According to the Plot Above, What Is the Equilibrium Position as Measured by the Motion Sensor?

Oscillatory Motion

When a mass is hung vertically from a spring, the jump stretches. The force on the mass due to the spring is proportional to the amount the bound is stretched. There is a signal at which the spring strength and the weight are equal in magnitude merely opposite in direction. This point is called the equilibrium position. If the mass is in any other position, there is a net forcefulness called the restoring forcefulness directed toward the equilibrium position.

The restoring force results in the mass eventually returning to the equilibrium position. The mass, nonetheless, picks upwardly some momentum and continues by the equilibrium position causing the same situation equally before only on the reverse side of the equilibrium position. The outcome is a repeated motion in which the mass passes through the equilibrium position, turns around, heads back toward the equilibrium position, passes through information technology, and and then on. Such beliefs is called oscillatory motion.

(Come across Affiliate xv in Fundamentals of Physics by Halliday, Resnick and Walker.)

The position of a mass oscillating on a jump can be described by the following equation

y(t) = y_eq + A cos ( 2 π t / T + φ ).

(Compare to eqs. 15-3 and fifteen-five). The term y_eq is needed in an experiment because the origin is determined by the location of the measuring device, thus the origin cannot be chosen to be the equilibrium position equally is typically washed in the theoretical business relationship. It is assumed that the statement (the stuff in the parentheses, a.m.a. "the phase") is in radians and NOT in degrees. Note that there are four parameters

• y_eq: the equilibrium position, determines the midpoint of the motion
• A: the amplitude, determines the size the the oscillations, i.eastward. the maximum deportation from the equilibrium position
• T: the period, determines how much fourth dimension goes by before the motion begins to echo itself
• φ: the stage constant, determines the part of the bike at the beginning, when t=0.

Experiment

• Plug a motion sensor into the PASCO betoken interface, the yellow plug should get into Digital Channel 1, the other into Digital Aqueduct 2.
• In Data Studio, double click on the Digital Channel 1 icon and cull Motion sensor. Set the trigger rate to 20 Hz. (You can obtain more data using 50 Hz, but the sensors sometimes accept trouble collecting data at that frequency.) For our sensors to work properly, the mass and the sensor should exist about 0.five meters apart.
• Place a mass on a spring hanging from a tall ringstand with a motion sensor positioned beneath as shown to a higher place. The bar from which the mass hangs should exist as far every bit possible from the motion sensor (i.e. at the elevation of the ringstand). Do not change its position during the experiment or yous will accept to retake all of your data.

• Part A. Mass Variations
  • Record the mass, don't forget to include the mass of the hanger. (Enter the sum in the commencement column of the Mass Variations table in the analysis department.)
  • Set the mass into oscillatory motion (stretch the leap approximately three centimeters and release) and brainstorm recording information. You merely need to record a few cycles. Attempt to ensure that the motion is vertical.
  • Inside Information Studio, make a table of the position data. If y'all do not accept a cavalcade for Time, and so click on the clock button to become the times equally well.
  • Copy the columns and paste them into an Excel spreadsheet.
  • Repeat this procedure with several different masses (at least three others, changing the mass by at least fifty g each time).
  • Utilize a balance to determine the mass of your spring.

  Mass of the spring (       )

• Office B. Amplitude variations
  • Stretch the spring 1 cm from its equilibrium position and release. Use a watch with a second hand (or mayhap a jail cell phone or an online stopwatch like this ane) to record the time required for ten complete oscillations. Dissever that number by 10 to obtain the period (the time for one complete oscillation) and enter it into the table below.
  • Repeat with amplitudes of 2 cm, 3 cm, 4 cm and 5 cm.

  Amplitude Variations

  Amplitude (cm)	Period (     )
  one
  ii
  three
  4
  5

Assay

1. For each of the masses, plot position versus time [an XY (Besprinkle) Chart in Excel]. On the aforementioned graph, plot a mathematical function that resembles every bit much as possible your data. Click here for some instructions for this procedure.
2. You lot should exist able to identify in your function the equilibrium position, the aamplitude, menstruation and phase constant associated with this oscillatory move. Include in your report a table of the masses and this information. The instructions in a higher place likewise tell you how to discover the "squares" which provides a numerical way to discover the parameters that give the "best fit" -- presumably the fit with the minimum or "least squares".  Perform this pace on at least one of your runs.

   Mass Variations

   Mass (    )	Eq. Pos. (    )	Amplitude (    )	Catamenia (    )	Ph. const. (    )	Squares (    )

3. Plot weight versus y_eq . Recall that to obtain weight in newtons, one multiplies the mass in kilograms past g (9.8 grand/due south²). Fit this information (i.e. add a trendline) to a directly line. Extract from the fit the force abiding of the spring k.
4. The formula

   T² = 4 π ² yard / k

   comes from squaring both sides of eq. 15-thirteen (Halliday/Resnick/Walker), which is an idealized equation that assumes the spring is massless. Make a graph of the period squared versus mass (T² versus m) Fit your data to a straight line.

5. The absolute value of x intercept of this graph represents the contribution of the bound's mass to the period. (Note that the constant b in y= chiliad x + b is the y intercept, non the x intercept.) Co-ordinate to your results, what fraction of the spring's mass contributes? How does this compare to the theoretical prediction of 1/three?

   Mass of spring (     )	ten intercept (     )	Fraction	Predicted Fraction	Per centum Mistake
   			1/iii

6. Compare your experimental periods to the theoretical values

   T = 2 π ( one thousand / k )^one/ii.

   and to a corrected theory that takes into account the upshot of the spring'south mass

   T_cor = ii π [ (m + 0.333 m_spring) / k ]^one/2.

   Exper. Period (    )	Theor. Period (    )	Percent Difference (    )	Corrected Theor. Menstruum (    )	Percent Deviation (    )

7. Plot menses versus mass (using the data in which the mass was varied -- Part A) and period versus amplitude (using the information in which the amplitude was varied but mass held fixed -- Role B). Make sure that the two plots have the same scale on the y axis. (If you lot need to change the scale on 1 or both, correct click on the numbers along the y axis and choose Format Axis, select the Scale tab (Excel 2003) or Centrality Option (Excel 2007), and enter values in the Minimum and Maximum textboxes. The Motorcar checkbox/radiobutton should not exist checked.) What can you conclude from this comparison of period'southward dependences on mass and amplitude?

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Source: http://www1.lasalle.edu/~blum/pl106wks/pl106_MassOnSpring.htm

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