Transcript power pt on SHM
Chapter
10 Simple harmonic motion
Adapted from Holt book on physics
Section 1
Simple Harmonic Motion
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Simple Harmonic Motion
Defn: any periodic motion that is the result of a restoring force that is proportional to displacement. Period = T (seconds per cycle) Frequency = f = 1/T (cycles per second) Angular frequency = = 2 f = 2 /T Physics 101: Lecture 22, Pg 2
Chapter
10 Section 1 Simple Harmonic Motion
Simple Harmonic Motion
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
10 Section 2 Measuring Simple Harmonic Motion
Amplitude in SHM
• In SHM, the maximum displacement from equilibrium is defined as the
amplitude
of the vibration.
– A
pendulum’s
amplitude = swing angle from the vertical ( Dq = radians) – For a
mass-spring system,
the amplitude is the maximum amount the spring is stretched or compressed from its equilibrium position. ( D x = meters)
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Chapter
10 Section 2 Measuring Simple Harmonic Motion
Measures of Simple Harmonic Motion
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Chapter
11 Section 2 Measuring Simple Harmonic Motion
Amplitude of a pendulum
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Chapter
11 Section 1 Simple Harmonic Motion
Simple Harmonic Motion
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Section 1 Simple Harmonic Motion
Hooke’s Law
• One type of
periodic motion
is the motion of a mass attached to a spring.
• The direction of the force acting on the mass (
F elastic
) is always opposite the direction of the mass’s displacement from equilibrium (
x
= 0).
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Chapter
11 Section 1 Simple Harmonic Motion
Hooke’s Law, continued
At equilibrium:
• The
spring force
and the mass’s
acceleration
become
zero.
• The
speed
reaches a
maximum.
At maximum displacement
: • The
spring force
and the mass’s
acceleration
a
maximum.
reach • The
speed
becomes
zero.
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Section 1 Simple Harmonic Motion
Hooke’s Law, continued
• Measurements show that the
spring force,
or
restoring force,
is
directly proportional
to the
displacement
of the mass.
• This relationship is known as
Hooke’s Law:
F elastic
= –
kx
spring force = –(spring constant displacement) • The quantity
k
is a positive constant called the
spring constant.
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Chapter
11
Spring Constant
Section 1 Simple Harmonic Motion
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Section 1 Simple Harmonic Motion
Sample Problem
Hooke’s Law
If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, what is the spring constant?
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Section 1 Simple Harmonic Motion
Sample Problem, continued
1. Define
Given:
m
= 0.55 kg
x =
–2.0 cm = –0.20 m
g
= 9.81 m/s 2 Diagram: Unknown:
k
= ?
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Section 1 Simple Harmonic Motion
Sample Problem, continued
2. Plan Choose an equation or situation:
When the mass is attached to the spring,the equilibrium position changes. At the new equilibrium position, the net force acting on the mass is zero. So the spring force (given by Hooke’s law) must be equal and opposite to the weight of the mass.
F
F
net
= 0 =
F elastic
elastic
= –
kx F g
–
kx
= –
mg
–
mg
= 0 +
F g
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Section 1 Simple Harmonic Motion
Sample Problem, continued
2. Plan, continued
Rearrange the equation to isolate the unknown:
kx
mg
0
kx
mg k
mg x
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Chapter
11 Section 1 Simple Harmonic Motion
Sample Problem, continued
3. Calculate Substitute the values into the equation and solve:
k
mg x
(0.55 kg)(9.81 m/s 2 ) –0.020 m
k
270 N/m
4. Evaluate
The value of
k
implies that 270 N of force is required to displace the spring 1 m.
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Section 1 Simple Harmonic Motion
The Simple Pendulum
• A
simple pendulum
consists of a mass called a bob, which is attached to a fixed string.
• At any displacement from equilibrium, the
weight
of the bob (
F g
) can be resolved into two components.
• The
x component
q
)
(
F g,x = F g sin
is the only force acting on the bob in the direction of its motion and thus is the
restoring force.
The forces acting on the bob at any point are the force exerted by the string and the gravitational force.
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Section 1 Simple Harmonic Motion
The Simple Pendulum, continued
• The magnitude of the restoring force (
F g,x = F g sin
q
)
is proportional to sin q .
