Lab #11: Simple Harmonic Motion of a Linear Oscillator



Physics 211: Lab

Simple Harmonic Motion of a Linear Oscillator

Reading Assignment:

Chapter 16 – Section 1- Section 9

Introduction:

Imagine a point P that oscillates back and forth with simple harmonic motion. The period, T, the frequency, f, and the angular frequency, ω, of point P are defined using the following equations.

[pic] [pic]

The linear motion of point P as a function of time is sinusoidal in nature. Assume that the position x= 0 is defined as the central location of the motion.

 

 

 

 

 

The position, velocity, and acceleration of point P can be described as a function of time as follows:

 

[pic]

[pic]

[pic]

 

Notice that a(t) is related to x(t) and can be rewritten as follows:

[pic]

In other words, the acceleration is not constant. It, too, varies in a sinusoidal manner.

 

The dynamics of simple harmonic motion are governed by Newton’s Second Law and can be written as:

[pic]

or, substituting for a(t):

[pic]

Since both m and ω are constants, notice that the Net Force, Fnet, is proportional to the position, x, of point P. The negative sign signifies that the direction of the Net Force is opposite the direction of the position. In other words, the Net Force, Fnet, is a restoring force because it attempts to bring point P back to its equilibrium position.

As a result, it should follow that any Net (Restoring) Force that varies as a function of position, x, should cause simple harmonic motion. Such a system is called a linear oscillator, since the force varies linearly with the position. A spring force acting on a mass, described by Hooke’s Law, satisfies this condition:

 

 

[pic]

 

 

In fact, it has been shown that a mass oscillating at the end of a spring (with no frictional forces) experiences pure simple harmonic motion. By combining the above two equations, an equation for the theoretical value of the angular frequency of the frictionless system is obtained:

[pic]

[pic]

 

[pic] (for a system that is not damped)

When describing simple harmonic motion the term used for ω is “angular frequency” instead of “angular velocity”. This often confuses students because the concepts are similar. There are many other examples of simple harmonic motion. (Read Section 16-5 and 16-6 of the text for an analysis of torsion pendulums, simple pendulums, and physical pendulums.)

 

Damping:

In the real world, most oscillations are eventually “damped” due to friction or viscous forces. A damping force causes the amplitude (and therefore, energy) of a given system to decrease over time. In the position equation, x(t), for a damped simple harmonic oscillator, xmax , is a decreasing function, not a constant. The angular frequency, ((, of the damped system is less than the natural angular frequency, ω, that the system would have if it was not damped. (Read Section 16-8 for a specific example of a damping force.)

 

Driven Oscillations:

Many systems that oscillate experience a driving force in addition to a restoring force. A driving force is a periodic force (with a unique period, frequency, and angular frequency of its own) that effectively pushes or pulls an object away from the equilibrium position. However, if the angular frequency of the driving force, ωd, is approximately the same as the natural angular frequency, ω, of the system, then the system experiences resonance.

[pic]

When resonance occurs, the amplitude, xm, of the system reaches a maximum.

There are many examples of resonance in everyday life. Engineers must be extremely careful not to design a structure that has a natural frequency that matches a potential driving force. The Tacoma Narrows Bridge disaster is an example of such an error. Troops usually march in step with each other as they march in formation. However, whenever crossing a bridge, they are taught to break step, just in case their cadence matches the natural frequency of the bridge. A wine glass has a natural frequency of vibration that can be heard if the top of the glass is moistened and rubbed. When a singer with a strong voice sings a note at the very same pitch (frequency), the sound waves in the air drive the glass to vibrate back and forth. If the force by the air molecules is strong enough and acts long enough, the glass will break. Musical instruments, a child on a swing, the motor in a car, etc. all exhibit resonance under certain conditions.

 

Most real examples of simple harmonic motion involve all three types of forces simultaneously: a restoring force, a damping force, and a driving force. Depending upon the nature of these forces, the resulting motion is sometimes more complicated than simple sinusoidal motion. In addition, most systems exhibit more than one natural frequency. The analysis of these more complicated systems is an advanced topic of study in physics. However, the basics of simple harmonic motion remain the same.

Physics 211: Lab

Simple Harmonic Motion of a Linear Oscillator

Goals:

• Determine the natural period, frequency, and angular frequency of a linearly oscillating system.

• Observe the relationships between the position, velocity, and acceleration of an object undergoing simple harmonic motion.

• Observe the resulting motion of a damped simple harmonic oscillator.

