12. Periodic Motion: Mass and Spring



12. Periodic Motion: Mass and SpringGuided InquiryDriving Question | ObjectiveWhat variables affect the period of oscillation of a mass and spring system? Experimentally determine the physical properties of a hanging mass and spring system that affect its period of oscillation.Design and Conduct Your ExperimentIt is your group’s responsibility to design and conduct an experiment whose data will support your answer to the driving question above. Use the answers to the guiding questions below to help guide your experiment design. After you have answered the guiding questions, write an outline of the equipment setup and procedure you will use to collect data, identifying the steps in sequence and the points at which each piece of equipment will be used.Suggested Materials and EquipmentAlthough you have the freedom to design your procedure using any reasonable equipment at your disposal, the following equipment is recommended for your experimental setup.Data collection systemMeter stickPASCO Smart Cart with hook1Hooked mass setPASCO Smart Cart Rod Stand Adapter2Springs of similar size (diameter and length), Table clamp or large base but varying spring constant (3), 1–15 N/mSupport rod, 60-cm or tallerSprings with similar spring constant and Support rod, 45-cm diameter, but of varying length (2), 0.1–0.3 mRight angle clamp1ap372ap40 PASCO Smart CartPASCO Smart CartRod Stand AdapterGuiding Questions1.How do you plan on constructing your mass and spring system? Will the system be oriented horizontally or vertically? Explain your choice. 2.What are four physical properties of your mass and spring system that can be changed, that you believe will affect its oscillation period?3.How do you plan on changing each of these properties within your experiment? Explain the process for changing each one.4.How would you assemble the equipment from the materials provided in preparation for making measurements? Explain the important points regarding your setup, and explain how you plan to measure the period of the system.5.The experimental process generally involves changing one variable, keeping others constant while recording data. For this experiment, which variables will you change and which variables will be held constant?Experimental DesignYour goal is to experimentally determine the physical properties of a mass and spring system that affect its period of motion. Use the responses to the Guiding Questions to help finalize your procedure and your equipment configuration.Once you are convinced that your procedure will accomplish the experiment's objectives, record your experimental setup and procedure in the following sections. SetupDraw and/or describe your experimental setup such that a third party could recreate the same setup in an attempt to reproduce your experiment?ProcedureOutline the procedure you will use in your experiment, listing all of the steps below. Your outline should be written such that a third party could follow the same procedure in an attempt to reproduce your experiment?Collect DataPerform your experiment and record all relevant data. Present your data below (or in an attached document) in a form that best suits the experiment format, such that a third party can understand your experimental results in an attempt to reproduce them.Analysis Questions1.For each part of your experiment, list each variable involved and state whether it was held constant, increased, or decreased.2.In your experiment, what variables (physical properties) affected the period of a mass and spring system and how did they affect the period?3.The mathematical equation describing the period Ts of a mass and spring system is:(2)where k is the spring constant of the spring, and m is the amount of hanging mass. Does your data support this mathematical relationship? Justify your answer.Synthesis Questions1.The motion of oscillating mass and spring systems follow cyclical patterns, so their motion is often described using sinusoidal functions with an angular frequency ω. The angular frequency is analogous with the angular velocity of something in circular motion and they share the same symbol ω. The angular frequency of a system can be determined with this equation:(3)What is the period for a mass and spring system whose angular frequency is 6.28 rad/s? Show calculations and all work.2.Use Equations 2 and 3 to derive a new expression for ω using just mass m and the spring constant k. Show your work here.3.The position versus time graph below shows the motion of an oscillating mass and spring system. This graph can be described using the equation: or: where A is the maximum displacement of the mass from equilibrium (both positive and negative). The corresponding graph for velocity versus time can be described with this equation:Use your knowledge of the graphical connection between position versus time and velocity versus time graphs to sketch the system’s corresponding velocity versus time graph in the blank axes below. Be sure to label both axes with a correct scale.4.Sketch the position versus time graph for an oscillating mass and spring system whose position at time t = 0 is equal to its maximum displacement of 8.0 cm, and takes 0.80 seconds to complete one cycle of motion. Sketch as much of the graph that will fit in the blank axes below, and identify on your sketch the points at which the system has maximum velocity.5.If the spring in the previous question has a spring constant of 25 N/m, what is the value of the mass? Show calculations and all work.6.A given spring has a spring constant k and period Ts. If you doubled the mass, what would the new period be? (Show all work, and put in terms of Ts). ................
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