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Physics 131: Lecture 23
Today’s Agenda Finish up Ch 12 Simple harmonic motion
D fi iti Definition Period and frequency Position, velocity, and accelerationPosition, velocity, and acceleration Period of a mass on a spring Vertical spring
Energy and simple harmonic motion Energy of a spring force
Physics 201: Lecture 1, Pg 1
Clicker Question 1:A circular merry-go-round of mass 200 kg and 2 m radius
rotates freely with an initial angular velocity of 0.5 rad/s y g yabout a vertical axis passing through its center. A 50 kg student is initially standing on the rim of the merry-go-round. Treat the merry-go-round as a solid disk and the y gstudent as a point particle. Student
i f
Physics 201: Lecture 1, Pg 2
initial final
Clicker Question 1:
i f
initial final
Physics 201: Lecture 1, Pg 3
Clicker Question 1:
Which of the following best describes what is conserved as theWhich of the following best describes what is conserved as the student walks from the rim of the merry-go-round to the center?
A. Only the angular momentum of the system is conserved.B Only the kinetic energy of the system is conservedB. Only the kinetic energy of the system is conserved.C. Both the angular momentum and the kinetic energy is
conserved.
Physics 201: Lecture 1, Pg 4
Periodic Motion
Motion that repeats itself over and over
Characterized by two pieces of information:
Period ( T ): time it takes to complete one cycle Unit: seconds
Frequency ( f ): number of cycles per unit of timeU it H d 1 Unit: Hz = seconds-1
T 1
Physics 201: Lecture 1, Pg 5
fT
Clicker Question 2:
An object is undergoing periodic motion and takes 10 sAn object is undergoing periodic motion and takes 10 s to undergo 20 complete oscillations. What is the period and frequency of the object?
(a) T = 10 s, f = 2 Hz(b) T = 2 s f = 0 5 Hz(b) T = 2 s, f = 0.5 Hz(c) T = 0.5 s, f = 2 Hz(d) T = 0.5 s, f = 20 Hz(d) T 0.5 s, f 20 Hz(e) T = 10 s, f = 0.5 Hz
Physics 201: Lecture 1, Pg 6
Simple Harmonic Motion
Particular type of periodic motion Particular type of periodic motion Very common type of motion Motion due to a springo o due o a sp g Motion of a pendulum (small angles) Motion of atoms in molecules
SHM requires a restoring force
Physics 201: Lecture 1, Pg 7
Restoring Force: Hooke’s Law F = -kx x is distance spring is displaced from its relaxed length k is the spring constant (how stiff the spring is) Restoring force is proportional to displacement Restoring force is opposite in direction to displacement
FSPRING
F = -k(-x) = +kx
F = -k(+x) = -kxFSPRING-x
Physics 201: Lecture 1, Pg 8
( )
+x
Simple Harmonic Motion
Simple Harmonic MotionLet’s investigate this SHM a bit:
t
TAy 2cos
A
A
A
Physics 201: Lecture 1, Pg 9
A
Position 2
Drawing of:
A lit d f ill ti
t
TAx 2cos
A = amplitude of oscillationT = period
A
tA
2
0
AT
Ax
02cos
2
Physics 201: Lecture 1, Pg 10
22cos2cos t
TATt
TAx
t
TA 2cos
SHM
Physics 201: Lecture 1, Pg 11
Final Exam Final Exam: Tuesday 12/15 at 2:00pm Final Exam: Tuesday 12/15 at 2:00pm Review Session: Thursday 12/4 12pm 2pm in SS2135 (Harlow and Thursday 12/4 12pm-2pm in SS2135 (Harlow and
Meyertholen) Exam Jam
We will keep office hours until the finalOffice hours: Meyertholen M 2 3 F 12 2 Office hours: Meyertholen M 2-3 F 12-2
Meyertholen extra office hours: Monday 12/12: 10-12pm and 1-4pm12pm and 1 4pm
Office hours: Harlow T 2-3 RF 10-11
Physics 201: Lecture 1, Pg 12 Drop in center: Please check schedule
Vertical Mass and SpringIf we include gravity, there are two forces acting on mass. With mass, new equilibrium position has spring stretched d. F = 0 Fy = 0
kd – mg = 0d = mg/k Let this point be y = 0
F = mak(d y) mg = mak(d - y) – mg = mak(mg/k) - ky – mg = ma
-k y = may
Same as horizontal! SHO
Physics 201: Lecture 1, Pg 13
New equilibrium position y = -d
Clicker Question 3: A block on a spring oscillates back & forth with simple harmonic motion of amplitude A A plot of displacement (x)harmonic motion of amplitude A. A plot of displacement (x) versus time (t) is shown below. At what points during its oscillation is the speed of the block biggest?
