TUTORIAL 1
SIMPLE HARMONIC MOTION
Instructor: Kazumi Tolich
About tutorials
2
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Tutorials are conceptual exercises that should be worked on in groups.
Each slide will consist of a series of questions that you should discuss
with the students sitting around you.
There will be a few clicker questions per session. Clicker questions are
shown in red.
There will be several teaching assistants wandering around the room
to assist you. If you are having difficulty, they will ask you leading
questions to help you understand the idea.
I. Hook’s law for spring
3
SIMPLE HARMONIC MOTION
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Mech
79
A block of mass m on a frictionless surface is attached to an ideal spring, as shown in figure 1.
I. The
Hooke’s
spring,law
withfor
a springs
spring constant k, is fixed to the wall. The dashed line indicates the position
A block
of
mass
m
on
surface
attached
to an ideal
massless
spring,
in
of the right edge ofa frictionless
the spring when
theisspring
is relaxed,
and
the block
is atasitsshown
equilibrium
figure 1. The spring, which has spring constant k, is fixed to the wall. The dashed line indicates
position. Figures 2 and 3 show the block held in place a distance A to the right and left of the
the position of the right edge of the spring when the spring is neither stretched nor compressed
respectively.
andequilibrium
the block isposition,
at its equilibrium
position.
Block is at rest
Block is at rest
Block is at rest
m
m
A
Figure 1
Figure 2
m
A
Figure 3
Figures 2 and 3 show the block held in place a distance A to the right and left of the equilibrium
position, respectively.
SIMPLE HARMONIC MOTION
Mech
79
I. Hook’s law for spring
I. Hooke’s law for springs
m
A block of mass m on a frictionless surface is attached to an ideal massless spring, as shown in
figure 1. The spring, which has spring constant k, is fixed to the wall. The dashed line indicates
the position of the right edge of the spring when the spring is neither stretched nor compressed
and the block is at its equilibrium position.
4
Figure 1
Block is at rest
Block is at rest
m
A
m
A
Figure 3
Figure 2
Block is at rest
Figures 2 and 3 show the block held in place a distance A to the right and left of the equilibrium
position, respectively.
m
m
m
A
Figure 1
A
A. In the boxes at right, draw arrows to represent
Figure 3
Figure
2
the
directions
of:
Arrows for figure 2
Arrows for figure 3
Position of block
Position of block
Force on block
by spring
Force on block
by spring
Figures 2 and 3 show the block held in place a distance A to the right and left of the equilibrium
A. position,
In therespectively.
boxes at right, draw arrows
to represent
the directions
=0
• the position
of the block,
x , taking xof:
when
the
block
is
at
its
equilibrium
position
and
¤ The position of the block, 𝐱, taking 𝐱 = 0 when the block is at the equilibrium
Arrows for figure 2
Arrows for figure 3
A. In the
boxes at right, draw arrows to represent
position.
• the force
on the block by
the spring, F.
the directions of:
Position of block
Position of block
⃗.
¤ The force on the block by theUse
spring,
the 𝐅
arrows
that you drew to explain why the
• the position of the block, x , taking x = 0 when
minus
sign
is
necessary
insign
the expression
the
block
is
at
its
equilibrium
position
and
¤ Use the arrows that you drew to explain why the minus
is necessary in
=
–kx
(Hooke’s
law
for
an
ideal
spring).
F
• the
force
on
the
block
by
the
spring,
F.
Force on block
Force on block
Hook’s law: 𝐅⃗ = −𝑘𝐱.
Use the arrows that you drew to explain why the
B.
by spring
by spring
minus sign
is necessary
in the expression
What
is the
net force
on the block when it is held in place as shown in
F = –kx (Hooke’s law for an ideal spring).
figure 2?
B. What is the net force on the block when it is held in place as shown in figure 2? Explain.
B. What is the net force on the block when it is held in place as shown in figure 2? Explain.
Quiz: T1-1 answer
5
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¨
¨
The net force is zero.
The block is not accelerating, so according to Newton's 2nd law, the
net force on it is zero.
