Simple Harmonic Motion (SHM) Slide 2 / 67 SHM and Uniform Circular - - PowerPoint PPT Presentation
Slide 1 / 67 Simple Harmonic Motion (SHM) Slide 2 / 67 SHM and Uniform Circular Motion There is a deep connection between Simple Harmonic Motion (SHM) and Uniform Circular Motion (UCM). Simple Harmonic Motion can be thought of as a one-
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There is a deep connection between Simple Harmonic Motion (SHM) and Uniform Circular Motion (UCM). Simple Harmonic Motion can be thought of as a one- dimensional projection of Uniform Circular Motion.
http://www.physics.uoguelph.ca/tutorials/shm/phase0.html
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Imagine we have a ball moving in Uniform Circular Motion and we shine a light on it. Now at the end where the image is cast we have a spring and mass system which will oscillate with the same period. This experiment will show that the shadow of the ball in UCM will match that of the mass in SHM.
x
x
x
t=0 T/2 T
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Period is defined as the time it takes for an object to complete one trip around a circular path, or to complete one
Period is represented by "T" The period of a system is normally measured in seconds (s). Usually we are given the total time it takes for a system to rotate around its central axis n number of times. To find the period we divide the total time by the number of times we completed one revolution.
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If it takes 50 seconds for an object to travel around a circle 5 times, what is the period of its motion?
A
B
10 s
25 s E 50 s
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2 If an object is traveling in circular motion and its period is 7.0s, how long will it take it to make 8 complete revolutions? A 7/8 s B 8/7 s
48 s
56 s E
112 s
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The number of revolutions that an object completes in a given amount of time is called the frequency of its motion. The symbol for frequency is "f" Frequency is measured in units of revolutions per unit time; we will usually use 1/seconds (s-1). Another name for s-1 is Hertz (Hz). Frequency can also be measured in revolutions per minute (rpm), etc. Often we are given the time (t) it takes for an object to make a number of revolutions (n). In that case,
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3 An object travels around a circle 50 times in 10s, what is the frequency (in Hz) of its motion? A 0.2 Hz B 1 Hz
5 Hz
25 Hz E 500 Hz
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4 If an object is traveling in circular motion with a frequency of 7.0 Hz, how many revolutions will it make in 20s? A 7/20 B 20/7
7
140 E 280
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Before we defined the values of Period and
side we see that each one is the reciprocal of the
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5 An object has a period of 4.0s, what is the frequency of its motion (in Hertz)? A 0.25 Hz B 0.5 Hz
1 Hz
2 Hz E 4 Hz
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6 An object is revolving with a frequency of 8.0 Hz, what is its period (in seconds)? A 0.125 s B 0.25 s
1 s
4 s E 8 s
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Also, recall from Uniform Circular Motion.... and
http:/ / njc.tl/ hn
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7 An object is in circular motion. The radius of its motion is 2.0 m and its period is 5.0s. What is its velocity? A B
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8 An object is in circular motion. The radius of its motion is 2.0 m and its frequency is 8.0 Hz. What is its velocity? A B
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· Displacement is measured from the equilibrium point · Amplitude is the maximum displacement (equivalent to the radius, r, in UCM). · A cycle is a full to-and-fro motion (the same as one trip around the circle in UCM) · Period is the time required to complete
· Frequency is the number of cycles completed per second (the same as frequence in UCM)
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9 The period of a mass-spring system is 4.0s and the amplitude of its motion is 0.50m. How far does the mass travel in 4.0s? A 0 m B 0.5 m
0.75 m
1.5 m E 2.0 m
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10 The period of a mass-spring system is 4.0s and the amplitude of its motion is 0.50m. How far does the mass travel in 6.0s? A 0.5 m B 0.75 m
1.0 m
3.0 m E 2.5 m
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There is a point where the spring is neither stretched nor compressed; this is the equilibrium position. We measure displacement from that point (x = 0 on the previous figure). The force exerted by the spring depends on the displacement:
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11 A spring whose spring constant is 20N/m is stretched 0.20m from equilibrium; what is the magnitude of the force exerted by the spring? A 0.4 N B 2 N
4 N
8 N E 10 N
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12 A spring whose spring constant is 150 N/m exerts a force of 30N on the mass in a mass-spring
A 0.2 m B 0.3 m C 1 m D 2 m E 3 m
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13 A spring exerts a force of 50N on the mass in a mass-spring system when it is 2.0m from
A 2.5 N/m B 5 N/m C 15 N/m D 25 N/m E 50 N/m
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In Hooke's Law the negative sign indicated that it is a restoring force, meaning the force wants to bring the system back to its original position. k is the spring constant The force is not constant, therefore the acceleration is not constant either.