a) When x = +A or -A (i.e. maximum displacement) b) When x = 0 (i e zero displacement)b) When x = 0 (i.e. zero displacement) c) The speed of the mass is constant
+Ax
A
tA
x
Physics 201: Lecture 1, Pg 14
-A
Clicker Question 4: A block on a spring oscillates back & forth with simple harmonic motion of amplitude A A plot of displacement (x)harmonic motion of amplitude A. A plot of displacement (x) versus time (t) is shown below. At what points during its oscillation is the magnitude of the acceleration of the block biggest?biggest?(a) When x = +A or -A (i.e. maximum displacement) (b) When x = 0 (i.e. zero displacement) ( ) ( p )(c) The acceleration of the mass is constant
x+A
tx
Physics 201: Lecture 1, Pg 15
-A
Angular Frequency vs. FrequencyFrequency = number of oscillations per second
f 2 f
q y p
f 22
f
d radsradt
s
2
Physics 201: Lecture 1, Pg 16
t
TAx 2cos tA cos
Example 1A block is pulled 0.30 m from its equilibrium position and let go.A block is pulled 0.30 m from its equilibrium position and let go. (a) Find the block’s equation of motion. (b) What is the block’s position after 0.5 s? The frequency of oscillation for the block/spring system is 0.77 Hz.The frequency of oscillation for the block/spring system is 0.77 Hz.
+ 0.30 m
srad /83.4Hz77.02 sT 311
tx s
rad83.4cosm30.0(a)Remember to use radians!
sT 3.1Hz77.0
Physics 201: Lecture 1, Pg 17
m22.0s5.082.4cosm30.0 sradx(b)
SHM and Circles
A peg on a rotating A peg on a rotating turntable undergoes periodic motion
There is a close
AA
There is a close relationship between circular motion and simple harmonicsimple harmonic motion
Physics 201: Lecture 1, Pg 18
SHM and Circles
Physics 201: Lecture 1, Pg 19
Position
Angular position: Angular position:
t
Call A the vector AApointing toward the peg
The x-position of A is
The x position of A is
tAAx coscos
Physics 201: Lecture 1, Pg 20
This is our equation for SHM!
Clicker Question 5:
What is the time derivative of the function f(t) = A CosWhat is the time derivative of the function f(t) = A Cos (t)?
(a) -A Cos (t)(b) - A Cos (t)(c) A Sin (t)(d) A Sin (t)(e) A Sin (t)(e) - A Sin (t)
Physics 201: Lecture 1, Pg 21
Velocity
The x component of v The x-component of v is: vv
vx
sinvvx
AA
AArv
tAv sint
tAv sin
Av
Physics 201: Lecture 1, Pg 22
Av max
Position and Velocity
t
x is zero when v = +A or - AcosAx x is zero when v = +A or - A v is zero when x = +A or -A
cosAx
sinAv
Physics 201: Lecture 1, Pg 23
Acceleration
The x component of The x-component of a is:
aa ax
aaAA
Ara 22
cosaax
Ara
tAa cos2
t
tAa cos
Aa 2
Physics 201: Lecture 1, Pg 24
Aamax tAv sin
Simple Harmonic Motion:Simple Harmonic Motion:
x(t) = [A]cos(t)v(t) = -[A]sin(t)
x(t) = [A]sin(t)v(t) = [A]cos(t)ORv(t) [A]sin(t)
a(t) = -[A2]cos(t)v(t) [A]cos(t)a(t) = -[A2]sin(t)
OR
xmax = Av = A
Period = T (seconds per cycle)Frequency = f = 1/T (cycles per second)vmax = A
amax = A2
Frequency f 1/T (cycles per second)Angular frequency = = 2f = 2/T
A
Physics 201: Lecture 1, Pg 25
tAtx cos
Clicker Question 6Object A is attached to spring A and is moving in simple harmonic motion. Object B is attached to spring B and is moving in simple harmonic motion. The period and the amplitude of object B are both two times the corresponding values for object A. How do the maximum speeds of the two objects compare?j p
a) The maximum speed of A is one fourth that of object B.b) The maximum speed of A is one half that of object B.c) The maximum speed of A is the same as that of object B.d) Th i d f A i t ti th t f bj t Bd) The maximum speed of A is two times that of object B.e) The maximum speed of A is four times that of object B.
Physics 201: Lecture 1, Pg 26
Simple Harmonic Motion:Simple Harmonic Motion:
x(t) = [A]cos(t) x(t) = [A]sin(t)v(t) = -[A]sin(t)a(t) = -[A2]cos(t)
v(t) = [A]cos(t)a(t) = -[A2]sin(t)
OR
xmax = Avmax = A
Period = T (seconds per cycle)Frequency = f = 1/T (cycles per second)
A l f 2 f 2 /Tamax = A2 Angular frequency = = 2f = 2/T
tAtx cos
Physics 201: Lecture 1, Pg 27