• When the maximum angle of displacement q is relatively small (<15 °), sin q is approximately equal to in radians.
q • As a result,
the restoring force is very nearly proportional to the displacement
.
• Thus, the pendulum’s motion is an excellent approximation of
simple harmonic motion.
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Chapter
11 Section 1 Simple Harmonic Motion
Restoring Force and Simple Pendulums
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Pendulum
g L T
2 2
L g
For “small oscillation”, period does not depend on •mass •amplitude Demos: M,A,L dependence Physics 101: Lecture 22, Pg 20
Concept Question
Suppose a grandfather clock (a simple pendulum) runs slow. In order to make it run on time you should: 1. Make the pendulum shorter 2. Make the pendulum longer CORRECT
g L T
2 2
L g
Physics 101: Lecture 22, Pg 21
Concept Question
A pendulum is hanging vertically from the ceiling of an elevator. Initially the elevator is at rest and the period of the pendulum is pendulum accelerates upward. The period of the pendulum will now be 1. greater than T 2. equal to T 3. less than T CORRECT T . Now the
g L T
2 2
L g
“Effective g” is larger when accelerating upward (you feel heavier) Physics 101: Lecture 22, Pg 22
Concept Question
If the amplitude of the oscillation (same block and same spring) was doubled, how would the period of the oscillation change? (The period is the time it takes to make one complete oscillation) 1. The period of the oscillation would double.
2. The period of the oscillation would be halved 3. The period of the oscillation would stay the same CORRECT
+2A
x t
-2A
Physics 101: Lecture 22, Pg 23
Potential Energy of a Spring
PE S
1 2 kx 2
Where x is measured from the equilibrium position
x=0
m
PE S x x 0
Physics 101: Lecture 22, Pg 24
Same thing for a vertical spring:
PE S
1 2 ky 2
Where y is measured from the equilibrium position
y
m
y=0 PE S y 0
Physics 101: Lecture 22, Pg 25
Chapter 11 Section 1 Simple Harmonic Motion Force and Energy in Simple Harmonic Motion
Physics 101: Lecture 22, Pg 27
Concept Question
In Case 1 a mass on a spring oscillates back and forth. In mass is doubled but the spring and the amplitude of the oscillation is the same as in Case 1. Case 2 , the In which case is the maximum kinetic energy of the mass the biggest? 1. Case 1 2. Case 2 3. Same CORRECT Physics 101: Lecture 22, Pg 28
Concept Question
PE = 1 / 2 kx 2 KE = 0 same for both PE = 0 KE = KE MAX same for both x=-A x=0 x=+A x=-A x=0 x=+A Physics 101: Lecture 22, Pg 29
Simple Harmonic Motion: Quick Review
x(t) = [A]cos( v(t) = -[A ]sin( t) a(t) = -[A 2 ]cos( t) x v a max max max = A = A = A 2 t) OR x(t) = [A]sin( t) v(t) = [A ]cos( t) a(t) = -[A 2 ]sin( t) Period = T (seconds per cycle) Frequency = f = 1/T (cycles per second) Angular frequency = = 2 f = 2 /T Physics 101: Lecture 22, Pg 31
Review: Period of a Spring
For simple harmonic oscillator = 2 f = 2 /T For mass M on spring with spring constant k k m T 2 m k Demos: A,m,k dependence Physics 101: Lecture 22, Pg 32
Chapter
11 Standardized Test Prep
Multiple Choice
Base your answers to questions 1 –6 on the information below
.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
1.
In what direction does the restoring force act?
A.
to the left
B.
to the right
C.
to the left or to the right depending on whether the spring is stretched or compressed
D.
perpendicular to the motion of the mass
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1 –6 on the information below
.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
2.
If the mass is displaced –0.35 m from its equilibrium position, the restoring force is 7.0 N. What is the spring constant?
F.
G.
–5.0 –2.0 10 –2 10 1 N/m N/m
H.
J.
5.0 2.0 10 10 1 –2 N/m N/m
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1 –6 on the information below
.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
2.
If the mass is displaced –0.35 m from its equilibrium position, the restoring force is 7.0 N. What is the spring constant?
F.
G.
–5.0 –2.0 10 –2 10 1 N/m N/m
H.
J.
5.0 2.0 10 10 1 –2 N/m N/m
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1 –6 on the information below
.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
3.
In what form is the energy in the system when the mass passes through the equilibrium point?