∑ 𝐅⃗ = 𝑚𝒂
SIMPLE HARMONIC MOTION
I. Hook’s law for spring
6
Mech
79
I. Hooke’s law for springs
A block of mass m on a frictionless surface is attached to an ideal massless spring, as shown in
figure 1. The spring, which has spring constant k, is fixed to the wall. The dashed line indicates
the position of the right edge of the spring when the spring is neither stretched nor compressed
and the block is at its equilibrium position.
Block is at rest
Block is at rest
Block is at rest
m
m
A
Figure 1
m
A
Figure 3
Figure 2
Figures 2 and 3 show the block held in place a distance A to the right and left of the equilibrium
position, respectively.
¤ Suppose
the hand in figure 2 were suddenly removed. After the hand is
Arrows
for figureon
2
Arrows
figure 3 by the spring be related to the
A. In the
boxes at right, draw
arrows to
represent the
removed,
how
would
force
the forblock
the directions of:
Position of block
Position of block
net
force
onx, the
taking xblock?
= 0 when
• the
position
of the block,
the block is at its equilibrium position and
C.
Use Newton’s second law to write an expression for the
acceleration, 𝐚, of the block in terms of 𝐱, 𝑘, and 𝑚 for an instant
after the block has been released.
• the force on the block by the spring, F.
Use the arrows that you drew to explain why the
minus sign is necessary in the expression
F = –kx (Hooke’s law for an ideal spring).
Force on block
by spring
Force on block
by spring
B. What is the net force on the block when it is held in place as shown in figure 2? Explain.
Mech
80
Simple harmonic motion
II. Simple harmonic motion
7
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A.
Mech Simple harmonic motion
II. Simple harmonic
motion
80
A pendulum is hung directly
above
the
block-spring system
II.𝑚
Simple
harmonic
motion above the blockA pendulum
with
a
mass
is
hung
directly
from section I. The block
andisthe
bob
both have
A pendulum
hungpendulum
directly above the
block-spring
system
spring
system
from
section
I.
The
length
of
the
pendulum
𝑙.
from
section
I.
The
block
and
the
pendulum
bob
both
mass m. The spring constant is k, and the length of the ishave
mass m. The spring constant is k, and the length of the
Thependulum
parameters
theparameters
systems
been
chosen
such
that
is l.ofThe
of the
systems
have
been
pendulum is have
l. The
parameters
of the systems
have
been
chosen
such
that
the
block
is
always
directly
below
the
that the
block
isbob.
always
below
thechosen
block issuch
always
directly
below
the directly
pendulum
bob.the
pendulum
pendulum bob.
A. In the spaces
below,
draw a on
vector
to represent
net
Draw a vector to represent
the net
force
the
blocktheand
force on each object when it is on the far left as shown.
A. bob
In the
spaces
a vector
to represent
the net
the
when
it isbelow,
on thedraw
far
left
as shown.
Net force on block
Net force on bob
force on each object when it is on the far left as shown.
Net force on block
Net force on bob
l
l
m
k
m
Equilibrium
k
position
m
m
B. Suppose that the masses of the block and the pendulum bob were doubled, and the block and
Equilibrium
bob were then released from rest at the far left position.
position
The following questions serve as a guide to help you determine whether the block would
remain directly below the pendulum bob at all times.
Quiz: T1-2 answer
Mech
80
8
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Simple harmonic motion
Simple harmonic
motion
The net force on the bob isII.tangent
to its path
A pendulum is hung directly above the block-spring system
toward the equilibrium position.
from section I. The block and the pendulum bob both have
𝐓
m. The spring constant is k, and the length of /0
the
The bob is not acceleratingmass
radially
or
pendulum
is l. from
The parameters
of the systems have been
chosenof
suchacceleration
that the block is always directly below the
toward the pivot. The direction
pendulum bob.
does not have any component in the radial
In the spaces below, draw a vector to represent the net
direction. The component ofA. the
in thewhen it is on the far left as shown.
forceweight
on each object
radial direction is balanced withNetthe
forcetension
on block in
Net force on bob
the string.
The weight of the bob has a component
tangent to the path toward the equilibrium
position.
𝐖/2
l
m
k
m
Equilibrium
position
B. Suppose that the masses of the block and the pendulum bob were doubled, and the block and
bob were then released from rest at the far left position.
