phy 400 - chapter 6 - rotational motion
TRANSCRIPT
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PHY400
Physics for Non-Majors
Assoc. Prof. Dr. Ahmad Taufek Abdul Rahman PhD (Medical Physics), University of Surrey, UK
M.Sc. (Radiation Health Physics), UTM
B.Sc. Hons. (Physics & Math), UTM
Chapter 6
ROTATIONAL MOTION
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6. Rotational Motion
– Angular Quantities
– Relationship Linear-Circular Motion
– Rotational Motion Equation
– Centripetal Force
Chapter 6
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Angular Quantities
The motion a rigid body can be analysed as the translational
motion of its centre of mass plus rotational motion about its centre
of mass.
For the ideal rotational motion of an object about a fixed axis, it
mean that all point / particle of the system move in circles
Chapter 6
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Angular Quantities Consider to the diagram, when system rotate, P is moving in circles
at uniform distance R from the axis of rotation O. The point moves
through an angle when its travel the distance l measured along
the circumference of its circular path.
is stated in radian, 1 radian is defined as the angle by a rotation
with l (length) is equal to R (radius)
= angles
l = length of circumference
R = radius
Chapter 6
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Angular Quantities
Angular displacement,
Chapter 6
, stated in radian
Derive that 360 = 2 radian
R
l
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Angular Quantities Consider a bicycle wheel rotate from one initial position, let say 1,
to some final position 2, therefore, the different between this two
positions stated as angular displacement (),
The angular displacement measured in radian (rad)
Chapter 6
= 2 - 1
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Angular Quantities Angular Velocity,
Consider a bicycle wheel rotate from one initial position, 1, to
some final position 2, thus the time rate of change of angular
displacement is defined as average angular velocity, .
The instantaneous angular velocity is the limit of this ratio as t
approaches zero (t 0)
The angular velocity has units of radian per second (rad/s)
Chapter 6
= / t
dt
d
t
0lim
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Angular Quantities Angular Acceleration,
Consider a bicycle wheel rotate from one initial position, 1, to
some final position 2, thus the time rate of change of angular
velocity is defined as average angular acceleration, .
The instantaneous angular acceleration is the limit of this ratio as
t approaches zero (t 0)
The angular acceleration has units of radian per second squared
(rad/s2)
Chapter 6
= / t
dt
d
t
0lim
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Angular Quantities
Chapter 6
dt
d
t
0lim
dt
d
t
0lim
= 2 - 1
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Angular Quantities
The frequency is the number of complete revolutions per
second:
Frequencies are measured in hertz:
The period is the time one revolution takes:
Chapter 6
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Linear vs. Angular Motion
At the end of this chapter, students should be able to: Relate
parameters in rotational motion with their corresponding quantities
in linear motion. Write and use;
Chapter 6
r
vrararvrθs ct
22 ; ; ;
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Linear vs. Angular Motion
Consider a point P located a distance R from the axis of rotation, if
the body rotates with angular velocity , any point will have a linear
velocity whose direction is tangent to its circular path
Chapter 6
Rvdt
dRv
RdlR
l
dt
dlv
2
1
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Linear vs. Angular Motion
If the angular velocity of a rotating object changes, the object as a
whole and each point/particle in it has an angular acceleration.
Each point also has a linear acceleration whose direction is
tangent to that point’s circular path.
Chapter 6
Radt
dRa
RdRv
dt
dva
tantan
tan
2
1
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Linear vs. Angular Motion
The total linear acceleration is equal to sum of tangential
acceleration (atan) and centripetal acceleration (aR)
Chapter 6
Ra
RR
R
R
va
aaa
R
R
R
2
2
22
tan
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Linear vs. Angular Motion
Chapter 6
Linear Quantities
s = Displacement
v = velocity
atan = acceleration
Angular Quantities
= Displacement
= velocity
= acceleration
Linear Quantities Vs. Angular Quantities
s = R
v = R
atan = R
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Rotational Motion Equation
At the end of this chapter, students should be able to write and
use equations for rotational motion with constant angular
acceleration;
Chapter 6
t 2
1
αtωω 0
2
02
1ttωθ
αθωω 22
0
2
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Example 1
a) Write out the definitions for all of the rotational quantities.
b) Write the equations relating the angular quantities to the
linear quantities.
c) Write down the equations for rotary motion with constant
acceleration.
d) Write the rotational versions of:
i. F = ma
ii. W = Fd
iii. E = ½ mv2
Chapter 6
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Example 2
The moon orbits the earth every 29 days. Find its angular velocity.
