1
Q
Lesson 6.1 Simple Harmonic Motion (Part 1)
What is Periodic Motion?
A
🔬 Periodic Motion: Objects vibrates or oscillate back and forth, over the same path, with each oscillation taking the same amount of time
Ex: uniform coil spring
- Ignore mass
- Horizontally mounted
- Equilibrium Position: Spring’s natural length
2
Q
Lesson 6.1 Simple Harmonic Motion (Part 1)
What is the restoring force?
A
Restoring Force: Mass moved left/right, compressing the spring or stretching the string, the spring exerts a force on the mass that acts in the direction of returning mass to the equilibrium position
- Directly proportional to the displacement
- Opposite side as displacement
3
Q
Lesson 6.1 Simple Harmonic Motion (Part 1)
What is Hooke’s Law and its relation to Restoring Force?
A
- F = -kx*
- (Only use when the spring is not compressed to where it is touching or stretched more than max elasticity)*
- k - spring constant (N/m)
- Directly proportional with force needed to stretch spring
To stretch the spring in direction x, an external force must be exerted on free end of the spring with a magnitude of at least:
Fext = +kx
4
Q
Lesson 6.1 Simple Harmonic Motion (Part 1)
What is displacement?
A
The distance x of the mass from the equilibrium point at any moment
5
Q
Lesson 6.1 Simple Harmonic Motion (Part 1)
What is amplitude?
A
Maximum distance (greatest distance from equilibrium point)
6
Q
Lesson 6.1 Simple Harmonic Motion (Part 1)
What is a cycle?
A
One cycle refers to the complete to-and-from motion from some point back to the same point
7
Q
Lesson 6.1 Simple Harmonic Motion (Part 1)
What is the period (T)?
A
Time required to complete one cycle
8
Q
Lesson 6.1 Simple Harmonic Motion (Part 1)
What is the frequency (f)?
A
Number of complete cycles per second Hertz (Hz), where 1 Hz = 1 cycle per second (s-1)
f = 1/T
9
Q
Lesson 6.1 Simple Harmonic Motion (Part 1)
How does vertical oscillation compare to horizontal oscillation?
A
- Same as horizontal spring
- The length of vertical spring with a mass m on the end will be longer at equilibrium than when that same spring is horizontal.
- Equilibrium: ΣF = 0 = mg - kx0
- Spring stretches an extra amount x0=mg/k to be in equilibrium
- If x is measured from this new equilibrium position, can be used directly with the same value of k
10
Q
Lesson 6.1 Simple Harmonic Motion (Part 1)
What is the potential energy of a string?
A
Potential Energy is stored in a stretched or compressed string:
Elastic Potential Energy:
PE = 1/2kx2
Total Mechanical Energy (sum of kinetic and potential energies):
E = 1/2mv2 + 1/2kx2
11
Q
Lesson 6.1 Simple Harmonic Motion (Part 1)
What is simple harmonic motion (SHM)?
A
Simple Harmonic Motion (SHM): Any oscillating system for which the net restoring force is directly proportional to the negative of the displacement (Also called a simple harmonic oscillator (SHO)).
- Only if friction is negligible - total mechanical energy stays constant
- As mass oscillates, the energy continuously changes from potential to kinetic energy
12
Q
Lesson 6.1 Simple Harmonic Motion (Part 1)
What is the total mechanical energy of a simple harmonic oscillator?
A
Total Mechanical Energy of a simple harmonic oscillator is proportional to the square of the amplitude
13
Q
Lesson 6.1 Simple Harmonic Motion (Part 1)
What happens in simple harmonic motion at the extreme points?
A
At extreme points: x = A and x= -A
- Energy is stored as potential energy
- Mass stops for an instant as it changes direction, so v=0 and
E = 1/2m(0)2 + 1/2kA2 = 1/2kA2
14
Q
Lesson 6.1 Simple Harmonic Motion (Part 1)
What happens in simple harmonic motion at the equilibrium points?
A
At equilibrium points: x=0
- Energy is exclusively kinetic
- Maximum speed during motion, so vmax and
E = 1/2mvmax2 + 1/2k(0)2 = 1/2mvmax2
15
Q
Lesson 6.1 Simple Harmonic Motion (Part 1)
What happens in simple harmonic motion at the intermediate points?