The following questions serve as a guide to help you determine whether the block would
remain directly below the pendulum bob at all times.
II. Simple harmonic motion
9
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¨
1.
Suppose that the masses of the block and the pendulum bob were doubled,
and the block and bob were then released from rest at the far left position.
The following questions serve as a guide to help you determine whether the
block would remain directly below the pendulum bob at all times.
As a result of doubling the masses of the block and pendulum bob, would
the following quantities increase by a factor of 2, decrease by a factor of
2, or remain the same?
¤
the magnitude of the net force on the block, 𝐹456, /9:;< , or the bob, 𝐹456, /:/ , when
it is at the far left
Quiz: T1-3 answer
10
¨
¨
increase by a factor of 2
The weight of the bob doubles, so does its component
tangent to the path.
𝐓/0
𝐖/2
II. Simple harmonic motion
11
As a result of doubling the masses of the block and pendulum bob,
1.
¤
¤
would the time it takes for each object to travel from the far left position to the equilibrium position, ∆𝑡/9:;< or ∆𝑡/:/ ,
increase, decrease, or remain the same?
Check that your answers regarding ∆𝑡/9:;< and ∆𝑡/:/ are consistent with the relationships below: (Hint:
For each object, what is the relationship between ∆𝑡 and the period of oscillation, 𝑇?)
2.
3.
would the magnitude of the acceleration of each object, 𝑎/9:;< or 𝑎/:/ , when it is at the far left increase by a factor of 2,
decrease by a factor of 2, or remain the same?
¤
𝑇/9:;< = 2𝜋 𝑚⁄𝑘 (period of oscillation for a mass on a spring)
¤
𝑇/:/ = 2𝜋 𝑙⁄𝑔 (period of oscillation for a simple pendulum)
Will the block and pendulum bob still move together after their masses have been doubled? If not,
describe what additional changes could be made so that the block and pendulum bob would again move
together.
Quiz: T1-4 answer
12
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¨
No
The period of oscillation is increased by 2 for the block, but it remains the
same for the bob.
E
F
¨
∆𝑡/9:;< = 𝑇/9:;< =
¨
∆𝑡/:/ = 𝑇/:/ =
E
F
E
F
E
F
E
G
2𝜋 𝑚⁄𝑘 = 𝜋 𝑚⁄𝑘 (depends on mass)
E
G
2𝜋 𝑙 ⁄𝑔 = 𝜋 𝑙 ⁄𝑔 (independent of mass)
• Tbob = 2π l/g (period of oscillation for a simple pendulum)
III. Energy of a simple harmonic oscillator
3. Will the block and pendulum bob still move together after their masses have been
doubled? If not, describe what additional changes could be made so that the block and
pendulum bob would again move together.
13
¨
The diagram at right is a plot of the total
III. Energy of a simple harmonic oscillator
energy of a The
horizontal
block-spring system as
diagram at right is a plot of the total energy of a
block-spring
system
asblock
a functionwith
of the
a function ofhorizontal
the
position
of
the
position of the block with respect to its equilibrium
The block oscillates
with aThe
maximum
distance
respect to itsposition.
equilibrium
position.
block
from equilibrium of A.
oscillates withA. aWhat
maximum
distance from
feature of the diagram shows that the total
energy of the system is conserved as the block
equilibrium of A.
oscillates?
E
Etot
-A
B. Determine what fraction of the total energy is potential energy when the block is at
(1) x = +A and (2) x = +A/2. (Hint: U = 1/2 kx2.)
+A
x
oscillation, T?)
• Tblock = 2π m/k (period of oscillation for a mass on a spring)
III. Energy of a simple harmonic oscillator
• Tbob = 2π l/g (period of oscillation for a simple pendulum)
14
A.
B.
C.
D.
What feature of the diagram shows
is conserved as the block oscillates?
