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DL 11-1 Overview
Timeline
(10 minute intervals; elapsed time)
Equipment
11-0 : Administrativa TA station
web browser
11-1-0: Introductory Remarks Overhead
11-1-1 : Oscillatory Motion
SG tasks
Each station
Spring
Weight sets (10.0 g to 1.0 kg)
Support bracket
Pasco box
Ibook
WC discussion Overhead
11-1-2 : SHM and Hooke’s Law
SG tasks
Each station
Spring
Weight sets
Support bracket
Meter stick
WC discussion Overhead
11-1-3 : SHM Equation
SG tasks
Spring
Weights set
Support bracket
Pasco box
Ibook
WC discussion Overhead
Activity 11-0: Administrativa
Take roll, mainly such that students know that they are in the right room for the right course.
Before DL actually starts is a good time to put up this information ("Physics 7C, DL section __, TB
114"), along with your name, your office hours and e-mail contact information. Do not let this eat up too
much time. It is okay to start the next activity right away once the majority of the students have arrived,
and deal individually with the few stragglers come in late (be understanding; this is their first DL of the
new quarter).
Deal with the actual on-line enrollment administrative tasks after DL has finished—it is probably
more better for you (and the students) if you don't have to deal with this during instruction time. Follow
the latest add and drop guidelines discussed in the TA meeting. Print out a hard copy for your own
records.
Activity 11-1-1: Oscillatory Motion
Concepts:
Oscillatory motion is a repetitive, periodic motion.
The motion is a displacement from an equilibrium position.
Period T is the time to repeat a cycle, not the time to come back to the same (non-maximum) position.
The period is independent of starting conditions.
The amplitude, A, corresponds to the maximum displacement from equilibrium. This does depend on
the starting conditions.
Graphing skills! Labeling axes. Reading meaningful information off graphs.
Simple Harmonic Motion (SHM) can be described with a sinusoidal function.
Y(t) = Asin(2t/T)
Briefly show on a set-up what parameters are associated with what on the mass-spring system and
how to measure these parameters.
Make sure that the students understand the basic concept of how to measure the period of an SHM
system—the amount of time to repeat a cycle of SHM, not the time to come back to the same (non-
maximum) position. Measure multiple periods.
In the WC discussion, have separate SGs demonstrate how they determined the dependence of the
period on a given parameter, and whether the period depends, does not depend, is proportional, or
inversely proportional to this parameter.
Some "real-world" thinking questions you can use to prompt and guide your students:
- Does the period of the mass-spring system depend on g? Would the period be any different on the
Moon or on the ISS? But wouldn't equilibrium point hang not as low at those places?
Activity 11-1-2: SHM and Hooke's Law
Concepts:
The force is restorative.
Using static forces we can determine the spring constant.
Using the spring constant and Newton’s 2
nd
law, we can determine acceleration.
The position of an SHM system can be expressed using a generalized sine function.
The y , v , and a of an SHM system are related via differential expressions.
The y and a of an SHM system are related via Hooke's Law.
Combining all of these we can relate the mass and spring constant to the period.
How do you characterize simple harmonic motion? How do the graphs connect to the actual
motion of the objects? How do the equations for the oscillations connect to the graphs? This activity
should answer these questions. Spend some time going through the answers after the groups have put
their results on the board. Make sure that they scale the maximum/minimum values of each graph.
t
y ( t )
+ A
– A
I II III IV
t
v ( t )
I II III IV
t
a ( t )
I II III IV
T
2
A
2
T
A
T
A
T
A
Answers:
2. (a) 0 (although not a given choice), II, IV
(b) I, III
3. (a) I, III
(b) 0 (although not a given choice), II, IV
4. (a) 0 (although not a given choice), II, IV
(b) I, III
Be sure to make the connections between the position, the velocity and the acceleration graphs
and expressions (and of course the force, via Hooke's Law). Start with Newton's Second Law (and not
Hooke's Law, which is what they are trying to prove). Then
F
= ma ; but a – y from the kinematic
graphs/equations, thus
F
= – ky , which is Hooke's Law!
Activity 11-1-2: SHM and Hooke's Law
()
The most general form of the equation that describes any object
undergoing SHM (simple harmonic motion) is given by:
.
Recall that how the chain rule and derivatives of harmonic functions
(sines and cosines) work:
d
dt
A sin t
A cos t
;
d
dt
A cos t
A sin t
.
The velocity and acceleration of SHM objects are just derivatives with
respect to time t of the y ( t ) position:
v t
d
dt
y t
;
a t
d
dt
v t
d
dt
y t
.