A.
elastic potential energy
B.
gravitational potential energy
C.
kinetic energy
D.
a combination of two or more of the above
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1 –6 on the information below
.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
3.
In what form is the energy in the system when the mass passes through the equilibrium point?
A.
elastic potential energy
B.
gravitational potential energy
C.
kinetic energy
D.
a combination of two or more of the above
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1 –6 on the information below
.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
4.
In what form is the energy in the system when the mass is at maximum displacement?
F.
elastic potential energy
G.
gravitational potential energy
H.
kinetic energy
J.
a combination of two or more of the above
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1 –6 on the information below
.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
4.
In what form is the energy in the system when the mass is at maximum displacement?
F.
elastic potential energy
G.
gravitational potential energy
H.
kinetic energy
J.
a combination of two or more of the above
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1 –6 on the information below
.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
5.
Which of the following does
not
affect the period of the mass-spring system?
A.
mass
B.
spring constant
C.
amplitude of vibration
D.
All of the above affect the period.
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1 –6 on the information below
.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
5.
Which of the following does
not
affect the period of the mass-spring system?
A.
mass
B.
spring constant
C.
amplitude of vibration
D.
All of the above affect the period.
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1 –6 on the information below
.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
6.
If the mass is 48 kg and the spring constant is 12 N/m, what is the period of the oscillation?
F.
G.
8 4 s s
H. J.
/2 s s
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 1 –6 on the information below
.
A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface.
6.
If the mass is 48 kg and the spring constant is 12 N/m, what is the period of the oscillation?
F.
G.
8 4 s s
H. J.
/2 s s
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 7 –10 on the information below
.
A pendulum bob hangs from a string and moves with simple harmonic motion.
7.
What is the restoring force in the pendulum?
A.
the total weight of the bob
B.
the component of the bob’s weight tangent to the
C.
motion of the bob the component of the bob’s weight perpendicular to the motion of the bob
D.
the elastic force of the stretched string
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 7 –10 on the information below
.
A pendulum bob hangs from a string and moves with simple harmonic motion.
7.
What is the restoring force in the pendulum?
A.
the total weight of the bob
B.
the component of the bob’s weight tangent to the
C.
motion of the bob the component of the bob’s weight perpendicular to the motion of the bob
D.
the elastic force of the stretched string
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 7 –10 on the information below
.
A pendulum bob hangs from a string and moves with simple harmonic motion.
8.
Which of the following does period of the pendulum?
not
affect the
F.
the length of the string
G.
the mass of the pendulum bob
H.
the free fall acceleration at the pendulum’s location
J.
All of the above affect the period.
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 7 –10 on the information below
.
A pendulum bob hangs from a string and moves with simple harmonic motion.
8.
Which of the following does period of the pendulum?
not
affect the
F.
the length of the string
G.
the mass of the pendulum bob
H.
the free fall acceleration at the pendulum’s location
J.
All of the above affect the period.
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 7 –10 on the information below
.
A pendulum bob hangs from a string and moves with simple harmonic motion.
9.
If the pendulum completes exactly 12 cycles in 2.0 min, what is the frequency of the pendulum?
A.
0.10 Hz
B.
0.17 Hz
C.
6.0 Hz
D.
10 Hz
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 7 –10 on the information below
.
A pendulum bob hangs from a string and moves with simple harmonic motion.
9.
If the pendulum completes exactly 12 cycles in 2.0 min, what is the frequency of the pendulum?
A.
0.10 Hz
B.
0.17 Hz
C.
6.0 Hz
D.
10 Hz
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 7 –10 on the information below
.
A pendulum bob hangs from a string and moves with simple harmonic motion.
10.
If the pendulum’s length is 2.00 m and
a g
= 9.80 m/s 2 , how many complete oscillations does the pendulum make in 5.00 min?
F.
1.76
G.
21.6
H.
106
J.
239
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter
11 Standardized Test Prep
Multiple Choice, continued
Base your answers to questions 7 –10 on the information below
.
A pendulum bob hangs from a string and moves with simple harmonic motion.
10.
If the pendulum’s length is 2.00 m and
a g
= 9.80 m/s 2 , how many complete oscillations does the pendulum make in 5.00 min?
F.
1.76
G.
21.6
H.
106
J.
239
Copyright © by Holt, Rinehart and Winston. All rights reserved.