A
- Energy is partially kinetic and potential energy
- Energy is conserved
1/2mv2 + 1/2kx2 = 1/2kA2
16
Q
Lesson 6.1 Simple Harmonic Motion (Part 1)
What is velocity? (as a function of position)
A
17
Q
Lesson 6.2 Simple Harmonic Motion (Part 2)
What is the period of a Simple Harmonic Oscillator?
A
Period of a Simple Harmonic Oscillator found to depend on
- stiffness of spring
- Mass of the oscillator
- Not depend on the amplitude
Increases alongside the mass and decreases alongside the spring constant
Force is directly proportional to the spring constant (so also acceleration)
18
Q
Lesson 6.2 Simple Harmonic Motion (Part 2)
What is the frequency of a Simple Harmonic Oscillator?
A
19
Q
Lesson 6.2 Simple Harmonic Motion (Part 2)
What is the time of a Simple Harmonic Oscillator?
A
20
Q
Lesson 6.2 Simple Harmonic Motion (Part 2)
How do you use derivatives to find distance, velocity, and acceleration?
A
where ω = 2πf
x(t) = Acos(ωt)
v(t) = x(t)’ = -Aω sin(ωt)
a(t) = x(t)’’ = -Aω2 cos(ωt)
21
Q
Lesson 6.2 Simple Harmonic Motion (Part 2)
Describe position as a function of time:
Describe in terms of the frequency and period.
A
- Object’s position on the x axis: x = Acos θ
- Mass rotating with angular velocity ω, so θ = ωt, where theta is in radians. Then x = Acos ωt
- Angular velocity is in radians, so ω =2πf.
Therefore (in terms of frequency):
x = Acos(2πft)
And (in terms of the period):
x = Acos(2πt)/T
22
Q
Lesson 6.2 Simple Harmonic Motion (Part 2)
What is sinusoidal motion?
A
At x=Acos ωt
Assume oscillation starts from rest (v = 0) at its maximum displacement (x = A) at t = 0
- Same shape as cosine curve shown - shifted to the right by a quarter cycle
- Sine and cosine curves are being sinusoidal
- Simple harmonic motion is therefore also sinusoidal
23
Q
Lesson 6.2 Simple Harmonic Motion (Part 2)
Velocity and Acceleration curves as derivates of distance:
A
24
Q
Lesson 6.2 Simple Harmonic Motion (Part 2)
Describe velocity as a function of distance:
A
- The magnitude of v is vmax sinθ
- And θ = ωt = 2πft
25
# Lesson 6.2 Simple Harmonic Motion (Part 2)
What is maximum velocity?
26
# Lesson 6.2 Simple Harmonic Motion (Part 2)
Describe acceleration as a function of time:
* Max speed is larger if the amplitude is larger
* Newton’s second law gives the acceleration as a function of time:
27
# Lesson 6.2 Simple Harmonic Motion (Part 2)
What is maximum acceleration?
28
# Lesson 6.2 Simple Harmonic Motion (Part 2)
**1.** (I) If a particle undergoes SHM with amplitude 0.21 m, what is the total distance it travels in one period?
29
# Lesson 6.2 Simple Harmonic Motion (Part 2)
**3.** (II) An elastic cord is 61 cm long when a weight of 75 N hangs from it but is 85 cm long when a weight of 210 N hangs from it. What is the “spring” constant k of this elastic cord?
30
# Lesson 6.2 Simple Harmonic Motion (Part 2)
**5.** (II) A fisherman’s scale stretches 3.6 cm when a 2.4-kg fish hangs from it. (a) What is the spring stiffness constant and (b) what will be the amplitude and frequency of oscillation if the fish is pulled down 2.1 cm more and released so that it oscillates up and down?
31
# Lesson 6.2 Simple Harmonic Motion (Part 2)
**7.** (II) A mass m at the end of a spring oscillates with a frequency of 0.83 Hz. When an additional 780-g mass is added to m, the frequency is 0.60 Hz. What is the value of m?
32
# Lesson 6.2 Simple Harmonic Motion (Part 2)
**9.** (II) Figure 11–51 shows two examples of SHM, labeled A and B. For each, what is (a) the amplitude, (b) the frequency, and (c) the period?
