Equations
A cylindrical water tank holds 10 000 L of water. The volume of the tank is given by the following equation.
hrV 2a) If the radius of the tank is 9 m. Then what is the height of the tank?b) What is the constant of proportionality for proportion between V and r?c) If the volume of the tank needs to be increased by a factor of 3 then what must the new radius be (assume h remains the same)?d) If the radius is increased by a factor of 5 then by what factor does the volume of the tank increase?
mhr
Vh
hrV
mr
mV
LV
03930.0
9
10
10000
2
2
3
a) b)
2
2
2
1235.0 rV
hrV
hrV
mh 1235.0Constants of proportionality can have units!!
mr
r
rV
mV
LV
59.15
1235.030
1235.0
30
30000
2
2
3
c) d)
25
)5
(
)(
1235.0
1235.0
1
2
2
1
1
1
2
2
1
2
1
2
21
22
1
2
V
V
r
r
V
V
r
r
V
V
r
r
V
V
homework: 12,15, 24, 20
complete proportion assignment and hand in
VECTORSVector quantities have magnitude and direction. A vector is displayed with an arrow.
[E 32o N]
[32o N of E]
22 m/s [58o]
[32o above the horizon]The product of a vector and a scalar is a vector with the same direction but with a different magnitude (unless multiplied by one). Multiplying by a negative scalar results in a vector in the opposite direction.
v
v3
v2
The component of a vector is the projection of a vector along an axis of a rectangular coordinate system.
Rectangular components are always perpendicular to each other
and are said to be orthoganal components.
To add vector quantities the arrows representing the vectors are first joined head to tail. Next the resultant vector is drawn from the tail of the first vector to the head of the last vector.
The result of adding vectors is called the vector addition, resultant vector, resultant, net vector or vector sum.
Remember that it does not matter which order vectors are added or how they are grouped.
Subtraction of vectors can be thought of in two equal ways.
A
B
AB
AB
B
A
tail to tail
Solving vector problems involves using trigonometry. Two general methods are used. The first involves the cosine law and the sine law. Remember . . .
cos2222 BABAR
sinsinsin
RBA
The second method utilizes vector components. This method is especially good for adding more than two vectors.
All answers should include:-a ruled vector drawing, properly labeled-a vector equation-proper vector notation**Remember sin-1(0.5) = 30o or 150o
DISPLACEMENT and VELOCITY AS VECTORS
Determine the resultant displacement of a dog that runs 12.6 m [N 21o W] then18.7 m [S 57o W].
homeworkMr. Anderson walked 25 m [S 32o E] then 41.2 m [N 51o W]. What was his resultant displacement?
Rd
21
ddd R
1
d]21[6.121 WNmd o
2
d
]57[7.182 WSmd o
o57o21
oR ddddd 78cos2 21
2
2
2
1
2
oRd 78cos)7.18)(6.12(27.186.12 22
2
mdR 26.20
78sinsin
2
Rdd
o78sin
26.20
sin
7.18
ooor 5.11553.64
o53.64
]53.85[26.20 WNmd oR
by looking at the diagram one can see the angle is acute
look at the diagram to see how this direction was calculated
COMPONENT METHOD
Determine the average velocity of a dog that runs 14.0 m [S 21.5o E] then 10.5 m [E] then 25.6 m [S 71.1o W] in 35 seconds.
homework Determine the average speed and velocity of a car that travels 12 km [SE], 20 km [S 10o W] and 10 km [S 80o W] in a total of 45 minutes.
1
d
]5.21[0.141 ESmd o
2
d
][5.102 Emd
3
d
]9.18[6.253 SWmd o
Rd
321
dddd R
Rxd
Ryd
yd1
xd1
xd3
yd3
21.5o
71.1o
Let left and down be positive.
xxxRx dddd 321
ooRx dddd 1.71sin5.21sin 321
ooRxd 1.71sin6.255.105.21sin0.14
mdRx 589.8
yyyRy dddd 321
ooRy ddd 1.71cos5.21cos 31
ooRyd 1.71cos6.255.21cos0.14
mdRy 32.21
Vector notation is not needed for components
Rdyd
xd
222
yxR ddd
222
32.21589.8
Rd
md R 99.22
y
x
d
d
tan32.21
589.8tan o94.21
]94.21[99.22 WSmd oR
t
dv
]94.21[6569.0 WSs
mv o
st 35
In the preceding question you do not have to draw in any right angled triangles, you do not need to show me Pythagoras theorem work or trigonometric ratio work. You need not show me the velocity calculation. All other work must be shown for full marks.
homework: text p.16 18,20
extra: p.17 8 p. 64 8 (only components), 16
Booklet p.6 1,2,6,7,12,15-17 (13,14 fun, tricky, not on test)
homework answer left and down are positive
Rd
1
d
2
d
3
d
kmdRx 836.4
kmdRy 92.29
]181.9[41.40
]181.9[31.30
WShr
kmv
WSkmd
o
oR
Trickier Displacement Problems1. A person walks with an average velocity
of 6.25 km/hr [N 38o E] for 90 minutes. The trip consisted of two stages. The first stage was a trip of 5.8 km [E]. What was the displacement of the second stage of the trip?
