Transcript MAC_Circular_Motion_1

Senior Mathematics C - Dynamics
Circular Motion
Dynamics
(notional time 30 hours) -
Focus
The approach used
throughout this topic should
bring together concepts from
both vectors and calculus.
Students should use a vector
and/or a calculus approach to
develop an understanding of
the motion of objects that are
subjected to forces.
Circular Motion 1
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Subject matter
• derivatives and integrals of vectors (SLEs 1, 2, 3,
13)
• Newton’s laws of motion in vector form applied to
objects of constant mass (SLEs 2–15)
• application of the above to: (SLEs 4–12, 14, 15)
• straight line motion in a horizontal plane with
variable force
• vertical motion under gravity with and without air
resistance
• projectile motion without air resistance
• simple harmonic motion (derivation of the
solutions to differential equations is not required)
• circular motion with uniform angular velocity.
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Suggested learning experiences
The following suggested learning experiences may be developed as individual student work, or may be
part of small-group or whole-class activities.
1.
Given the position vector of a point as a function of time such as r (t) = t i + t2 j + sin t k determine
the velocity and acceleration vectors.
2.
Given the displacement vector of an object as a function of time, by the processes of differentiation,
find the force which gives this motion.
3.
Given the force on an object as a function of time and suitable prescribed conditions, such as
velocity and displacement at certain times, use integration to find the position vector of the object.
4.
Investigate the motion of falling objects such as in situations in which:
resistance is proportional to the velocity, by considering the differential equation
resistance is proportional to the square of velocity, by considering the differential equation
where k is a positive constant.
5.
Model vertical motion under gravity alone; investigate the effects of the inclusion of drag on the
motion.
6.
Develop the equations of motion under Hooke’s law; verify the solutions for displacement by
substitution and differentiation; relate the solutions to simple harmonic motion and circular motion
with uniform angular velocity.
7.
From a table of vehicle stopping distances from various speeds, calculate (a) the reaction time of
the driver and (b) the deceleration of the vehicle, which were assumed in the calculation of the
table.
8.
Model the path of a projectile without air resistance, using the vector form of the equations of
motion starting with a = -g j where upwards is positive.
9.  Use the parametric facility of a graphing calculator to model the flight of a projectile.
10. Investigate the flow of water from a hose held at varying angles, and model the path of the water.
11. Investigate the motion of a simple pendulum with varying amplitudes.
12. Investigate the angle of lean required by a motorcycle rider to negotiate a corner at various speeds.
13. Use the chain rule to show that the acceleration can be written as
if the velocity, v, of a particle moving in a straight line is given as a function of the distance, x.
Investigate the speed required for a projectile launched vertically to escape from the earth’s
gravitational field, ignoring air resistance but including the variation of gravitational attraction with
distance.
15. Use detectors or sensors to investigate problems, e.g. rolling a ball down a plank.
16.  Use spreadsheets to investigate problems.
14.
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Suggested learning experiences
The following suggested learning
experiences may be developed as individual
student work, or may be part of small-group
or whole-class activities.
1.
Given the position vector of a point as a
function of time, determine the velocity and
acceleration vectors.
2.
Given the displacement vector of an
object as a function of time, by the
processes of differentiation, find the force
which gives this motion.
12.
Investigate the angle of lean required by
a motorcycle rider to negotiate a corner at
various speeds.
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How Circular Motion was taught in 1951.
Please view video clip at:
http://www.youtube.com/watch?v=1qjcRbXHXBk
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Mr. Stephen Kulik, a teacher at the Orange County High
School of the Arts, introduces his Block 5 2006-2007
class to the idea of circular motion.
Please view video clip at:
http://www.youtube.com/watch?v=4JVsZgn1Nf8
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The Preferred Approach
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Position Vector r
r = (r cos q) i + (r sin q) j
For uniform circular motion, dq/dt = w
and r = (r cos wt) i + (r sin wt) j
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Velocity Vector v
v
r = (r cos wt) i + (r sin wt) j
v = r = (-rw sin wt) i + (rw cos wt) j
.