33
# Lesson 6.2 Simple Harmonic Motion (Part 2)
**11.** (II) At what displacement of a SHO is the energy half kinetic and half potential?
34
# Lesson 6.2 Simple Harmonic Motion (Part 2)
**13.** (II) A 1.65-kg mass stretches a vertical spring 0.215 m. If the spring is stretched an additional 0.130 m and released, how long does it take to reach the (new) equilibrium position again?
35
# Lesson 6.2 Simple Harmonic Motion (Part 2)
**15.** (II) A 0.25-kg mass at the end of a spring oscillates 2.2 times per second with an amplitude of 0.15 m. Determine (a) the speed when it passes the equilibrium point, (b) the speed when it is 0.10 m from equilibrium, (c) the total energy of the system, and (d) the equation describing the motion of the mass, assuming that at t = 0, x was a maximum.
36
# Lesson 6.2 Simple Harmonic Motion (Part 2)
**17.** (II) If one oscillation has 3.0 times the energy of a second one of equal frequency and mass, what is the ratio of their amplitudes?
37
# Lesson 6.2 Simple Harmonic Motion (Part 2)
**19.** (II) A mass resting on a horizontal, frictionless surface is attached to one end of a spring; the other end is fixed to a wall. It takes 3.6 J of work to compress the spring by 0.13 m. If the spring is compressed, and the mass is released from rest, it experiences a maximum acceleration of 12 m/s2. Find the value of (a) the spring constant and (b) the mass.
38
# Lesson 6.2 Simple Harmonic Motion (Part 2)
**21.** (II) At t = 0, an 885-g mass at rest on the end of a horizontal spring (k = 184 N/m) is struck by a hammer which gives it an initial speed of 2.26 m/s. Determine (a) the period and frequency of the motion, (b) the amplitude, (c) the maximum acceleration, (d) the total energy, and (e) the kinetic energy when x = 0.40A where A is the amplitude.
39
# Lesson 6.3 Simple Pendulum
What is a simple pendulum?
🔬 **Simple Pendulum** consists of a small object suspended from the end of a lightweight cord
* The cord does not stretch
* Mass can be ignored
* Negligible friction resembles the simple harmonic motion
40
# Lesson 6.3 Simple Pendulum
Describe the displacement of a pendulum along the arc given by *s = lΘ.*
Displacement (s) of the pendulum along the arc given by *s = lΘ*
* ***Θ*** is the angle (in radians) that the cord makes with the vertical
* ***l*** is the length of the cord
*If the restoring force is proportional to s or* ***Θ****, the motion will be simple harmonic*
41
# Lesson 6.3 Simple Pendulum
What is the restoring force in a simple pendulum? How does it change for small angles?
**Restoring Force**: net force on the bob = the components of the weight tangent to the arc
F = -mg sinΘ
* Opposite direction to the angular displacement
* Proportional to sine of Θ and not Θ itself - motion is not SHM
* If Θ is very small, then sine of Θ is nearly equal to Θ
Hence, for very small angles:
**F = -mgsin Θ ≈ -mgΘ**
If s = lΘ or Θ = s/l:
**F ≈ -mg/l \* s**
* Thus for _small displacement_, the motion is modelled as being approximately simple harmonic*
* The approximate equation fits Hooke’s Law (F=-kx) where s is in the place of x and k=mg/l
42
# Lesson 6.3 Simple Pendulum
What is the period of a small pendulum?
43
# Lesson 6.3 Simple Pendulum
Period of a Pendulum for the acceleration due to Gravity:
44
# Lesson 6.3 Simple Pendulum
What is the frequency of a small pendulum?
* The period and frequency do not depend on mass
* The period does not depend on the amplitude
45
# Lesson 6.3 Simple Pendulum
**27.** (I) A pendulum has a period of 1.85 s on Earth. What is its period on Mars, where the acceleration of gravity is about 0.37 that on Earth?
46
# Lesson 6.3 Simple Pendulum
**28.** (I) How long must a simple pendulum be if it is to make exactly one swing per second? (That is, one complete oscillation takes exactly 2.0 s.)