2. A plane flies 265 km [N 52o W], then it flies 189 km [N 88o W] followed by an unknown stage. If the resultant displace-ment was 512 km [S 82o W] then what was the displacement of the third leg?
]38[375.9
)5.1])(38[25.6(
ENkmd
hrENhr
kmd
tvd
t
dv
o
o
When showing work I need to see units once at the beginning and end of a calculation.
Rd ]38[375.9 ENkmd oR
21 dddR
2d
1d
][8.51 Ekmd
o52
oRR ddddd 52cos2 1
22
1
2
2
]5956.0[388.72 WNkmd o
1.
Use cos law twice because angle is too close to 90. This angle is off a bit ffrom cos law calc
2.
2d
1d
3d
Rd
321 ddddR
]52[2651 WNkmd o
]88[1892 WNkmd o
]82[512 WSkmd oR
Vector subtraction is a bit trickier to show in a 4 vector drawing but all of the component possibilities end up being the same concept.
213
dddd R
xxRxx dddd 213 ooo
Rx dddd 88sin52sin82sin 213
down and left are positive
rearranging negatives direction positives
kmd x 3.1093
yyRyy dddd 213
oooRy dddd 88cos52cos82cos 213
kmd y 0.2413
]34.24[6.2643 WSkmd o
VELOCITY AND ACCELERATION AS VECTORS
tavv
12
vvv 12
These are vector equations!!!
A car is moving 60 km/hr [W], soon after it is moving 45 km/hr [SSW]. What was the car’s change in velocity?
v
12 vvv
1v
][601 Whr
kmv
2v][452 SSW
hr
kmv
67.5o
ovvvvv 5.67cos2 12
2
1
2
2
2
]82.45[65.59 EShr
kmv o
A car is travelling 50 km/hr [E] when it accelerates at 30 km/hr2 [NE] for 30 minutes. What is the car’s final velocity?
tavv
12 ][501 Ehr
kmv
][15
5.0][302
NEhr
kmta
hrNEhr
kmta
1
v
2
v
ta
135o
otavtavv 135cos2 1
22
1
2
2
]07.80[53.612 ENhr
kmv o
A car is travelling 80 km/hr [SE] after it accelerated at 40 km/hr2 [S20oE] for 45 minutes. What was the car’s initial velocity?
tavv
21 ][802 SEhr
kmv
]20[30
75.0]20[402
EShr
kmta
hrEShr
kmta
o
o
2v
1v
ta
otavtavv 25cos2 2
22
2
2
1
]50.58[31.541 EShr
kmv o
25o
obtuse
homework booklet p. 6-8 18-20
extra
text p.29 25-29 p.31 15,16
booklet booklet p. 6-8 3,5,9,10
CONSTANT ACCELERATION KINEMATIC EQUATIONS
- one-dimensional motion thus vector notation is not required but you still must set up a direction system
- if the force of gravity is the net unbalanced force then the object is accelerating at g
g = 9.81 m/s2 [down]
davv 221
22
tavv 12 tvv
d
)2
( 12
21 2
ta
tvd
22 2
ta
tvd
-these problems are a review of grade 11 kinematics
- two of the equations are quadratic equations of elapsed time and you might be required to use the quadratic formula to solve for elapsed time (one of the answers is extraneous)
-when an object is thrown upwards its maximum height occurs when its velocity is zero
A ball is thrown upwards from a height of 20 m with a velocity of 20 m/s [up].
a) What height does the ball reach?
b) What time does the ball reach max. height?
c) How long does the ball take to reach the ground?
d) What velocity does the ball hit the ground with?
20 m
At maximum height v2 is 0
For the whole flight the ball accelerates at 9.81 m/s2 [down]
The initial velocity of the ball is 20 m/s [up]
a)
m 40.39 ofheight a reached ball the
39.20 81.9
2
vv 0
2v v20
up is
2
21
22
2
21
221
mds
ma
adv
das
mv
sta
vvt
t
vva
039.2
12
12
b) c)
sorst
tt
tt
tatvd
md
8308.0908.4
02020905.4
2
81.92020
2
20
2
2
2
1
d)
s
mv
tavvt
vva
15.282
12
12
problems
TEXT
p.27 9-22, 24 p.31 10-13 p.37 9, 11-16
p.40 3,5-8,10 (tough) p.65 18-23
Booklet
-all of linear motion!
set up half, solve half
PROJECTILE MOTION
An object that moves through the air without a propulsion system is called a projectile and it experiences projectile motion.