3. Will the block and pendulum bob still move together after their masses have been
doubled? If not, describe what additional changes could be made so that the block and
bob would
again of
move
together.
thatpendulum
the total
energy
the
system
Determine what fraction of the total energy is potential energy, 𝑈 =
III. Energy of a simple harmonic oscillator
1⁄2 𝑘𝑥 G , when the block is at 𝑥 =The+𝐴
and 𝑥 = + 𝐴⁄2. Plot the
diagram at right is a plot of the total energy of a
horizontal
potential energy stored in the spring
as ablock-spring
function system
of 𝑥.as a function of the
E
Etot
position of the block with respect to its equilibrium
maximum
distance
kinetic position.
energyTheofblock
theoscillates
block,with
𝐾,aas
a function
from equilibrium of A.
On the same axes, plot the
of 𝑥. (Hint: What function must be added to the potential energy to
A. What feature of the diagram shows that the total
equal the total energy of the system?)
Label
so asthat
you
energy
of theyour
systemgraphs
is conserved
the block
oscillates?
can easily distinguish the kinetic energy 𝐾 from the potential energy 𝑈.
Is the time that the block takes to move from 𝑥 = 0 to 𝑥 = + 𝐴⁄2
-A
greater than, less than, or equal to the time that the block takes to move
from 𝑥 = + 𝐴⁄2 to 𝑥 = +𝐴? Explain
how your
answer
is total
consistent
B. Determine
what fraction
of the
energy is potential energy when the block is at
(1) x = +A and (2) x = +A/2. (Hint: U = 1/2 kx .)
with your graph of the kinetic energy.
+A
2
On the same axes above, plot the potential energy stored in the spring, U, as a function of x.
x
Quiz: T1-5 answer
15
¨
¨
¨
¨
Less than
As the block moves from 𝑥 = 0 to 𝑥 = + 𝐴⁄2, the kinetic energy is always greater
than its kinetic energy as the block moves from 𝑥 = + 𝐴⁄2 to 𝑥 = +𝐴.
So, the speed of the block is also always greater than its as the block moves
from 𝑥 = + 𝐴⁄2 to 𝑥 = +𝐴.
The faster it moves, the less time it takes to cover the same distance.
oscillation, T?)
• Tblock = 2π m/k (period of oscillation for a mass on a spring)
III. Energy of a simple harmonic oscillator
• Tbob = 2π l/g (period of oscillation for a simple pendulum)
16
3. Will the block and pendulum bob still move together after their masses have been
doubled? If not, describe what additional changes could be made so that the block and
pendulum bob would again move together.
Consider two points, P and Q, to the right of the block’s
equilibrium position. Point P has position 𝑥 = + 𝐴⁄2. Point Q is
the point for which the kinetic energy and the potential energy of
III. Energy of a simple harmonic oscillator
the system are each equal to half
the total energy.
The diagram at right is a plot of the total energy of a
E.
1.
2.
3.
horizontal
system
as a function
of the
Is point Q to the left of, to the right
of, orblock-spring
at the same
position
as point
position of the block with respect to its equilibrium
P? Mark the locations of points Pposition.
and QThe
onblock
the oscillates
𝑥-axis above.
with a maximum distance
from equilibrium of A.
When the block is at point P, is the kinetic energy of the system greater
A. What feature of the diagram shows that the total
than, less than, or equal to the potential
energy? Explain how you can
energy of the system is conserved as the block
oscillates?
tell from the graph.
Calculate the ratio of the kinetic energy to the potential energy when
the block is at point P (i.e., at 𝑥 = + 𝐴⁄2).
E
Etot
-A
+A
B. Determine what fraction of the total energy is potential energy when the block is at
(1) x = +A and (2) x = +A/2. (Hint: U = 1/2 kx2.)
On the same axes above, plot the potential energy stored in the spring, U, as a function of x.
x
Quiz: T1-6
17
¨
3
¨
The total mechanical energy is constant and given by 𝐸 = 𝑈 + 𝐾 = 𝑘𝐴G .
¨
E
G
This can be obtained at the turning point, where 𝐾 = 0, and all the mechanical
energy is due to the potential energy.
E
O G
G
G
E
= 𝑘𝐴G .
¨
Potential energy at 𝑥 = + 𝐴⁄2 is 𝑈 = 𝑘 +
¨
Kinetic energy at 𝑥 = + 𝐴⁄2 is 𝐾 = 𝐸 − 𝑈 = 𝑘𝐴G − 𝑘𝐴G = 𝑘𝐴G .