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The maximum force exerted on the mass is when the spring is most stretched or compressed (x = -A or +A): F = -kA (when x = -A or +A) The minimum force exerted on the mass is when the spring is not stretched at all (x = 0) F = 0 (when x = 0)
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14 At which location(s) is the magnitude of the force on the mass in a mass-spring system a maximum? A x=A B
All of the above
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15 At which location(s) is the magnitude of the force
A x=A B x=0 C x=-A D A & C E all of the above
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If the spring is hung vertically, the only change is in the equilibrium position, which is at the point where the spring force equals the gravitational force. The effect of gravity is cancelled out by changing to this new equilibrium position.
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If we return to the relationship between Uniform Circular Motion and Simple Harmonic Motion we can explain the displacement for the spring system. Which can also be written as:
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As stated in previous chapters velocity is the rate of change of position, so by taking the derivative of the position equation for SHM we can calculate the velocity at a specific point in time.
Acceleration is defined at the rate of change of velocity, therefore by taking the derivative of the velocity equation we just solved for we can find the acceleration of the system in SHM at any point in time.
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t t t
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16 What is the acceleration when x = 0? A
It varies
T/4
T/2 3T/4
T x (displacement) v (velocity) a (acceleration)
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17 What is the acceleration when x=A? A
a<0 B a=0 C a>0 D
It varies
T/4
T/2 3T/4
T x (displacement) v (velocity) a (acceleration)
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18 What is the acceleration when x=-A? A
It varies
T/4
T/2 3T/4
T x (displacement) v (velocity) a (acceleration)
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19 What is the velocity when x=0? A v<0 B v=0
v>0
A or C
T/4
T/2 3T/4
T x (displacement) v (velocity) a (acceleration)
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20 What is the velocity when x=A? A v<0 B v=0
v>0
A or C
T/4
T/2 3T/4
T x (displacement) v (velocity) a (acceleration)
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21 Where is the mass when acceleration is at a maximum?
A
x=A
B
x=0
C
x=-A
D A or C
T/4
T/2 3T/4 T
x (displacement) v (velocity) a (acceleration)
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22 Where is the mass when velocity is at maximum? A x=A B
T/4
T/2 3T/4
T x (displacement) v (velocity) a (acceleration)
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23 Which of the following represents the position as a function of time? A x=4 cos(2t) B x=2 cos(2t)
C
x=8 cos(2t)
D
x=2 sin(2t)
T/4
T/2
3T/4
T
x (displacement) v (velocity) a (acceleration)
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24 Which of the following represents the velocity as a function of time? A v= -12 sin(2t) B v=-12 cos(2t)
v=-4 sin(2t)
v=-4 cos(2t)
T/4
T/2 3T/4 T
x (displacement) v (velocity) a (acceleration)
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25 Which of the follwing represents the acceleration as a function of time? A
a=-8 sin(2t)
B a=-8 cos(2t) C a=-4 sin(2t) D a=-4 cos(2t)
T/4
T/2 3T/4
T
x (displacement) v (velocity) a (acceleration)
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Any vibrating system where the restoring force is proportional to the negative of the displacement is in simple harmonic motion (SHM), and is often called a simple harmonic oscillator. Also, SHM requires that a system has two forms of energy and a method that allows the energy to go back and forth between those forms.