Projectiles follow a parabolic curved path called a trajectory. The shape of the parabola depends on the initial velocity given to the projectile.
When analyzing projectile motion the projectile’s velocity must be resolved into its horizontal and vertical components.
v
yv
xv
cos
vvx
sin
vvy
Analysis of projectile motion allows two conclusions to be drawn
•The horizontal component of a projectile’s velocity is constant.
•The vertical component of a projectile’s velocity experiences acceleration due to gravity.
A ball is thrown off an 85 m cliff such that it’s initial velocity is 65 m/s [28oabove the horizon].
a) What are the horizontal and vertical components of the initial velocity?b) What is the maximum height of the ball?c) What is the horizontal range of the ball?d) What is the final velocity of the ball when
it hits the ground?
discuss what happens when an object is launched horizontally
85 m
horizontal range
maximum height
1v
081.9)
52.3039.57
sincos
2865
positive is up a)
22
1
111
1
y
yx
yx
o
vs
mab
s
mv
s
mv
vvvv
aths
mv
ground. theabove m 132.48 of
height maximum a reaches ball the
48.47
)81.9(2
)52.30(0
2
2
2
21
22
21
22
md
d
a
vvd
davv
yy
yy
The horizontal range is the horizontal displacement of the projectile when the projectile lands on the ground.
To calculate horizontal range one needs the horizontal velocity and the amount of time the projectile is in the air.
storst
tt
tt
tat
m
086.2308.8
08552.30905.4
2
81.952.3085
2vd
85d c)
2
2
2
1yy
y
md
d
tvd
x
x
xx
8.476
)308.8)(39.57(
The final velocity of the ball (just before it hits the ground) is the sum of the projectile’s horizontal and vertical components.
The horizontal component velocity remains constant but the vertical component velocity changes with time.
s
mv
v
tavvt
vvad
y
y
yy
yy
98.50
)308.8)(81.9(52.30
)
2
2
12
12
vx
v2y
2v
]61.41[76.76v2 bths
m o
SPECIAL HORIZONTAL RANGE EQUATION
xd
1
v
0 yd
cos
motionx
1
v
dt
v
dt
t
dv
x
x
x
xx
2)(sin0
20
2d
motiony
2
1
2
1
2
1y
tatv
tatv
tatv
y
y
02
sin
0
)2
sin(0
2)(sin0
motiony
1
1
2
1
tav
ort
tavt
tatv
dy is zero at this time.
cos1
v
dt
remember
x
0cos2
sin
02
)cos
(
sin
02
sin
1
1
1
1
1
v
dav
v
da
v
tav
x
x
a
vd
vda
vv
da
x
x
x
2sin
cossin2
sincos2
2
1
2
1
1
1
in this final equation substitute g for a and get rid of all + and – since they mean nothing to the x motion
g
vd x
2sin2
1
Problems text p. 51 7
Booklet projectile 7, 8, 25
UNIFORM CIRCULAR MOTION
Chapter 3 p. 122-127
homework: p. 126 5-11
extra: p. 127 1-7
p. 159 1-6
booklet Centripetal 1-13
CENTRIPETAL ACCELERATION
centrum- centre, petere- to seek (Newton)
An object travelling with a constant speed in a circle is accelerating because the object’s velocity is changing in direction. The object is said to have “uniform” circular motion and it is undergoing centripetal acceleration.
Please read p. 123-124
1v
2
v
1r
2
r
1v
2v
v
1r
2
r
r
These two triangles are similar and for any time interval this would be so.
r
rvv
r
r
v
v
ct
at
v
0lim
This proof is not on any test.
t
r
r
va
tr
rva
tc
tc
0
0
lim
limThis is velocity!
For the final equation remove the absolute value notation.
r
va c
2
[towards the center]
The direction of the acceleration is constantly changing so this is an instantaneous acceleration.
Notice on the first diagram that half way through the interval v is pointing at the center of the circle.
For uniform circular motion . . .
T
rv
2 T is the period
of revolution.
2
24
T
rac
224 rfac f is the frequency.
rememberT
f1
An increase in the velocity of an object that is centripetally accelerating (with no change to r) is called tangential ac-celeration. A parallel push causes tangential acceleration and velocity’s magnitude changes. A perpendicular push causes centripetal acceleration and only the direction of velocity changes.
What is the centripetal acceleration of a stone being whirled in a circle, at the end of a 1.5 m string, on a smooth sheet of ice, with a frequency of 1.25 Hz?
2
22
22
53.92
)25.1)(5.1(4
4
s
ma
Hzma
rfa
c
c
c
The planet Mercury moves in an approximately circular path at an average distance of 5.8 x 1010 m, accelerating centripetally at 0.04 m/s2. What is its period of rotation about the Sun?
sT
smmx
T
a
rT
T
ra
cc
6
2
1022
22
2
2
10566.7
04.0
)108.5(4
44