Download Lecture 6 Circular motion

© Copyright FIRST EDUCATION 2011
0430 860 810 Nick Zhang
Lecture 6 Circular motion
Instantaneous velocity and speed
For an object travelling in the uniform circular motion, its instantaneous velocity is not constant because the
direction of the object is continuously changing, however, its instantaneous speed is constant.
constant speed = average speed =
distance travelled
circumference of the circle
2π r
=
=
∆t
period
T
where
r is the radius of the circle and T is the time taken for the object to complete a revolution.
If the object travelling in the uniform circular motion completes a full revolution, the average velocity over
the entire revolution should be zero because the displacement is zero.
Example
A toy car completes a single lap of a circular track in 15 s with an average speed of 1.3 m/s. Assume that the
speed of the toy car is constant.
a. What is the radius of the track?
b. What is the magnitude of the toy car's instantaneous velocity halfway through the lap?
c. What is the average velocity of the toy car after half of the lap has been completed?
d. What is the average velocity of the toy car over the entire lap?
As all objects with changing velocities are experiencing an acceleration, this means that all objects that are
moving in a circle are accelerating.
According to Newton's Second Law of Motion, an acceleration can be caused only by a non-zero net force.
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Centripetal acceleration is always directed towards the centre of the circular path and is given by
a=
where
v 2 4π 2 r
= 2
r
T
a = centripetal acceleration (ms-2)
v = speed (ms-1)
r = radius of circle (m)
T = period of motion (s)
This formula provides a way of calculating the centripetal acceleration of a mass moving with uniform
circular motion having speed v and radius r.
The net force involved in circular motion is called centripetal force which is always directed towards the
centre of the circle.
Fnet = ma =
where
mv 2 m ⋅ 4π 2 r
=
r
T2
Fnet = net force or centripetal force on the object (N)
m = mass (kg)
v = speed (ms-1)
r = radius of circle (m)
T = period of motion (s)
Example
A motorcyclist is riding around a circle of radius of 100 m. The surface is flat and horizontal. The
motorcyclist is travelling at a constant speed of 30.0 ms-1. The motorcycle with rider has a mass of 250 kg.
What is the magnitude of the net force on the motorcycle with rider?
Draw an arrow to show the direction of the acceleration on the motorcycle.
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Examples of centripetal force
As long as an object is in uniform circular motion, the net force (centripetal force) acting on the object must
be towards the centre of the circle. The following are some typical examples.
1. Tension
Example
When travelling around a roundabout, Nick notices that the dice suspended from the rear-vision mirror
swing out. If Nick is travelling at 10 m/s and the roundabout has a radius of 5.0 m, what angle will the string
connected to the dice (mass 100 g) make with the vertical?
θ
2. Friction
When a car travels around a corner, the sideways frictional forces contribute to the centripetal force. If the
sideways frictional forces are not sufficient, the net force on the car will be less than the centripetal force
required to keep the car moving in a circle.
N
Fr
W
Example
A car of mass 1200 kg travels around a bend with a radius of 10 m. The total sideways friction on the wheels
is 14400 N. The road is not banked. Calculate the maximum constant speed at which the car can be driven
around the bend without skidding off the road.
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Cyclists and motorcyclists also rely on sideways frictional forces to enable them move around corners. To
increase the size of the sideways frictional forces, they often lean into the corner. The reaction force exerted
by the ground will have a component towards the centre of the circular motion.
Rsin( θ )
reaction force (R)
normal
reaction
force
sideways friction
θ
Rcos( θ )
Example
A cyclist rounds a bend. The surface of the bend is horizontal. The cyclist is forced to lean at an angle of 20º
to the vertical to just take the bend successfully. The total sideways frictional force on the tyres is 360 N.
The cycle had a mass of 20 kg. What is the mass of the cyclist?
Going around the circular banked track
N
Fnet = Fr cos(θ ) + N sin(θ )
Fnet
Fr
θ
W
Example
In designing a bicycle track at a racing track, the designer wants to bank the track on a particular corner so
that the bicycles will go around the corner with no sideways frictional force required between the tyres and
the track at 10 m/s. Figures below show the banked track and the bicycle.
On Figure B draw two arrows to show the two forces acting on the bicycle and rider (treated as a single
object).
Figure A
Figure B
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The circular path of the bicycle has a constant radius of 100 m, and the bicycle will be travelling at a
constant 10 m/s.
What should be the value of the angle of the bank, θ , so that the bicycle travels around the corner with no
sideways frictional force between the tyres and the track?
Example
A demonstration at a show involves a bike being ridden around a circular banked track.
The horizontal path the bike takes is a circle with radius of 20 m, and the bike travels at a constant speed of
15 m/s.
The bike and rider have a total mass of 300 kg. Ignore retarding friction.
Figure A
Figure B
What is the magnitude of the net force on the bike and rider?
Figure B shows the only two forces on the bike and rider. On Figure B draw an arrow to show the direction
of the net force on the bike and rider. Explain how the forces shown cause this net force.
What is the correct angle of the banked track, θ , to achieve this net force?
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3. Normal reaction force
For a person in the Gravitron it is the normal reaction force from the wall provides the centripetal force
which produces a circular motion.
Fr
N
Wall
W
Example
Sarah, who weighs 50 kg, is at a fair ground and takes a ride on the Rotor. The Rotor is a large cylinder
which, when spinning, pin riders against the internal wall. Figure A is a sketch of the situation. Initially
Sarah is standing on the floor, and the Rotor speeds up. She feels the wall pushing harder and harder against
her back. When the friction between the wall and her back is large enough, the floor is lowered. Sarah is
held against the wall, and does not slip down.
The diameter of the Rotor is 16 m.
Figure A
Figure B
Figure B is a force diagram showing all the forces acting on Sarah: force of wall on Sarah (C), weight force
(W) and friction force (F).
When the Rotor is moving just fast enough to stop Sarah slipping, what is the magnitude of force F?
The Rotor continues to speed up so that its final speed at the wall is 11 m/s.
What is the magnitude of the force (C) of the wall on Sarah?
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Non-uniform circular motion (vertical circular track)
1. Energy transformations
Let's review the conservation of mechanical energy.
If the gravitational force and the restoring force are the only forces doing work on an object, the kinetic
energy and the potential energy of the object interconvert while the total mechanical energy of the system
(comprising the object and the earth or spring) remains constant.
Example
A skateboarder with mass 50 kg enters the half-pipe at point A, as shown in Figure below. Assume the
frictional forces are negligible.
A
4.0 m
B
What is the skateboarder's speed at point B?
What is the centripetal force on the skateboarder at point B?
What is the apparent weight of the skateboarder at point B?
2. Newton's laws of motion
N
W
N
W
N
W
By performing the force analysis, we could find the the net force which provides the centripetal force for the
circular motion at the top or bottom of the circular track.
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Example
A ride at a fun fair involves a car travelling around a vertical circle of radius 15 m. The arrow indicates the
direction of travel as shown in Figure below.
What is the minimum speed that the car can have at position A if it is not to leave the rails and start to fall?
Sarah, who has a mass of 60 kg, is a passenger in the car.
What will be Sarah's apparent weight at position A when the car is travelling at this minimum speed?
Explain your answer.
As the car passes point B at the bottom of the circle, it has a speed of approximately 5.5 m/s.
What is Sarah's apparent weight at this point? Show your working.
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