Note that v . r = 0, and that v is perpendicular to r.
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Acceleration a is given by
dv
dt
A.
d x
B. dt
.
C. v
..
D. x
E. All of the above.
2
2
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Acceleration Vector a
a
r = (r cos wt) i + (r sin wt) j
v = r = (-rw sin wt) i + (rw cos wt) j
a = v = r = (-rw2 cos wt) i + (-rw2 sin wt) j
.
. ..
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Acceleration Vector a
| v | = v = rw
a
w=
v/
r
| a | = a = rw2 = v2/r
r = (r cos wt) i + (r sin wt) j
v = r = (-rw sin wt) i + (rw cos wt) j
a = v = r = (-rw2 cos wt) i + (-rw2 sin wt) j
.
. ..
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Centripetal Force Fc
Fc
a = rw2 = v2/r
Fc = ma = mrw2 = mv2/r
Fc = ma = (-mrw2 cos wt) i + (-mrw2 sin wt) j
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Student Exercise
Design the banking of a
road around a curve
with radius 250 metres
in an area where the
speed limit is 100 km/hr.
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Design the banking of a road around a curve with radius
250 metres in an area where the speed limit is 100 km/hr.
j
i
n = (n sin q) i + (n cos q) j
W = -mg j
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Design the banking of a road around a curve with radius
250 metres in an area where the speed limit is 100 km/hr.
j
i
Fnet = (n sin q) i + (n cos q) j - mg j
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Design the banking of a road around a curve with radius
250 metres in an area where the speed limit is 100 km/hr.
j
i
Fnet = (n sin q) i + (n cos q) j - mg j
Fc = (mv2/r) i
Fnet = Fc
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Design the banking of a road around a curve with radius
250 metres in an area where the speed limit is 100 km/hr.
j
i
Fnet = (n sin q) i + (n cos q) j - mg j
Fc = (mv2/r) i
Fnet = Fc
n cos q – mg = 0 -> n cos q = mg
n sin q = mv2/r
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Design the banking of a road around a curve with radius
250 metres in an area where the speed limit is 100 km/hr.
j
i
n cos q = mg
n sin q = mv2/r
n sin q
n cos q
=
Circular Motion 1
mv2
rmg
20
Design the banking of a road around a curve with radius
250 metres in an area where the speed limit is 100 km/hr.
j
i
tan q
=
Circular Motion 1
v2
rg
21
Design the banking of a road around a curve with radius
250 metres in an area where the speed limit is 100 km/hr.
j
i
tan q =
(100 x 1000/3600)2
250 x 9.8
tan q = 0.31494079113126732174…
q = 17.5o
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Coming Attractions
Please view video clip at:
http://www.youtube.com/watch?v=GMWCsfVNHjg
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Circular Motion 2
1. A 2 kg ball on a string is rotated about a circle of radius
10 m. The maximum tension allowed in the string is 50 N.
What is the maximum speed of the ball?
2. During the course of a turn, an automobile doubles its
speed. How much additional frictional force must the
tyres provide if the car safely makes around the curve?
3. A satellite is said to be in geosynchronous orbit if it
rotates around the earth once every day. For the earth,
all satellites in geosynchronous orbit must rotate at a
distance of 4.23 x 107 metres from the earth's centre.
What is the magnitude of the acceleration felt by a
geosynchronous satellite?
4. The maximum lift provided by a 500 kg aeroplane is
10000 N. If the plane travels at 100 m/s, what is its
shortest possible turning radius?
5. A popular daredevil trick is to complete a vertical loop on
a motorcycle. This trick is dangerous, however, because
if the motorcycle does not travel with enough speed, the
rider falls off the track before reaching the top of the
loop. What is the minimum speed necessary for a rider to
successfully go around a vertical loop of 10 metres?
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References:
http://www.youtube.com/watch?v=4JVsZgn1Nf8
http://www.youtube.com/watch?v=1qjcRbXHXBk
http://www.youtube.com/watch?v=GMWCsfVNHjg
http://www.ux1.eiu.edu/~cfadd/1150/05UCMGrav/Cu
rve.html