"Simple Harmonic Motion" (9 класс)

Слайд #1

VIBRATION (or OSCILLATION)
any periodic motion moving at a distance about the equilibrium position and repeat itself over and over for a period of time
Examples: Spring moving up and down; Simple pendulum

Слайд #2

Observations of vibrational motion
When you walk, your arms and legs swing back and forth. These motions repeat themselves.







The back-and-forth motion of an object that passes through the same positions is an important feature of vibrational motion.
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Слайд #3

Equilibrium position

Слайд #4

Restoring force

Слайд #5

Amplitude

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Period
𝑇= 𝑡 𝑡𝑜𝑡𝑎𝑙 # 𝑜𝑓 𝑜𝑠𝑐𝑖𝑙𝑙𝑎𝑡𝑖𝑜𝑛𝑠
𝑇 𝑠𝑝𝑟𝑖𝑛𝑔 =2𝜋 𝑚 𝑘

Слайд #7

Frequency
𝑓= # 𝑜𝑓 𝑜𝑠𝑐𝑖𝑙𝑙𝑎𝑡𝑖𝑜𝑛𝑠 𝑡 𝑡𝑜𝑡𝑎𝑙
𝑓 𝑠𝑝𝑟𝑖𝑛𝑔 = 1 2𝜋 𝑘 𝑚

Слайд #8

Sample Problem
(p.729 #22) A spring with a cart at its end vibrates at frequency 6.0 Hz. (a) Determine the period of vibration. (b) Determine the frequency if the cart’s mass is doubled while the spring constant remains unchanged, and (c) the frequency if the spring constant doubles while the cart’s mass remains the same.

Слайд #9

RECALL: Hooke’s Law
𝐹 𝑔 𝛼 ∆𝑦

𝑭 𝒈 =𝒌∆𝒚
−𝐹 𝑔
𝐹 𝑠
𝑭 𝒔 = −𝑭 𝒈
𝑭 𝒔 =−𝒌∆𝒚

Слайд #10

Hooke’s Law
𝑭 𝒔 =−𝒌∆𝒚
k  spring constant
𝒌= 𝑭 ∆𝒚
SI Unit: N/m
Stiffness of a spring (Greater value for k, the stiffer the spring is)

Слайд #11

Sample Problem
A load of 50 N attached to a spring hanging vertically stretches the spring 5.0 cm. The spring is now placed horizontally on a table and stretched 11.0 cm. What force is required to stretch the spring this amount?

Слайд #12

Sample Problem
(p.729 #19) You exert a 100 – N pull on the end of a spring. When you increase the force to 120 N, the spring’s length increases 5.0 cm beyond its original stretched position. What is the spring constant of the spring and its original spring and its original displacement?

Слайд #13

Elastic Potential Energy ( 𝑼 𝒔 )
𝑬𝑷𝑬= 𝟏 𝟐 𝒌( ∆𝒚) 𝟐
Elastic potential energy is Potential energy stored as a result of deformation of an elastic object, such as the stretching of a spring.
SI Unit: Joules

Слайд #14

What does the slope represent?
What does the area under the graph represent?
Hooke’s Law from a Graphical Point of View

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What does the slope represent?
𝑚= 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛
𝑚= 𝐹 𝑠 ∆𝑦

Slope  spring constant


Hooke’s Law from a Graphical Point of View

Слайд #16

What does the area under the graph represent?
Hooke’s Law from a Graphical Point of View
𝐴𝑟𝑒𝑎= 𝑏ℎ 2
𝐴𝑟𝑒𝑎= (∆𝑦)( 𝐹 𝑠 ) 2
𝐴𝑟𝑒𝑎= (∆𝑦)(𝑘∆𝑦) 2
𝐴𝑟𝑒𝑎= 𝑘 (∆𝑦) 2 2




Elastic Potential Energy

Слайд #17

SAMPLE PROBLEM
p. 223 #19
A door spring is difficult to stretch. (a) What maximum force do you need exert on a relaxed spring with a 1.2 𝑥 10 −4 N/m to stretch it 6.0 cm from its equilibrium position? (b) How much does the elastic potential energy of the spring change?

Слайд #18

SAMPLE PROBLEM
p. 223 #19
(c) Determine its change in EPE as it returns from the 6.0 cm stretch position to a 3.0 cm stretch position. (d) Determine its change in EPE as it returns from the 3.0 cm stretch position back to its equilibrium position.

Слайд #19

Forces and acceleration for a cart on a spring
𝐹 𝑛𝑒𝑡 = −𝐹 𝑠
𝑚 𝑎 =− −k∆𝑦
𝒂 = 𝒌∆𝒚 𝒎

Слайд #20

Energy of Vibrational Systems
Kinetic energy and elastic potential energy
Energy is conserved.
1 2 𝑚 𝑣 𝑚𝑎𝑥 2 = 1 2 𝑘 𝐴 2 = 1 2 𝑚 𝑣 2 + 1 2 𝑘 (∆𝑦) 2

Слайд #21

Relationship between the amplitude of the vibration and the cart's maximum speed





This makes sense conceptually:
When the mass of the cart is large, it should move slowly.
If the spring is stiff, the cart will move more rapidly.
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Слайд #22

Sample Problem
A spring with a 1.6 x 104 N/m spring constant and a 0.1-kg cart at its end has a total vibrational energy of 3.2 J.
Determine the amplitude of the vibration.
Determine the cart's maximum speed.
Determine the cart's speed when it is displaced 0.010 m from equilibrium.
What would the amplitude of the vibration be if the energy of the system doubled?
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Слайд #23

Sample Problem
(p. 729 #28) A spring with a spring constant 2.5 𝑥 10 4 𝑁 𝑚 has a 1.4 kg cart at its end. (a) If its amplitude of vibration is 0.030 m, what is the total energy of the system? (b) What is the maximum speed of the cart? (c) If the energy is tripled, what is the new amplitude and what will be the maximum speed of the cart?
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Слайд #24

A weakly damped system continues to vibrate for many periods.
In an overdamped system, the vibrating system takes a long time to return to the equilibrium position, if it ever does.
In a critically damped system, the vibrating object returns to equilibrium in the shortest time possible.

Слайд #25

The simple pendulum
A pendulum is a vibrating system in which the motion is very apparent.
Consider a simplified model of a pendulum system that has a compact object (a bob) at the end of a comparatively long and massless string and that undergoes small-amplitude vibrations.
This idealized system is called a simple pendulum.
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Слайд #26

The simple pendulum
Two objects interact with the bob of the pendulum.
The string S exerts a force that is always perpendicular to the path of the bob.
Earth exerts a downward gravitational force.

Слайд #27

Period of a simple pendulum
𝑇 𝑝𝑒𝑛𝑑𝑢𝑙𝑢𝑚 =2𝜋 𝑙 𝑔
l  length of the string
g  acceleration due to gravity

Слайд #28

Sample Problem
A simple pendulum has a period of 1 sec on Earth. What would be its period be on the moon (where g is 1/6 of the earth’s g)?