47
# Lesson 6.3 Simple Pendulum
**29.** (I) A pendulum makes 28 oscillations in exactly 50 s. What is its (a) period and (b) frequency?
48
# Lesson 6.3 Simple Pendulum
**30.** (II) What is the period of a simple pendulum 47 cm long (a) on the Earth, and (b) when it is in a freely falling elevator?
If the pendulum’s period is infinity on an elevator in freefall it will not oscillate back and forth and will be pushed into the opposite direction of gravity (so onto the ceiling), which would go on for infinity or until the elevator is no longer in freefall.
49
# Lesson 6.3 Simple Pendulum
**31\*.** (II) Your grandfather clock’s pendulum has a length of 0.9930 m. If the clock runs slow and loses 21 s per day, how should you adjust the length of the pendulum?
If the pendulum’s period is infinity on an elevator in freefall it will not oscillate back and forth and will be pushed into the opposite direction of gravity (so onto the ceiling), which would go on for infinity or until the elevator is no longer in freefall.
50
# Lesson 6.4 Oscillations and Resonance
What is damped harmonic motion?
**Damped Harmonic Motion** - The amplitude of any real oscillating spring slowly decreases in time until the oscillations stop altogether
* Damping due to resistance in air and internal resistance
* Energy dissipated to thermal energy - decreases the amplitude
* SHM is easier to work with, so if sampling is not large, the oscillations can be thought of as SHM
51
# Lesson 6.4 Oscillations and Resonance
What are heavily damped systems?
*No longer resemble SHM*
## Footnote
Three common cases exist:
**(A)** _Underdamped_ - the system makes several oscillations become coming to rest
**(B)** _Critical Damping_ - The displacement reaches zero in a short time (shock absorbers in cars)
**(C)** _Overdamped_ - Damping is so large there are no oscillations and the system take a long time to come to a rest
52
# Lesson 6.4 Oscillations and Resonance
What is force oscillation?
**Force Oscillation**: System has an external force applied to it which then has its own frequency
* Mass then oscillates at the external frequency of the external force:
53
# Lesson 6.4 Oscillations and Resonance
For forced oscillation with only light damping...
* The amplitude of oscillation is found to depend on the difference between f and f0
* Maximum when the frequency of the external force equals the natural frequency of the system: f = f0
* When the external frequency f is near natural frequency, f ≈ f0, the amplitude can become large is damping is small
54
# Lesson 6.4 Oscillations and Resonance
What is resonance and resonance frequency?
**Resonance**: Effect of increasing amplitude at f = f0
**Resonant Frequency**: Natural oscillation frequency f0 of a system
55
# Quiz 6.2
A sewing machine needle moves in simple harmonic motion with a frequency of 2.5 Hz and an amplitude of 1.27 cm.
* *(a)** How long does it take the tip of the needle to move from the highest point to the lowest point in its travel?
* *(b)** How long does it take the needle tip to travel a total distance of 11.43 cm?
**(a)** 0.20 s
**(b)** 0.90 s
56
# Quiz 6.2
If a floating log is seen to bob up and down 15 times in 1.0 min as waves pass by you, what are the frequency and period of the wave?
57
# Quiz 6.2
A guitar string is set into vibration with a frequency of 512 Hz. How many oscillations does it undergo each minute?
1. 26.8
2. 30,700
3. 512
4. 1610
5. 8.53
the frequency is 512 1/s \* 60s/1min = **30,700 oscillations/min** (Option 2)
58
# Quiz 6.2
A point on the string of a violin moves up and down in simple harmonic motion with an amplitude of 1.24 mm and a frequency of 875 Hz.
* *(a)** What is the maximum speed of that point in SI units?
* *(b)** What is the maximum acceleration of the point in SI units?
(a) 6.82 m/s
(b) 3.75 × 104 m/s2
59
# Quiz 6.2
The quartz crystal in a digital watch has a frequency of 32.8 kHz. What is its period of oscillation?
1. 95.8 µs
2. 30.5 µs
3. 15.3 µs
4. 0.191 ms
5. 9.71 µs
**2.** 30.5 µs
60
# Quiz 6.2
An object is undergoing simple harmonic motion of amplitude 2.3 m. If the maximum velocity of the object is 10 m/s, what is the object's angular frequency?
