Download Circular Motion 2

Worked Examples
Uniform Circular Motion
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?
You may wish to view the video clip at:
http://www.youtube.com/watch?v=Q5FRbieCU1A
(optional)
Uniform Circular Motion
2
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?
The centripetal force in this case is provided
entirely by the tension in the string. If the
maximum value of the tension is 50 N, and the
radius is set at 10 m we only need to plug
these two values into the equation for
centripetal force:
T = Fc =
implies that v =
thus v =
= 15.8
Uniform Circular Motion
m/s
or 57 km/hr
3
1 km/hr is equivalent to:
A.
B.
C.
D.
E.
0.1666… m/sec
0.2777… m/sec
0.6 m/sec
1.6 m/sec
3.6 m/sec
Uniform Circular Motion
5
1 km/hr is equivalent to:
A.
B.
C.
D.
E.
0.2777… m/sec
Uniform Circular Motion
6
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?
You may wish to view the video clip at:
http://www.youtube.com/watch?v=qiIYEmpWaaQ
Uniform Circular Motion
8
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?
Since Fc varies with v2, an
increase in velocity by a factor
of two must be accompanied by
an increase in centripetal force
by a factor of four.
Uniform Circular Motion
9
Which of the following statements are true of an
object moving in a circle at a constant speed?
A. The object experiences a force which has a
component directed parallel to the direction of
motion.
B. Inertia causes objects to move in a circle.
C. Because the speed is constant, the acceleration is
zero.
D. The acceleration and the net force vector are
directed perpendicular to each other.
E. If the net force acting upon the object is suddenly
reduced to zero, then the object would suddenly
depart from its circular path and travel tangent to the
circle.
F. The acceleration of the object is directed tangent to
Uniform Circular Motion
11
the circle.
Which of the following statements are true of an
object moving in a circle at a constant speed?
A. B. C. D. E. If the net force acting upon the
object is suddenly reduced to zero,
then the object would suddenly
depart from its circular path and
travel tangent to the circle.
F. Uniform Circular Motion
12
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 4.23×107 metres
from the earth's centre.
What is the magnitude
of the acceleration felt
by a geosynchronous
Please view the video clip at:
satellite?
http://www.youtube.com/watch?v=FsfcIEmR_b0
Uniform Circular Motion
14
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 4.23×107 metres
from the earth's centre. What is the magnitude of the acceleration
felt by a geosynchronous satellite?
The acceleration felt by any object in uniform circular
motion is given by a =
. We are given the radius but
must find the velocity of the satellite. We know that in
one day, or 86 400 seconds, the satellite travels
around the earth once. Thus:
v=
thus a =
=
=
=
= 3 076 m/s
=
Uniform Circular Motion
0.224 m/s2
15
Identify the type of force which causes the following
object (in bold type) to travel along a circular path.
An eraser is tied to a string swung in a
horizontal circle.
a.
b.
c.
d.
e.
f.
g.
h.
Gravity
Normal
Tension
Applied
Friction
Spring
Electrical
Magnetic
Uniform Circular Motion
17
Identify the type of force which causes the following
object (in bold type) to travel along a circular path.
An eraser is tied to a string swung in a
horizontal circle.
a.
b.
c.
d.
e.
f.
g.
h.
Tension
Uniform Circular Motion
18
Identify the type of force which causes the following
object (in bold type) to travel along a circular path.
The moon orbits the earth.
a.
b.
c.
d.
e.
f.
g.
h.
Gravity
Normal
Tension
Applied
Friction
Spring
Electrical
Magnetic
Uniform Circular Motion
19
Identify the type of force which causes the following
object (in bold type) to travel along a circular path.
The moon orbits the earth.
a.
b.
c.
d.
e.
f.
g.
h.
Gravity
Uniform Circular Motion
20
Identify the type of force which causes the following
object (in bold type) to travel along a circular path.
A car makes a sharp right-hand turn along a
level roadway.
a. Gravity
b. Normal
c. Tension
d. Applied
e. Friction
f. Spring
g. Electrical
h. Magnetic
Uniform Circular Motion
21
Identify the type of force which causes the following
object (in bold type) to travel along a circular path.
A car makes a sharp right-hand turn along a
level roadway.
a. b. c. d. e. Friction
f. g. h. Uniform Circular Motion
22
Identify the type of force which causes the following
object (in bold type) to travel along a circular path.
A roller coaster car passes through a loop.
Consider the car at the bottom of the loop.
a. Gravity
b. Normal
c. Tension
d. Applied
e. Friction
f. Spring
g. Electrical
h. Magnetic
Uniform Circular Motion
23
Identify the type of force which causes the following
object (in bold type) to travel along a circular path.
A roller coaster car passes through a loop.
Consider the car at the bottom of the loop.
a. b. Normal
c. d. e. f. g. h. Uniform Circular Motion
24
The maximum lift provided by a
500 kg aeroplane is 10 000 N. If
the plane travels at 100 m/s, what
is its shortest possible turning
radius?