- Use your expressions for y ( t ), v ( t ),
and a ( t ) to complete the v ( t ) and a ( t )
graphs at right (note that here, =
0). Scale the maximum and
minimum values for v ( t ) and a ( t ) on
the vertical axes of their graphs.
2. At what time(s) (I, II, III, IV) is the object:
(a) momentarily located at the equilibrium point?
(b) momentarily located furthest from the equilibrium point?
3. At what time(s) (I, II, III, IV) is the object:
(a) momentarily stationary?
(b) moving at its maximum speed?
4. At what time(s) (I, II, III, IV) is the motion of the object:
(a) momentarily steady ( a = 0, or ∆ v = 0)?
(b) changing ( a ≠ 0, or ∆ v ≠ 0)?
5. Does the net force
F
on this object "obey" Hooke's Law (that is,
force –displacement)? Clearly explain your reasoning, using
the graphs above, and/or your expressions for y ( t ), v ( t ), and a ( t ).
t
y ( t )
+ A
– A
I II III IV
t
v ( t )
I II III IV
t
a ( t )
I II III IV
Activity 11-1-2: SHM and Hooke's Law
()
Any force that tries to "restore itself" ( i.e., proportional to
displacement, towards equilibrium) is said to obey Hooke's Law.
Most generally: "restoring" F = – k ·(some distance).
What does the negative sign in Hooke's Law mean?
- If the net force on an object "obeys" Hooke's Law, then the object
must undergo SHM:
Hooke's Law
F
SHM
y ( t )
- If an object is undergoing SHM, then its net force must "obey"
Hooke's Law:
Hooke's Law
F
SHM
y ( t )
Which of these two statements did you prove in this
activity—the forward (Hooke's Law SHM)
statement, or the reverse (Hooke's Law SHM)
statement?
- "Constant phase" = related to where you start timing the cycle
(the moment as it goes down through 0? When it is at its lowest
point?)
- The period of a mass' oscillation = time [s] to repeat one cycle of
motion.
T 2
m
k
.
Activity 11-1-3: SHM Equation
Concepts
- The position of an object undergoing SHM motion can be described by:
- Avoiding the use of the word "origin." What physicists mean by "origin" ( y = 0) is often different than
some students' colloquial use of "origin" (whatever the initial value of y is at t = 0, i.e., where y
"originates" from).
This is the students' introduction to graph-reading and the SHM equation of motion. Let them
deduce all the values and units for themselves. This activity sets the foundation for the ever more
intricate harmonic wave function they will discuss later in this Block.
Students need to make sure their calculators are in radians mode! Many students may not have
ever used the trigonometric functions on their calculators.
Ask each group to demonstrate their SHM system with a changed parameter, in addition to
explaining how their how their y ( t ) graph changes as a result.
- Doubling the mass m will increase the period by sqrt(2).
- Doubling the spring constant is tricky! Try adding two springs in parallel. Period is reduced by sqrt(2).
- Doubling the amplitude A will only change the vertical scale of the graph.
- Doubling the constant phase offset from +/2 to + will mean that the mass will start at the
equilibrium position at t = 0 s (which means it needs to be pushed down to get it started), instead of
starting from its maximum displacement angle at t = 0 s.
So during the whole class discussion, make sure students can "act" out the motion of their mass-
spring systems for each of these parameter changes, as well as understanding how their SHM graphs
change! It is important for students to see the relations between a y(t) graph and the motion it
represents, especially with regard to the initial starting position at t = 0.
Activity 11-1-3: SHM Equation
( 30 minutes)
Consider the specific case of a m = 0.2 kg mass on your spring:
1. Acquire a plot of the
vertical position and
draw it on the board.
What are the values and
units of these SHM
parameters?
For the following situations (2)-(5), in which the physical parameters
are changed, first draw a plot of your expected vertical displacement
versus time, then acquire a plot and draw a new curve in a new color
on the same time graph. Scale (rescale) your axes as necessary. Be
prepared to demonstrate the motions.
2. The amplitude A is halved.
3. The mass m is doubled.
4. The parameter is doubled.
5. The spring constant k is doubled.
Value? Units?
(a) A
(b) T
(c)
(d) B
Activity 11-1-3: SHM Equation
()
"Drawing a scaled and labeled y(t) graph" tips:
- Once all your SHM parameters are known, it is easier to draw a sine
curve first before scaling your axes with just period T intervals (and
half-intervals):
t [s]
[°]
t [s]
[°]
- Even if all your SHM parameters are known, drawing a sine curve
after scaling your axes is more difficult (try this!):
t [s]
[°]
t [s]
[°]
Physics 7C Fall 2002: DL 11-1 16
- The rotational inertia of a ring hanging from its edge is I = 1/2 mD
2
, where D is the diameter of the ring
(average of inner and outer diameters of the real rings).
a) Draw an extended force diagram showing the forces acting on the ring when it by an angle
from its equilibrium position. Choose the point of support of the ring as the axis of rotation to
calculate torques about. Calculate the net torque as a function of the sin. Make the small
angle approximation for sin.
b) Substitute your expressions for the net torque and I into the angular expression of Newton’s 2nd
Law: = I
- A bowling ball that is cut in half and set on the round side will oscillate from side to side if it is displaced
slightly. Why?