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There are two types of energy in a mass-spring system. The energy stored in the spring because it is stretched or compressed: Us = 1/2 kx2 AND The kinetic energy of the mass: KE = 1/2 mv
2
Etotal = 1/2 kx2 + 1/2 mv2
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When the mass is at the limits of its motion (x = A or x = -A), the energy is all potential: Etotal = 1/2 kx2 When the mass is at the equilibrium point (x=0) the spring is not stretched and all the energy is kinetic: Etotal = 1/2 mv2 But the total energy is constant. Etotal = 1/2 kx2 + 1/2 mv2
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26 At which location(s) is the kinetic energy of a mass-spring system a maximum? A x=A B
All of the above
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27 At which location(s) is the spring potential energy (US) of a mass-spring system a maximum? A x=A B
All of the above
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28 At which location(s) is the total energy of a mass- spring system a maximum? A x=A B
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29 At which location(s) is the kinetic energy of a mass-spring system a minimum? A x=A B
All of the above
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There are two types of energy in a simple pendulum. Gravitational potential energy when the object is above its lowest point in the path U=mgh And Kinetic Energy KE=1/2 mv2
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At any moment the total energy of the system is constant. Etotal = U + KE Etotal = mgh + 1/2 mv2
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30 At which location(s) is the kinetic energy of a simple pendulum a maximum? A x=h B
All of the above
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31 At which location(s) is the gravitational potential energy of a simple pendulum a maximum? A x=h B
All of the above
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32 At which location(s) is the total energy of the system a maximum? A x=h B
At all locations
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when θ=0
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33 What is the period of a spring-mass system set into SHM if the mass of the object is 625g and the spring constant is 2.5 N/m? A B
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34 What is the frequency of a spring-mass system set into SHM with a mass of 4 kg and a spring constant of 64 N/m? A B
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For small values of θ, sin(θ)=θ
L
x
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35 What is the period of a pendulum with a string 10 m long on the surface of the earth? (Assume g=10 m/s2) A B
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36 What is the frequency of a pendulum whose string is 5 m long on planet where the surface gravity is 25 m/s2? A B
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37 If a spring system with a period of T and a pendulum with the same period T are brought to another planet with a greater surface gravity, which system will have a greater period? A The mass spring system B The pendulum
Both of them will still have the same period
Not enough information
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cg
d
θ m g s i n ( θ ) mg mgcos(θ) dsin(θ) dcos(θ)
P
An object of mass m is pivoted about point P. P is a distance d away from the objects center of mass. The lever arm in this scenario is dsin(θ) and the force applied is mg.
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cg
d
θ m g s i n ( θ ) mg mgcos(θ) dsin(θ) dcos(θ)
P
In the Spring-Mass system ω2 is For the physical Pendulum ω2 is At extremely small values of θ, sin(θ)=θ
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We proved that ω2 for a Physical Pendulum is equal to from what we know about the mass-spring system. Knowing this we can find the period and frequency of its motion. and
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38 In a physical pendulum, what would happen to the period if the mass was divided by 4?
A
Increased 2 times
B
Increased 4 times
C
Decreased 2 times
D
Decreased 2 times
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39 In a physical pendulum, what would happen to the frequency if the mass was increased by 36?
A
Decreased 36 times
B
Decreased 36 times
C
Increased 6 times
D
Decreased 6 times
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40 In a physical pendulum, what would happen to the period if the mass was increased by 16?
A
Increased 16 times
B
Increased 4 times
C
Decreased 16 times
D
Decreased 4 times
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41 In a physical pendulum, what would happen to the frequency if the mass was decreased by 9?
A
Increased 9 times
B
Decreased 9 times
C
Increased 3 times
D
Decreased 3 times
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