2.5 x 10-6 rad/s
Chapter 6
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Example 3
a) Find the radial acceleration of an 82 kg person at the earth’s
equator.
b) Calculate the centripetal force.
c) Calculate the normal force exerted by the earth
0.0337 m/s2 2.76 N 801 N
Chapter 6
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Example 4
a) Determine the tangential velocity of the earth about the sun.
(REarth’s Orbit=1.5 x 108 km)
b) The tangential velocity at the equator of a large sphere of radius
35 cm increases from 2.2 m/s to 8.4 m/s in 18 rotations. Find the
acceleration.
c) A tangential force of 22.5 N applied to a 4.5 kg sphere results in
an angular acceleration of 78.2 rad/s2. Find the radius of the
sphere.
2.99 x 104 m/s
8. 2.37 rad/s2
9. 16.0 cm
Chapter 6
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Example 5
A motorbike moving at a constant speed 20.0 m s1 in a circular track
of radius 25.0 m. Calculate
a. the centripetal acceleration of the motorbike,
b. the time taken for the motorbike to complete one revolution.
Chapter 6
2s m 16.0 ca s 7.85T(towards the centre of the circular track)
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Example 5
A boy whirls a marble in a horizontal circle of radius 2.00 m and at
height 1.65 m above the ground. The string breaks and the marble
flies off horizontally and strikes the ground after traveling a
horizontal distance of 13.0 m. Calculate
a. the speed of the marble in the circular path,
b. the centripetal acceleration of the marble while in the circular
motion.
(Given g = 9.81 m s-2)
Chapter 6
1s m 22.4 u2s m 512 ca (towards the centre of the horizontal circle)
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Centripetal Force From Newton’s second law of motion, a force must be associated
with the centripetal acceleration. This force is known as the
centripetal force and is given by
Chapter 6
amFF nett
cc amF
mvmrr
mvF 2
2
c
caa
vrr
va 2
2
c
where cFF
and
and
force lcentripeta :cFwhere
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Centripetal Force • The centripetal force is defined as a force acting on a body
causing it to move in a circular path of magnitude
and its always directed towards the centre of the circular path.
• Its direction is in the same direction of the centripetal
acceleration as shown in Figure.
Chapter 6
r
mvF
2
c
ca
cF
cF
cF
ca
ca
v
v
v
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Centripetal Force • If the centripetal force suddenly stops to act on a body in the
circular motion, the body flies off in a straight line with the
constant tangential (linear) speed as show in Figure.
Chapter 6
cF
ca v
cF
cF
ca
ca
v
v
v
v
0Fc
0Fc
0ac
0ac
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Centripetal Force • Note :
– In uniform circular motion, the nett force on the system is
centripetal force.
– The work done by the centripetal force is zero but the
kinetic energy of the body is not zero and given by
Chapter 6
222 mr2
1mv
2
1K
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Example 6
Figure shows a conical pendulum with a bob of mass 80.0 kg on a
10.0 m long string making an angle of 5.00 to the vertical.
a) Sketch a free body diagram of the bob.
b) Determine
i. the tension in the string,
ii. the speed and the period of the bob,
iii. the radial acceleration of the bob.
(Given g =9.81 m s2)
Chapter 6
N 788T 1s m 0.865 v2s m 0.859 ras 6.33T
(towards the centre of the horizontal circle)
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Example 7
A car of mass 2000 kg rounds a circular turn of radius 20 m. The road
is flat and the coefficient of friction between tires and the road is 0.70.
a) Sketch a free body diagram of the car.
b) Determine the maximum car’s speed without skidding.
(Given g = 9.81 m s-2)
Chapter 6
1s m 11.7 v
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Example 8
A ball of mass 150 g is attached to one end of a string 1.10 m long.
The ball makes 2.00 revolution per second in a horizontal circle.
a) Sketch the free body diagram for the ball.
b) Determine
i. the centripetal acceleration of the ball,
ii. the magnitude of the tension in the string.
Chapter 6
r
2s m 173 ca (towards the centre of the horizontal circle) N 26.0T
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Thank You & All the Best
“The actions which the Messenger of Allah, may Allah bless him and grant him peace, loved most were those which were done most constantly.”
Narrated by A’isha (ra) in Al-Muwatta, vol 9, number 92b