1. 3.5 rad/s
2. 4.0 rad/s
3. 4.8 rad/s
4. 4.3 rad/s
4. 4.3 rad/s
61
# Quiz 6.2
A sewing machine needle moves up and down in simple harmonic motion with an amplitude of 1.27 cm and a frequency of 2.55 Hz. What are the **(a)** maximum speed and **(b)** maximum acceleration of the tip of the needle?
**(a)** 20.3 cm/s
**(b)** 326 cm/s2
62
# Quiz 6.2
An object that hangs from the ceiling of a stationary elevator by an ideal spring oscillates with a period T. If the elevator accelerates upward with acceleration 2g, what will be the period of oscillation of the object?
1. T
2. 4T
3. T/4
4. T/2
5. 2T
1. T
63
# Quiz 6.2
The figure shows a graph of the position x as a function of time t for a system undergoing simple harmonic motion. Which one of the following graphs represents the velocity of this system as a function of time?
graph b
64
# Quiz 6.2
A simple harmonic oscillator oscillates with frequency f when its amplitude is A. If the amplitude is now doubled to 2A, what is the new frequency?
1. 2f
2. f/4
3. 4f
4. f/2
5. f
5. f
65
# Quiz 6.2
The figure shows a graph of the position x as a function of time t for a system undergoing simple harmonic motion. Which one of the following graphs represents the acceleration of this system as a function of time?
graph a
66
# Quiz 6.2
The total mechanical energy of a simple harmonic oscillating system is
1. a minimum when it passes through the equilibrium point.
2. zero when it reaches the maximum displacement.
3. zero as it passes the equilibrium point.
4. a non-zero constant.
5. a maximum when it passes through the equilibrium point.
4. a non-zero constant
67
# Quiz 6.2
If a pendulum makes 12 complete swings in 8.0 s, what are its **(a)** frequency and **(b)** period?
**(a)** 1.5 Hz
**(b)** 0.67 s
68
# Quiz 6.2
If the frequency of a system undergoing simple harmonic motion doubles, by what factor does the maximum value of acceleration change?
1. 2
2. 4
3. 2/π
2. 4
69
# Quiz 6.2
A leaky faucet drips 40 times in 30.0 seconds.
**What is the frequency of the dripping?**
1. 0.63 Hz
2. 1.3 Hz
3. 1.6 Hz
4. 0.75 Hz
Frequency = Number # oscillations / time
Frequency = 40 / 30.0s = **1.3 Hz** (Option 2)
70
# Module 6 Exam
A ball swinging at the end of a massless string, as shown in the figure, undergoes simple harmonic motion. At what point (or points) is the magnitude of the instantaneous acceleration of the ball the greatest?
## Footnote
A and B
B
A and C
C
A and D
A and D
71
# Module 6 Exam
Grandfather clocks are designed so they can be adjusted by moving the weight at the bottom of the pendulum up or down. Suppose you have a grandfather clock at home that runs fast. Which of the following adjustments of the weight would make it more accurate? (There could be more than one correct choice.)
1. Decrease the amplitude of swing by a small amount.
2. Lower the weight.
3. Add more mass to the weight.
4. Remove some mass from the weight.
5. Raise the weight.
2. Lower the weight
72
# Module 6 Exam
A pendulum of length L is suspended from the ceiling of an elevator. When the elevator is at rest the period of the pendulum is T. How does the period of the pendulum change when the elevator moves upward with constant velocity?
1. The period increases if the upward acceleration is more than g/2 but decreases if the upward acceleration is less than g/2.
2. The period becomes zero.
3. The period decreases.
4. The period does not change.
5. The period increases.
4. The period does not change.
73
# Module 6 Exam
A pendulum of length L is suspended from the ceiling of an elevator. When the elevator is at rest the period of the pendulum is T. How would the period of the pendulum change if the supporting chain were to break, putting the elevator into freefall?
1. The period decreases slightly.
2. The period does not change.
3. The period increases slightly.
4. The period becomes infinite because the pendulum would not swing.
5. The period becomes zero.
4. The period becomes infinite because the pendulum would not swing
74
# Module 6 Exam
A simple pendulum and a mass oscillating on an ideal spring both have period T in an elevator at rest. If the elevator now moves downward at a uniform 2 m/s, what is true about the periods of these two systems?
