Download L6b--10-18-

Displacement, velocity and acceleration all
Pre-reading for lecture 6b:
oscillate
Simple Harmonic Motion
Physical Sciences 2: Lecture 6b
October 18, 2016
In this lecture we study one of the most important topic in physics: simple harmonic motion
For
this oscillator, x=A at t=0, so the solution is
(SHM). This type of motion is everywhere you look; it’s in a guitar string, a child swinging on a
swing, and in every atom and molecule!
x(t)=A cos(ωt)
Try the simulation:
http://canu.ucalgary.ca/map/content/shm/basic/simulate
An object undergoes simple harmonic motion when it is subjected to a restoring force F that is
proportional to the object’s displacement x. The simplest example of SHM is a mass on a spring
that obeys Hooke's law:
Spring$(k)&
Fs&
Mass$(m)&
Wall$
Consider a spring on a horizontal frictionless surface, as shown above. According to Hooke's law,
the force required to stretch or compress the spring is proportional to how far the spring is
stretched/compressed: 𝐹 = −𝑘𝑥, where 𝑘 is the spring constant and 𝐹 is the force necessary to
displace the spring an amount x from its equilibrium. The negative sign insures the spring force F
always points in a direction opposite to displacement x. If a mass m is attached to the end of the
spring and released at x=A, the only unbalanced force acting on the mass is 𝐹 = −𝑘𝑥 = 𝑚𝑎! .
The resulting motion can be expressed as 𝑥 𝑡 = 𝐴𝑠𝑖𝑛(𝜔𝑡). This motion is shown below:
x%(cm)%
Amplitude$
t%(sec)&
T%
1
Physical Sciences 2: Lecture 6b
October 18, 2016
The maximum distance from equilibrium is called the amplitude A of motion.
The amount of time required for the mass to make one full oscillation (i.e. time for the motion to
repeat itself) is the period of motion T, measured in seconds.
The number of oscillations the mass will make in one period is the oscillation frequency:
𝑓 = (# 𝑜𝑠𝑐𝑖𝑙𝑙𝑎𝑡𝑖𝑜𝑛𝑠) 𝑡𝑖𝑚𝑒, with units of Hertz (where 1 Hz = 1 oscillation/sec).
The period and frequency are related with 𝑓 = 1/𝑇.
Another quantity we use for frequency is the angular frequency 𝜔. It is related to 𝑓 and T as
follows: 𝜔 = 2𝜋𝑓 = 2𝜋 𝑇 radians/sec.
The angular frequency 𝜔 for a mass on a spring is given by 𝜔 = 𝑘 𝑚. This relationship is very
important. It says a large mass will oscillate more slowly than a lighter mass. It also says the
higher the stiffness of the spring (i.e. larger k), the higher the oscillation frequency. Is this
consistent with your intuition?
Another important observation is that the frequency of oscillation does NOT depend on the
amplitude of motion A. If a mass oscillates back and forth with large/small amplitude, the
frequency of oscillation is the same. Turns out this play a crucial role in being able to use
harmonic oscillations to keep track of time (i.e. pendulum, quartz crystal watch).
The only horizontal force acting on the mass is Fspring, which is a conservative force. If you recall
from the energy module 4, the mechanical energy of the spring-mass system is therefore
conserved!
Suppose the total mechanical energy of the system is given by 𝑬𝑚𝑒𝑐ℎ. When the mass is at
𝑥 = ±𝐴 the speed is zero and the kinetic energy is zero as well; all the energy at 𝑥 = ±𝐴 is
!
stored in elastic/spring potential energy, given by 𝑈 = ! 𝑘𝑥 ! .
!
This tells us the total mechanical energy of the mass-spring system is 𝑬𝑀𝑒𝑐ℎ = ! 𝑘𝐴! .
Similarly, when the mass is zipping through the equilibrium position (x = 0), there is no potential
energy and the energy is entirely kinetic. At all other points in the motion there is both P.E. and
K.E. If the mass is at position x with a velocity v, we have:
𝟏
𝟏
𝑬𝑀𝑒𝑐ℎ = 𝟐 𝒌𝒙𝟐 + 𝟐 𝒎𝒗𝟐
The description of simple harmonic motion so far assumes no friction between the mass and the
surface. We therefore expect the mass to oscillate forever with undiminished amplitude (or
mechanical energy). The introduction of friction dampens the motion; the amplitude of motion
decays with time. The time dependence of the amplitude is given by the decreasing exponential
function: 𝑥 𝑡 = 𝑨𝒆!𝒕 𝝉 𝑐𝑜𝑠 𝜔𝑡 . The larger the dissipation, the faster the amplitude decays over
time. The parameter 𝜏, with units of time, is called the decay time constant. It is used to represent
how quickly the oscillations “dies out.” We will be concerned with three regime of damping: (a)
under-damped, (b) over-damped, and (c) critically damped.
2
Physical Sciences 2: Lecture 6b
October 18, 2016
•
Learning objectives: After this lecture, you will be able to…
1. …describe what we mean by simple harmonic oscillation and be able to identify real
world systems that undergo simple harmonic oscillations.