• Verify the conditions necessary for a driven oscillating system to achieve resonance.

• Observe the resulting motion of a driven simple harmonic oscillator.

• Apply the concepts of resonance and damping to the shocks of a car.

Equipment List:

1.2 meter track with clamp

3-point stand with vertical rod

Spring (k = 3.85 N/m)

Cart (500 grams)

Ultrasonic motion detector

20 volt DC Harrison power supply

Voltage Sensor

Mechanical Oscillator/Driver

 

Activity 1: Analysis of the Simple Harmonic Motion of a Cart / Spring System

There are two purposes for this activity. The first is to determine the natural frequency of a cart/spring system. The second is to compare the position, velocity, and acceleration graphs of an object undergoing simple harmonic motion.

 

Setting Up the Graphs:

1. Set up Data Studio™ to read the data collected by the ultrasonic motion detector located at the base of the track. Check to make sure that the motion detector is oriented towards the cart and is sending out a narrow beam signal.

 

2. Create a graphing window that contains the following graphs: Position vs. Time, Velocity vs. Time, and Acceleration vs. Time.

 

3. Attach the 500 gram cart to the spring (k = 3.85 N/m) located on the inclined track. Using the ruler along the side of the track, make note of the equilibrium position of the cart.

 

4. Using the ruler located along the length of the track, estimate the distance from the motion detector to the closest end of the cart when the cart is at rest in the equilibrium position. Record this value in the table below.

|Distance of Cart from Motion Detector at Equilibrium |  |

|(meters) | |

 

5. Stretch the spring some initial length (but do not over stretch the spring!) and then release the cart from rest. Press Record to collect data for approximately five cycles of simple harmonic motion, and then press Stop.

 

6. Notice that the Position vs. Time graph does not oscillate about the origin of position (the x-axis) of the graph. Why not? What does Data Studio™ define as “Position”? How could this graph be altered so that “Position” was redefined as “the location of the cart measured from equilibrium”?

 

7. Using the Experiment Calculator, create a new calculation called “Position from Equilibrium”. State the calculation formula in the table below.

 

|Calculation Name |Short Name |Units |Formula |

|Position From Equilibrium |X |meters |  |

 

8. In the graphing window, change the plot of the Position vs. Time to Position from Equilibrium vs. Time by clicking on the y-axis icon. Notice that the graphs differ by a vertical shift. The Position from Equilibrium vs. Time graph should oscillate around the x-axis.

 

Determine the Angular Frequency of the System:

9. Using a method of your choice, measure the amount of time that it takes the cart to complete 1 cycle.

 

10. From your measurement, record and/or calculate the period, frequency, and angular frequency of the resulting simple harmonic motion of the cart/spring system in the table below.

 

|Period – T |Frequency – f |Angular Frequency – ω |

|(s) |(Hz) |(rad/s) |

|  |  |  |

 

11. Determine the theoretical value for the natural angular frequency, ω, of the cart (mass) and spring system. Explain your calculation. (See the Introduction or Section 16-3 in the text. Note: This calculation assumes that the system is frictionless.)

|(Theoretical) | |

|Natural Angular Frequency - ω |Explanation of Calculation |

|(rad/s) | |

|  |  |

 

12. What is the value of the % error between the experimental value of ω and the theoretical value of ω (assumed to be “true”)? Clearly show your calculations. Is the experimental value (which is slightly damped) less than the theoretical value (which assumes that it is not damped, and therefore “natural”) as predicted?

Comparing the Graphs of Simple Harmonic Motion:

13. Analyze the relationships between position, velocity, and acceleration by answering the following questions:

Note #1: Recall that the motion detector defines the positive direction to be pointing up the incline.

Note #2: The Analyze Tool located in the bottom left hand corner of the graphing window is a helpful resource for analyzing the graphs. This tool looks like a cross hair.

 

• What is the location and direction of the cart when the velocity is at a maximum or minimum value?

• What is the location of the cart when the velocity is equal to zero?

• When the acceleration of the cart is at a maximum or minimum value, what is the velocity and location of the cart?

 

Activity 2: Damped Simple Harmonic Motion

The purpose of this Activity is to observe the behavior of an object undergoing damped simple harmonic motion.

 

1. Stretch the spring some initial length (but do not over stretch the spring!), press Record, and then release the cart from rest. Allow the cart to oscillate back and forth until it comes completely back to rest before pressing Stop.