1. Both periods would remain the same.
2. Both periods would decrease.
3. The period of the pendulum would decrease but the period of the spring would stay the same.
4. The period of the pendulum would increase but the period of the spring would stay the same.
5. Both periods would increase.
1. Both periods would remain the same
75
# Module 6 Exam
If a floating log is seen to bob up and down 15 times in 1.0 min as waves pass by you, what are the frequency and period of the wave?
**Frequency** = num # oscillations / time = 15 / 60 sec = 0.25 Hz
**Period** = 1/f = 1/0.25 = 4 sec
76
# Module 6 Exam
The quartz crystal in a digital watch has a frequency of 32.8 kHz. What is its period of oscillation?
1. 9.71 µs
2. 0.191 ms
3. 30.5 µs
4. 95.8 µs
5. 15.3 µs
3. 30.5 µs
77
# Module 6 Exam
\*11\* A sewing machine needle moves in simple harmonic motion with a frequency of 2.5 Hz and an amplitude of 1.27 cm.
* *(a)** How long does it take the tip of the needle to move from the highest point to the lowest point in its travel?
* *(b)** How long does it take the needle tip to travel a total distance of 11.43 cm?
\*\*\*\*\*\*\*\*\*\*\*
78
# Module 6 Exam
If a pendulum makes 12 complete swings in 8.0 s, what are its (a) frequency and (b) period?
**Frequency** = num # oscillations / time = 12 / 8.0sec = 1.5 Hz
**Period** = 1/f =1 / 1.5Hz = 0.67sec
79
# Module 6 Exam
A point on the string of a violin moves up and down in simple harmonic motion with an amplitude of 1.24 mm and a frequency of 875 Hz.
* *(a)** What is the maximum speed of that point in SI units?
* *(b)** What is the maximum acceleration of the point in SI units?
**(a)** 6.82 m/s
**(b)** 3.75 × 104 m/s2
80
# Module 6 Exam
The position of a cart that is oscillating on a spring is given by the equation x = (12.3 cm) cos[(1.26 s-1)t]. When t = 0.805 s, what are the **(a)** velocity and **(b)** acceleration of the cart?
**(a)** -13.2 cm/s
**(b)** -10.3 cm/s2
81
# Module 6 Exam
An air-track cart is attached to a spring and completes one oscillation every 5.67 s in simple harmonic motion. At time t = 0.00 s the cart is released at the position x = +0.250 m. What is the position of the cart when t = 29.6 s?
1. x = 0.342 m
2. x = -0.218 m
3. x = 0.0461 m
4. x = 0.218 m
5. x = 0.210 m
3. x = 0.0461 m
82
# Module 6 Exam
An object is oscillating on a spring with a period of 4.60 s. At time t = 0.00 s the object has zero speed and is at x = 8.30 cm. What is the acceleration of the object at t = 2.50 s?
1. 14.9 cm/s2
2. 11.5 cm/s2
3. 0.784 cm/s2
4. 1.33 cm/s2
5. 0.00 cm/s2
1. 14.9 cm/s2
83
# Module 6 Exam
The position of an object that is oscillating on an ideal spring is given by x = (17.4 cm) cos[(5.46 s-1)t]. Write an expression for the acceleration of the particle as a function of time using the cosine function.
**a** = - (519 cm/s2) cos[(5.46 s-1)t]
84
# Module 6 Exam
A 0.150-kg air track cart is attached to an ideal spring with a force constant (spring constant) of 3.58 N/m and undergoes simple harmonic oscillations. What is the period of the oscillations?
1. 1.14 s
2. 1.29 s
3. 2.57 s
4. 0.263 s
5. 0.527 s
2. 1.29 s
85
# Module 6 Exam
\* 19 \* A 1.15-kg beaker (including its contents) is placed on a vertical spring scale. When the system is sent into vertical vibrations, it obeys the equation y = (2.3 cm)cos(17.4 s-1 t). What is the force constant (spring constant) of the spring scale, assuming it to be ideal?
(Possible Answer): **796.49 N/m**
86
# Module 6 Exam
An object of mass m = 8.0 kg is attached to an ideal spring and allowed to hang in the earth's gravitational field. The spring stretches 2.2 cm before it reaches its equilibrium position. If it were now allowed to oscillate by this spring, what would be its frequency?