2. …articulate the concept of restoring force and be able to justify the presence of the
minus sign in Hooke’s law.
3. …describe simple harmonic motion mathematically with a sinusoid of a given
amplitude, frequency, period, and phase.
4. …convert a position-versus-time graph to a velocity and acceleration versus time
graph.
5. …describe the motion of a simple harmonic oscillator given the graph for U(x) and total
mechanical energy.
6. …describe the concept of a damped harmonic oscillator along with the three types of
damping: overdamped, underdamped, and critically damped.
7. ...describe how the graph of 𝑥 𝑡 = 𝐴𝑒 !
time constant 𝜏.
3
!
! cos (𝜔𝑡)
changes for different values of the
Physical Sciences 2: Lecture 6b
October 18, 2016
L6b: Am I getting it? #1
The figure represents the motion of a mass on a spring. Let’s check that we got the basics
right:
x%(cm)%
t%(sec)&
22 sec22 sec
1. What is the period of oscillation? (Pick closest answer)
(a)
(b)
(c)
(d)
(e)
(f)
3 sec
4.5 sec
6.5 sec
9.5 sec
15.5 sec
22 sec
2. What is the amplitude of oscillation?
(a)
(b)
(c)
(d)
0.25 cm
0.5 cm
1 cm
2cm
3. What is the approximate position of the mass at time t=5 sec?
(a)
(b)
(c)
(d)
(e)
-0.1 cm
-0.5 cm
-1 cm
1 cm
2 cm
4. What is the first time at which v=0? (Pick closest answer)
(a)
(b)
(c)
(d)
(e)
0 sec
1.5 sec
3 sec
5 sec
6.5 sec
5. At which of these times does the kinetic energy reach its maximum value?
(a) 0 sec
(b) 1.5 sec
(d) 5 sec
(e) 6.5 sec
(c) 3 sec
4
Physical Sciences 2: Lecture 6b
October 18, 2016
Activity 1: where does 𝒙 𝒕 = 𝐀𝒄𝒐𝒔 𝝎𝒕 come from?
Spring$(k)&
A mass attached to a spring with spring
constant k is at rest a distance x=A away
from the equilibrium position (x=0).
At position x=A the force on the block of
mass m is 𝑭 = −𝒌𝑨.
Fs&
Mass$(m)&
Wall$
1. Our goal is to describe the motion of the mass m. Which object(s) should you include
in your free body diagram?
(a) only the mass m
(b) only the spring
(c) the spring and the mass
(d) the wall-spring-mass system
(e) all 4 choices above lead to same answer
***Check with neighboring groups before moving on
2. Draw a free body diagram for your system
3(a) Write down Newton’s 2nd law ( 𝑭𝒙 = 𝑚𝒂𝒙 ). It will yield an equation that contains x(t).
(b) Replace the acceleration 𝒂𝒙 by 𝒂𝒙 = 𝒅𝟐 𝒙 𝒅𝒕𝟐 . This equation is now a differential equation.
(c) Can you check that 𝑥 𝑡 = A𝑐𝑜𝑠 𝜔𝑡 is a solution? Hint: It’s a solutions only if 𝜔! = 𝑘/𝑚
Bonus: Check that 𝑥 𝑡 = A𝑠𝑖𝑛 𝜔𝑡 is also a solution
Check that 𝑥 𝑡 = At ! + 𝐵𝑡 + 𝐶 is NOT a solution, no matter the values for A, B, and C.
5
Physical Sciences 2: Lecture 6b
October 18, 2016
Activity 2A: x(t), v(t), and a(t) versus time graphs
•
The figure shows the same position-Vs-time graph of a particle undergoing simple
harmonic motion with an amplitude A=20cm.
x(t)$
1&
2&
t&(sec)$
5&
4&
3&
(a) Draw the corresponding velocity-Vs-time graph.
(i) What is the amplitude of the v(t) graph? (ii) What is its frequency in Hz and rad/sec?
v(t)$
1&
2&
4&
3&
t&(sec)$
5&
(b) Draw the corresponding acceleration-Vs-time graph.
(i) What is the amplitude of the a(t) graph? (ii) What is its frequency in Hz and rad/sec?
a(t)$
1&
2&
4&
3&
5&
t&(sec)$
(c) If the period T is doubled without changing the amplitude A:
(i) Does the maximum velocity vmax of mass m change? If so, how?
(i) Does the maximum acceleration amax of mass m change? If so, how?
(d) If the amplitude A is doubled without changing the period T:
(i) Does the maximum velocity vmax of mass m change? If so, how?
(i) Does the maximum acceleration amax of mass m change? If so, how?
6
Physical Sciences 2: Lecture 6b
October 18, 2016
Activity 2B: Energy in Simple Harmonic Motion
1. The figure shows a graph of the potential energy of a block oscillating on a spring. The
horizontal (red) line represents the block’s total mechanical energy E.