¨
The ratio is
R
S
=
T
VOW
U
X
VOW
U
=3
P
E
E
Q
G
P
P
oscillation, T?)
• Tblock = 2π m/k (period of oscillation for a mass on a spring)
III. Energy of a simple harmonic oscillator
• Tbob = 2π l/g (period of oscillation for a simple pendulum)
18
3. Will the block and pendulum bob still move together after their masses have been
doubled? If not, describe what additional changes could be made so that the block and
pendulum bob would again move together.
Suppose that the block were replaced by a
block with half the mass and released from rest
III. Energy of a simple harmonic oscillator
at 𝑥 = +𝐴.
The diagram at right is a plot of the total energy of a
F.
1.
2.
horizontal block-spring system as a function of the
position of the block with respect to its equilibrium
position. The block oscillates with a maximum distance
from equilibrium of A.
E
Describe any resulting changes in the three energy
graphs (𝐸6:6Z9 , 𝑈, 𝐾). Explain.
A. What feature of the diagram shows that the total
the system isof
conserved
as the block
Is it possible to determineenergy
the ofperiod
oscillation
oscillates?
of a mass-spring system using information from
-A
energy graphs alone? If so, describe the steps you
would take to determineB. the
period.
If ofnot,
state
Determine
what fraction
the total
energy is potential energy when the block is at
(1) x = +A and (2) x = +A/2. (Hint: U = 1/2 kx .)
what other information you would need.
Etot
+A
2
On the same axes above, plot the potential energy stored in the spring, U, as a function of x.
x
Quiz: T1-7 answer
19
¨
¨
No
From the amplitude and the total mechanical energy, you can determine the
E
E
spring constant since at the turning point, 𝐸 = 𝑈 + 𝐾 = 𝑘𝐴G + 0 = 𝑘𝐴G .
G
¨
But to determine the oscillation period, you need to know the mass:
𝑇/9:;< = 2𝜋 𝑚⁄𝑘.
G
III. Energy of a simple harmonic oscillator
20
Simple harmonic motion
Mec
83
Suppose instead that the original block were released from rest
⁄2released
at 𝑥 = − 𝐴⁄2 and movedG.between
the positions
𝑥 =block
−𝐴
and from rest at x = -A/2 and moved
Suppose instead
that the original
were
between the positions x = -A/2 and x = +A/2.
𝑥 = + 𝐴⁄2.
G.
1.
2.
3.
4.
E
1. As
result of
this change,
woulddecrease,
the total energy
As a result of this change, would
thea total
energy
increase,
or
increase, decrease, or remain the same?
remain the same? Explain.
Explain.
On the axes at right, graph the total energy, 𝐸6:6Z9,45[ , potential
energy 𝑈45[ , and kinetic energy 𝐾45[ for the new motion.
Does any part of the new set of graphs exactly coincide with any part
of the previous set of graphs? Explain.
Is the time that the block takes to move from 𝑥 = 0 to 𝑥 = + 𝐴⁄2
2. to
Onthe
the time
axes atthat
right,
graph
the total
greater than, less than, or equal
the
block
tookenergy
to move
Etotal, new, potential energy Unew, and kinetic energy
from 𝑥 = 0 to 𝑥 = + 𝐴⁄2 beforeKthe
amplitude of oscillation was
new for the new motion.
reduced? Explain.
Etot, old
-A
+A
3. Does any part of the new set of graphs exactly coincide with any part of the previous set
of graphs? Explain.
x
Quiz: T1-8 answer
21
¨
¨
¨
¨
Greater than
The total mechanical energy is reduced to 𝐸45[ =
E
𝑘
G
O G
+
G
E
F
= 𝐸:9\ .
The potential energy curve between 𝑥 = − 𝐴⁄2 and 𝑥 = + 𝐴⁄2 remains
the same.
So, the kinetic energy between 𝑥 = − 𝐴⁄2 and 𝑥 = + 𝐴⁄2 is always
smaller in the new case than the old case, implying that the block is moving
slower in the new case.