 

2. Copy the graphing window into the Template by using “paste special”. Paste it as if it was a picture.

 

3. Answer the following questions regarding the damped motion of the cart / spring system:

 

• How does the period of the resulting simple harmonic motion change over time as damping occurs? Support your reasoning by analyzing data from the graphs. Clearly explain your analysis.

 

• How does the amplitude of the position change over time as damping occurs? (For example: Is it constant? Is it changing linearly? Is it changing exponentially? Is it increasing? Is it decreasing?)

 

• What does the change in amplitude suggest about the total energy loss of the system over time? (i.e. What type of dissipating forces are likely acting on the system?)

 

Activity 3: Forced Oscillations & Resonance

The purpose of this Activity is to observe the behavior of an object undergoing forced (driven) simple harmonic motion and to verify the conditions necessary for resonance to occur.

 

1. Place the cart on the track so that it is attached to the spring and at rest in its equilibrium position.

 

2. Important: Do not turn on the power supply until the following conditions have been satisfied:

 

• The Voltage knob should be turned completely in the counter-clockwise position (its lowest value) whenever the power supply is turned on and off. Note: There are two adjustment knobs. The black one provides coarse-tuning, and the red one provides fine-tuning. Both should be turned completely counter-clockwise.

• The Current knob should be turned completely in the clockwise direction (its highest value). This setting should not be changed throughout the entire experiment.

• Check the connections between the power supply and the Oscillator/Driver to be sure that the Red leads are attached to the + side of the power supply and the Black leads are attached to the – side of the power supply.

 

3. Set up the equipment:

• Set up Data Studio( to read the data collected from the Voltage Sensor.

• Display the Voltage Sensor data on the Digits display window. (This window will display the voltage supplied to the Mechanical Oscillator/Driver whenever Record is pressed.)

• Change the Digits display window from “0.0” to “0.00” so that more digits are displayed. This is done by double- clicking anywhere on the display window.

• Turn on the 20-volt DC power supply.

• Slowly increase the voltage in the circuit that controls the Oscillator/Driver so that the Oscillator/Driver begins to turn. Note that the Oscillator/Driver creates a periodic pulling motion on the cart / spring system.

 

4. Slowly increase the voltage in the circuit that controls the Oscillator/Driver so that the Oscillator/Driver is caused to rotate at different frequencies. Using the value displayed on the Digits display window, set the voltage of the system to the settings listed in the table below.

 

5. At each voltage setting (and therefore frequency), first allow the system to damp itself into a regular pattern. Then, using Data Studio™, press Record to collect data on the resulting motion. At each setting, determine the period, angular frequency, and amplitude. Place your data in the following table.

 

|Voltage Setting |Period |Angular Frequency |Amplitude |

|(volts) |(sec) |(radians/s) |(meters) |

|1.5 |  |  |  |

|2.0 |  |  |  |

|2.5 |  |  |  |

|3.0 |  |  |  |

|3.5 |  |  |  |

|4.0 |  |  |  |

 

6. Use Excel™ to make a plot of Amplitude vs. Angular Frequency. (Place Amplitude on the y-axis and Angular Frequency on the x-axis.) Label the axes.

 

7. Copy and paste the Excel™ plot into the Template.

 

8. Using the graph in Excel™, predict the approximate angular frequency of the Oscillator/Driver that will make the cart/spring system “resonate”. Turn the Voltage knob to the corresponding value from your graph, and then use the fine-tuning adjustments to find resonance. Ask your TA to help you if you have trouble locating it.

 

9. Once resonance is found, obtain a graph of the motion of the cart/spring system at this resonant frequency: Start the cart from rest at its equilibrium position. Press Record and allow the cart to oscillate back and forth until it reaches an apparent maximum amplitude limit for the given system. Press Stop.

 

10. Remember: Turn the Voltage knob completely counter-clockwise before turning the power supply off!

 

11. Copy the graphing window of Position from Equilibrium vs. Time, Velocity vs. Time, and Acceleration vs. Time into the Template by using “paste special”. Paste it as if it was a picture.

 

12. Determine the period, frequency, and angular frequency of the driven system at resonance and place the values in the table below.

 

|Resonant Period – T |Resonant Frequency – f |Resonant Angular Frequency – ω |

|(s) |(Hz) |(radians/s) |

|  |  |  |

 

13. Compare these values with the natural period, frequency, and angular frequency of the cart/spring system determined in Activity 1. What condition is necessary to cause a system to resonate?

 

 

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|X = +Xmax |

|X= 0 |

|X= -Xmax |

|P |

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