I have found that the key is in the argument of the wave function. I have modified the script that Henrik provided to use the TriangleWave function. I'm not even certain I understand how dividing by the "time" parameter in the argument of the wave is able to create the wave-like feature (without it, an arc is created, not a wave), but it seems to work perfectly. See below:

Manipulate[

 ParametricPlot[
  {
   (1 + 0.2*TriangleWave[(20*t)/time])*Cos[t],
   (1 + 0.2*TriangleWave[(20*t)/time])*Sin[t]
   },
  {t, 0, time},
  AspectRatio -> Automatic,
  Axes -> None,
  PlotRange -> {  -1.2*1, 1.2*1  },
  PerformanceGoal -> "Quality",
  ImageSize -> Medium
  ],

 {
  {time, 0.1},
  0.1,
  2 Pi
  },
 TrackedSymbols :> {time}
 ]

Yes, I perfectly agree - I could not have found better words!

Hi Henrik,

Thanks for your post and reply. Your solution does draw a spring compressing and expanding along an arc. However, I'm looking to do this in 2D, not 3D. Is there a way to permanently set the 3D picture from the side so that it appears 2D? I'm not sure a 3D parametric plot will be compatible with the rest of the script, though. The function is a SHO with the position mapped by Cosine and Sine functions, to give the appearance that the mass is oscillating in 2D along an arc. I'm trying to get a TriangleWave (which looks like a spring) to compress and contract with the mass along the 2D arc. When I plot the TriangleWave with Polar plot, I can get the spring to bend along the 2D arc, but I cannot get it to compress and expand. When I plot the TriangleWave with ParametricPlot, I can get it to expand and contract, but not bend along the arc.

The function I'm using to do the oscillation is:

\[Theta]1[m_, k_, t_] = \[Pi]/2 + 1/6 \[Pi] Cos[(Sqrt[2] Sqrt[k] t)/Sqrt[m]];

The graphics I've tried are (just click 'Play' to see what I'm talking about):

Manipulate[

 Graphics[{

   {
    Directive[  RGBColor[0.5, 0.5, 0.5], AbsoluteThickness[3], Opacity[1]  ],
    Circle[
     {0, 0},
     1,
     {0, Pi}
     ]
    },

   {(* Mass-1 *)
    Directive[  RGBColor[0, 0, 1], Opacity[0.75]  ],
    Disk[
     {
      1*Cos[  \[Theta]1[10, 2, time]  ],
      1*Sin[  \[Theta]1[10, 2, time]  ]
      },
     1/15
     ]
    (* Mass-1 *)},
   {(* Spring-1 *)
    PolarPlot[
      {
       1 + (1/(4*\[Pi]))*TriangleWave[  2*\[Pi]*sex ]
       },
      {  sex, 0, \[Theta]1[10, 2, time]  },

      PlotStyle ->
       {
        Directive[RGBColor[0, 0, 0.35], AbsoluteThickness[2.5], 
         Opacity[0.75]]
        },
      Axes -> None
      ][[1]],
    ParametricPlot[
      {
       (1/(4*\[Pi]))*sex*\[Theta]1[10, 2, time],
       1 + (1/(4*\[Pi]))*TriangleWave[  2*\[Pi]*sex ]
       },
      {  sex, 0, \[Theta]1[10, 2, time]  },

      PlotStyle ->
       {
        Directive[RGBColor[0, 0, 0.35], AbsoluteThickness[2.5], 
         Opacity[0.75]]
        },
      Axes -> None
      ][[1]]
    (* Spring-1 *)},
   {(* Spring-2 *)
    PolarPlot[
      {
       1 + (1/(4*\[Pi]))*TriangleWave[  2*\[Pi]*sex ]
       },
      {  sex, \[Theta]1[10, 2, time], Pi  },

      PlotStyle ->
       {
        Directive[RGBColor[0, 0, 0.35], AbsoluteThickness[2.5], 
         Opacity[0.75]]
        },
      Axes -> None
      ][[1]]
    (* Spring-2 *)}

   (* Graphics *)}, ImageSize -> Medium],

 "", "",
 (* Axes *)
 "", "",

 Style["Axis Parameters", Bold, FontSize -> 14, FontFamily -> "Times"],
 {
  {time, 10^-20, Style["time", FontSize -> 16, FontFamily -> "Times"]},
  10^-20,
  120,
  ControlType -> Trigger,
  DefaultDuration -> 120.0,
  DisplayAllSteps -> True,
  AnimationRate -> 2.0
  },
 {
  {time, 10^-20, Style["", FontSize -> 16, FontFamily -> "Times"]},
  10^-20,
  120,
  ControlType -> Slider,
  ImageSize -> Medium,
  Appearance -> "Labeled"
  },

 ControlPlacement -> Left,
 SaveDefinitions -> True,
 TrackedSymbols :> {time}

 (* Manipulate *)]

I'm open to the idea of using 3D Param Plot, but I don't know how to set the viewing from the side. Any more thoughts are very welcomed! I'm at a loss as to how to fix this.

Best,

Tim