Please view the video clip at:
http://www.youtube.com/watch?v=fmrH-LEZCf0&NR=1
Uniform Circular Motion
26
The maximum lift provided by a 500 kg aeroplane is 10 000 N.
If the plane travels at 100 m/s, what is its shortest possible turning
radius?
Again, we use the equation
Fc =
. Rearranging, we find
that r =
. Plugging in the
maximum value for the lift of the
plane, we find that
rmin =
= 500m
Uniform Circular Motion
27
What is the acceleration of a piece of dust on an old-fashioned
record album, 15 cm from the centre, if the record is spinning at
33.3 rpm?
a) 0.46 m/s2
b) 0.523 m/s2
c) 0.91 m/s2
d) 1.046 m/s2
e) 1.82 m/s2
Uniform Circular Motion
29
What is the acceleration of a piece of dust on an old-fashioned
record album, 15 cm from the centre, if the record is spinning at
33.3 rpm?
a) b) -
c) d) -
e) 1.82 m/s2
Uniform Circular Motion
30
A popular daredevil trick is to complete a
vertical loop on a motorcycle or in a small
car. This trick is dangerous, however,
because if the motorcycle / car does not
travel with enough speed, the rider / driver
falls off the track before reaching the top of
the loop. What is the minimum speed
necessary for a rider /
driver to successfully
go around a vertical
loop of 10 metres
Please view the video clip at:
radius?
http://www.youtube.com/watch?v=GMWCsfVNHjg
Uniform Circular Motion
32
A popular daredevil trick is to complete a vertical loop on a
motorcycle or in a small car. This trick is dangerous, however, because if
the motorcycle / car does not travel with enough speed, the rider / driver
falls off the track before reaching the top of the loop. What is the
minimum speed necessary for a rider / driver to successfully go around a
vertical loop of 10 metres radius?
• During the entire trip, the rider /driver experiences two
different forces: the normal force from the track, and the
gravitational force. At the top of the loop, both these
forces point down, or towards the centre of the loop.
Thus the combination of these forces provides the
centripetal force at that point. At the minimum speed of
the motorcycle / car, however, s/he experiences no
normal force. One can see this by envisioning that if the
rider / driver had gone any slower, s/he would have fallen
off the track. Thus, at the minimum speed, all the
centripetal force is provided by gravity.
Uniform Circular Motion
33
A popular daredevil trick is to complete a vertical loop on a
motorcycle or in a small car. This trick is dangerous, however, because if
the motorcycle / car does not travel with enough speed, the rider / driver
falls off the track before reaching the top of the loop. What is the
minimum speed necessary for a rider / driver to successfully go around a
vertical loop of 10 metres radius?
•
During the entire trip, the rider /driver experiences two different forces: the normal force
from the track, and the gravitational force. At the top of the loop, both these forces point
down, or towards the centre of the loop. Thus the combination of these forces provides the
centripetal force at that point. At the minimum speed of the motorcycle / car, however, s/he
experiences no normal force. One can see this by envisioning that if the rider / driver had
gone any slower, s/he would have fallen off the track. Thus, at the minimum speed, all the
centripetal force is provided by gravity.
• Plugging into our equation for centripetal force, we see that
mg =
• Rearranging the equation,
vmin =
=
= 9.9 m/sec = 9.9 x 3 600 / 1 000 km/hr = 36 km/hr
Thus the rider must be travelling at least 36 km/hr to make it around the loop.
Uniform Circular Motion
34
What is the tension in a 0.500 metre rope which carries
a 2.50 kg bucket of water in a vertical circle with a
velocity of 3.00 m/s when the bucket is at the bottom of
its swing?
a)
b)
c)
d)
e)
2.5 N
20.5 N
24.5 N
45 N
69.5 N
Uniform Circular Motion
36
What is the tension in a 0.500 metre rope which carries
a 2.50 kg bucket of water in a vertical circle with a
velocity of 3.00 m/s when the bucket is at the bottom of
its swing?
a)
b)
c)
d)
e)
69.5 N
Uniform Circular Motion
37
The QSA Senior
Mathematics C
Syllabus suggests
the following
learning experience:
Please view the video clip at:
http://www.youtube.com/watch?v=iAYkvp82klM
“Investigate the angle of lean
required by a motorcycle rider
to negotiate a corner at
various speeds.”
Uniform Circular Motion
39
References / Sources:
http://www.youtube.com/watch?v=Q5FRbieCU1A
http://www.youtube.com/watch?v=qiIYEmpWaaQ
http://www.youtube.com/watch?v=FsfcIEmR_b0
http://www.youtube.com/watch?v=fmrHLEZCf0&NR=1
http://www.youtube.com/watch?v=GMWCsfVNHjg
http://www.youtube.com/watch?v=iAYkvp82klM
http://www.physicsclassroom.com/reviews/circles/c
pmans1.cfm#1
Uniform Circular Motion
40