- How could you use a pendulum to accurately measure the force of gravity? Would this work on the
moon? What would the length of a pendulum be on the Earth that gives a period of 2 s? What would the
length of a pendulum be on the Moon that gives a period of 2 s?
- Derive an expression for the velocity of the pendulum bob, the angular velocity of the ring, and the
vertical velocity of the mass on the spring. This is a total of six equations.
02.09.
Physics 7C Fall 2002: DL 11-1 17
(Optional exercises—solutions are given below)
1. A 0.20 kg pendulum swings with no air resistance nor friction from its
highest point 1 to point 2, where it hits a barrier, and then swings
onwards onto its highest point 3. Circle the most correct answer
below, then briefly explain your answer.
I. The time for the pendulum to swing from point 1 to point 2 takes longer than the time
for the pendulum to swing from point 2 to point 3.
II. The time for the pendulum to swing from point 1 to point 2 takes the same amount of
time for the pendulum to swing from point 2 to point 3.
III. The time for the pendulum to swing from point 1 to point 2 takes shorter than the
time for the pendulum to swing from point 2 to point 3.
2. Astronauts on the surface of the Moon set up a pendulum with a string that is 0.5 m long.
They time the period of one oscillation to be 3.48 sec. What is the gravitational constant
g
Moon
3. How long should a pendulum string be in order to make a T = 1 sec "cuckoo" clock? How
long should a pendulum string be in order to make a T = 2 sec "grandfather" clock?
4. What causes the "restoring force" for a simple pendulum?
I. Rope tension force of the ceiling on the pendulum bob.
II. Radial component of the gravitational force of the Earth on the pendulum bob.
III. Tangential component of the gravitational force of the Earth on the pendulum
bob.
IV. Rope tension force of the pendulum bob on the ceiling.
V. None of the above.
5. Consider two identically constructed mass/spring systems. System "A" has four times the
total energy of system "B". What can be concluded about the simple harmonic motion of
these systems?
I. The frequency of system "A" is 4 greater than system "B."
II. The frequency of system "A" is 2 greater than system "B."
III. The amplitude of system "A" is 4 greater than system "B."
IV. The amplitude of system "A" is 2 greater than system "B."
V. There is not enough information is provided to conclude any of the above.
02.09.
20 cm
10 cm
1
2
3
barrier
Physics 7C Fall 2002: DL 11-1 19
Write the equations describing the position of these SHM objects as functions of time, given the
graphs (8)-(10) below.
y [m]
0
2
t [s]
–0.
+0.
1 3 4 5 6 7 8
–0.
+0.
y [m]
0
2
t [s]
–0.
+0.
1 3 4 5 6 7 8
–0.
+0.
y [m]
0
2
t [s]
–0.
+0.
1 3 4 5 6 7 8
–0.
+0.
Exercise solutions
1. The correct answer is (I).
g
Moon
= 1.63 m / s
2
3. For the clock with a 1 second period L = 0.25 m. For the clock with a 2 second period
L = 0.99 m. Compare these to the actual physical sizes of cuckoo and grandfather clocks.
4. The correct answer is (III).
5. The correct answer is (IV).
6. The correct answer is (III).
y t 0. 3 msin
2 t
- 31 s
y t 0. 2 msin
2 t
- 0 s
y t 0. 4 msin
2 t
10 s
y t 0. 2 msin
2 t
- 0 s
0. 2 m
02.09.
Physics 7C Fall 2002: DL 11-1 20
Announcements
Get the Physics 7C Block notes (in their entirety) from the Physics 7C web page
(http://physics7.ucdavis.edu) for free , or purchase them from Navin's Copy Shop, or photocopy
them from the Reserve Desk at Shields Library.
The first day of lecture will be on Thursday, October 4, in Roessler 66.
Practice Quiz 1 will be given during lecture on Thursday, October 4, and will cover the
material in Block 1. It will be a portion of an actual Physics 7C Quiz given in a previous
quarter. While the Practice Quiz does not count towards your course grade, it is strongly
suggested that you study for it and use the Practice Quiz as an in-class diagnostic tool to gauge
your understanding of the material covered so far in Block 1.
02.09.