(a) What is the spring’s equilibrium length? Is it zero?
(b) Where are the turning points of the motion? Explain how you identify them.
(c) What is the block’s maximum kinetic energy?
(d) Draw the block’s kinetic energy as a function of position.
(e) What will be the turning point if the total energy of the block is doubled?
Bonus: What does the following statement mean? 1/2 ∙ 𝑚𝑣!"# ! = 1/2 ∙ 𝑘𝐴!
Can you interpret it? Hint: A is the maximum amplitude of motion
7
this oscillator, x=A at t=0, so the solution is
x(t)=A
cos(ωt)
Physical Sciences
2: Lecture
6b
October 18, 2016
1. The
figure shows a
graph of the potential energy of a block oscillating on a spring. The horizontal line
represents the block’s total mechanical energy E.
the simulation: Activity 3: Damped Simple Harmonic Motion
x&(cm)$
://canu.ucalgary.ca/map/content/shm/basic/simulate/page2.html
(a) What is the spring’s equilibrium length?
Envelope&$
This figure below shows a graph of the
1. The
A$
figure shows a
graph of the potential
!!
!! energy of a block oscillating on a spring. The horizontal line
position-versus-time of a block
represents the block’s !
total mechanical energy E.
! ! oscillating
= !! ! cos!(!!)!!!!!!!!!!
on a spring. There is friction between the
(a) What is the spring’s equilibrium length?
block and the surface it slides on. The
!!
(b) Where are the turning points of the motion? Explain how you identify them.
! ! = !! ! cos!(!!)
block is continually losing energy to
(b) Where are the turning points of the motion? Explain how you identify them.
friction and the(c)motion
is damped.
Kine@c&energy&(J)$
What is the
Spring$(k)&
(d)
10&
(c)
Fs&
(d)
5&
Mass$(m)&
Wall$
2&
4&
DA$
6&
Kine@c&energy&(J)$
5&
t&(sec)$
Draw the
block’s
kinetic
UnderDdamped&response$
energy as a
function of
position.
10&
10&
8&
block’s
maximum
kinetic energy?
What is the
block’s
maximum
kinetic energy?
Draw the
block’s kinetic
energy as a
function of
position.
x&(cm)$
2&
4&
8&
6&
10&
x&(cm)$
(a) For the damped oscillation above, is the time constant 𝜏 greater than or less than the time in
which the oscillation amplitude decays to half of its initial value? Explain
(b) If you increase the amount of friction between the block and the table it slides on, what
happens to the time constant 𝜏? How does6it affect the above graph? Explain.
6
(c) Example: A small earthquake starts a skyscraper vibrating back and forth. The amplitude of
the vibration of the top of the skyscraper is 2.5 m at the moment the quake stops, and 8.0 sec later,
it is 0.5 m. !!
What is the time constant 𝜏 for this damped oscillation? Hint: Use the expression
𝑥 𝑡 = 𝐴𝑒 ! cos (𝜔𝑡)
Bonus: A critically damped oscillator decays back to equilibrium faster than an over-damped oscillator.
If you want your shock absorbers to go back to equilibrium quickly, too much friction/dissipation is NOT
good. You want to add just the right amount
of friction/dissipation to get critical damping.
This may seem a little surprising. Can you
argue why too much damping isn’t good in
this case?
8
Physical Sciences 2: Lecture 6b
October 18, 2016
Displacement, velocity and accelera
oscillate
Activity 4: How do you weigh yourself in space?
Astronauts lose bone density in space and
the medical staff in Houston continuously
monitors their weight. If there is no
gravity, how do you weigh astronauts in
space?
The device they use is called The Body
Mass Measurement Device and it has
been used since Skylab in the 1970’s. It is
used everyday by astronauts on the int’l
space station.
For this oscillator, x=A at t=0, so the s
x(t)=A cos(ωt)
Try the simulation:
http://canu.ucalgary.ca/map/content/shm/ba
Spring$
The components of this important device
are:
(1) a mass m attached to a spring
(2) a spring (with spring constant k)
between the floor and the mass m
(3) the mass of the astronaut mastro
Can you try and figure out how this device can measure the mass of the astronaut?
Spring$(k)&
Fs&
Mass$(m)&
Wall$
Note: It has recently become possible to weigh individual DNA molecules using the same technique! It’s done with a
nano-oscillator approximately 100nm long, vibrating at ~106 Hertz.
Bonus: The spring constant is 606 N/m and the mass m (without the astronaut) oscillates with a
period of 0.901 sec. What will be the period of oscillation with a 80kg astronaut?
9
Physical Sciences 2: Lecture 6b
October 18, 2016
This page has been left blank
10
Physical Sciences 2: Lecture 6b
October 18, 2016
One-Minute Paper
Your name: _____________________________
Names of your group members:
TF: _____________________________
_________________________________
_________________________________
_________________________________
•
Please tell us any questions that came up for you today during lecture. Write “nothing” if
no questions(s) came up for you during class.
•
What single topic left you most confused after today’s class?
•
Any other comments or reflections
11