Physics 106 Laboratory: Mass on a Spring

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Oscillatory Motion

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

mass on spring Spring Equilibrium

The restoring force results in the mass eventually returning to the equilibrium position. The mass, however, picks up some momentum and continues past the equilibrium position causing the same situation as before but on the opposite 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 it, and so on. Such behavior is called oscillatory motion.

(See Chapter 15 in Fundamentals of Physics by Halliday, Resnick and Walker.)

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

y(t) = yeq + A cos ( 2 π t / T + φ ).

The term yeq 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 as is typically done in the theoretical account. It is assumed that the argument (the stuff in the parentheses, a.k.a. "the phase") is in radians and NOT in degrees. Note that there are four parameters.

Oscillation Parameters

Experiment

Analysis

  1. For each of the masses, plot position versus time [an XY (Scatter) Chart in Excel]. On the same graph, plot a mathematical function that resembles as much as possible ("fits") your data. Click here for some instructions for this process.
  2. You should be able to identify in your function the equilibrium position, the amplitude, period and phase constant associated with this oscillatory motion. Include in your report a table of the masses and this information.

    Mass Variations
    Mass
    without spring
    contribution
    (    )
    Eq. Pos.
    (    )
    Amplitude
    (    )
    Period
    (    )
    Ph. const.
    (    )
    Squares
    (    )
               
               
               
               


  3. Plot weight versus yeq. Recall that to obtain weight in Newtons, one multiplies the mass in kilograms by g (9.8 m/s2). Fit this data (i.e. add a trendline) to a straight line. Extract from the fit the force constant of the spring k.

    Force constant (     )            


  4. The formula

    T2 = 4 π2 m / k


    comes from squaring both sides of T = 2 π m/k 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 spring's mass to the period. (Note that the constant b in y= m x + b is the y intercept, not the x intercept.) According to your results, what fraction of the spring's mass contributes? How does this compare to the theoretical prediction of 1/3?

    Mass of spring (     ) x intercept (     ) Fraction Predicted
    Fraction
    Percent
    Error
          1/3  


  6. Compare your experimental periods to the theoretical values

    T = 2 π ( m / k )1/2.

    and to a corrected theory that takes into account the effect of the spring's mass

    Tcor = 2 π [ (m + 0.333 mspring) / k ]1/2.

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


  7. Plot period versus mass (using the data in which the mass was varied) and period versus amplitude (using the data in which the amplitude was varied but mass held fixed). Make sure that the two plots have the same scale on the y axis. (If you need to change the scale on one or both, right click on the numbers along the y axis and choose Format Axis, select Axis Options, and enter values in the Minimum and Maximum textboxes. The Auto checkbox/radiobutton should not be checked.) What can you conclude from this comparison of period's dependences on mass and amplitude?

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