1. 1.6 Hz
2. 0.28 x 10-3 Hz
3. 0.52 Hz
4. 3.4 Hz
4. 3.4 Hz
87
# Module 6 Exam
A 2.0-kg block on a frictionless table is connected to two springs whose opposite ends are fixed to walls, as shown in the figure. The springs have force constants (springconstants) k1 and k2. What is the oscillation angular frequency of the block if k1 = 7.6 N/m and k2 = 5.0 N/m?
1. 3.5 rad/s
2. 0.40 rad/s
3. 0.56 rad/s
4. 2.5 rad/s
4. 2.5 rad/s
88
# Module 6 Exam
A block attached to an ideal spring of force constant (spring constant) 15 N/m executes simple harmonic motion on a frictionless horizontal surface. At time t = 0 s, the block has a displacement of -0.90 m, a velocity of -0.80 m/s, and an acceleration of +2.9 m/s2 . The mass of the block is closest to
1. 4.7 kg
2. 9.4 kg
3. 2.6 kg
4. 2.3 kg
1. 4.7 kg
89
# Module 6 Exam
**\* 23 \*** A 0.39-kg block on a horizontal frictionless surface is attached to an ideal spring whose force constant (spring constant) is 570 N/m. The block is pulled from its equilibrium position at x = 0.000 m to a displacement x = +0.080 m and is released from rest. The block then executes simple harmonic motion along the horizontal x-axis. When the position of the block is x = 0.057 m, its kinetic energy is closest to
1. 1 J.
0. 95 J.
1. 0 J.
0. 90 J.
0. 84 J.
*Might be ...* **0.9 J**
1/2kA2 = 1/2kx2 = 1/2mv2
90
# Module 6 Exam
An object of mass 6.8 kg is attached to an ideal spring of force constant (spring constant) 1720 N/m. The object is set into simple harmonic motion, with an initial velocity of v0 = 4.1 m/s and an initial displacement of x0 = 0.11 m. Calculate the maximum speed the object reaches during its motion.
91
# Module 6 Exam
A 0.50-kg box is attached to an ideal spring of force constant (spring constant) 20 N/m on a horizontal, frictionless floor. The box oscillates in simple harmonic motion and has a speed of 1.5 m/s at the equilibrium position.
* *(a)** What is the amplitude of vibration?
* *(b)** At what distance from the equilibrium position are the kinetic energy and the potential energy the same?
**(a)** 0.24 m **(b)** 0.17 m
92
# Module 6 Exam
A 0.50-kg object is attached to an ideal spring of spring constant (force constant) 20 N/m along a horizontal, frictionless surface. The object oscillates in simple harmonic motion and has a speed of 1.5 m/s at the equilibrium position. What are **(a)** the total energy and **(b)** the amplitude of vibration of the system?
**(a)** 0.56 J
**(b)** 0.24 m
93
# Module 6 Exam
What is the length of a simple pendulum with a period of 2.0 s?
1. 1.6 m
2. 20 m
3. 1.2 m
4. 0.87 m
5. 0.99 m
5. 0.99 m
94
# Module 6 Exam
\* 28\* A simple pendulum has a period T on Earth. If it were used on Planet X, where the acceleration due to gravity is 3 times what it is on Earth, what would its period be?
1. T/3
2. T
3. square 3 T
4. T/square 3
5. 3T
95
# Module 6 Exam
An astronaut has landed on Planet N-40 and conducts an experiment to determine the acceleration due to gravity on that planet. She uses a simple pendulum that is 0.640 m long and measures that 10 complete oscillations 26.0 s. What is the acceleration of gravity on Planet N-40?
1. 4.85 m/s2
2. 1.66 m/s2
3. 2.39 m/s2
4. 3.74 m/s2
5. 9.81 m/s2
4. 3.74 m/s2
96
# Module 6 Exam
An astronaut has landed on an asteroid and conducts an experiment to determine the acceleration of gravity on that asteroid. He uses a simple pendulum that has a period of oscillation of 2.00 s on Earth and finds that on the asteroid the period is 11.3 s. What is the acceleration of gravity on that asteroid?
1. 1.74 m/s2
2. 5.51 m/s2
3. 0.307 m/s2
4. 0.0544 m/s2
5. 1.66 m/s2
3. 0.307 m/s2
97
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