Clear[u1, u2, g, f, r]
ρ := 1
m := 1
f[x_] := 
      Sin[x] /; 2  π ≤ x ≤ 
          3  π
f[x_] := 
      0 /; (x > 3  π || x < 2  π)
r[x_] := ((ρ (m Cos[x] - ρ Sin[x]) (UnitStep[-3 π + x] - UnitStep[-2 π + x]))/((m^2) + (ρ^2))) + 
              ((ρ (E^((((2 π - x) ρ)/m))) m ((E^(((π ρ)/m))) UnitStep[-3 π + x] + UnitStep[-2 π + x]))/((m^2) + (ρ^2)))
u1[x_] := 
      If[(0 ≤ x && 
            x - t ≥ 0), (1/2) (f[x + t] + f[x - t]), 
        0]
u2[x_] := 
      If[(0 ≤ x && 
            x - t ≤ 
              0), (1/2) (f[x + t] - f[-x + t]) + r[t - x], 
        0]
g[x_] := u1[x] + u2[x]
\(a[x_] := 
      Plot[g[x], {x, 0, 15}, 
        AspectRatio -> Automatic, 
	
        PlotStyle -> {RGBColor[1, 0.7, 0], 
            Thickness[0.005]}, 
	
        PlotRange -> {{\(-1\), 15}, {\(-1.5\), 1.5}}, 
        DisplayFunction -> Identity]\)
b[x_] := 
      Show[Graphics[Disk[{0, r[t]}, 0.2], 
          AspectRatio -> 
            Automatic, 
       PlotRange -> {{-1, 15}, {-1.5, 1.5}}], 
        DisplayFunction -> Identity]
c[x_] := 
      Show[{a[x], b[x]}, 
        DisplayFunction -> $DisplayFunction]
mass = Table[c[x], {t, 0, 20, 0.1}]

Clear[u1, u2, g, f, r]
ρ := 1
m := 100
f[x_] := 
      Sin[x] /; 2  π ≤ x ≤ 
          3  π
f[x_] := 
      0 /; (x > 3  π || x < 2  π)
r[x_] := ((ρ (m Cos[x] - ρ Sin[x]) (UnitStep[-3 π + x] - UnitStep[-2 π + x]))/((m^2) + (ρ^2))) + 
              ((ρ (E^((((2 π - x) ρ)/m))) m ((E^(((π ρ)/m))) UnitStep[-3 π + x] + UnitStep[-2 π + x]))/((m^2) + (ρ^2)))
u1[x_] := 
      If[(0 ≤ x && 
            x - t ≥ 0), (1/2) (f[x + t] + f[x - t]), 
        0]
u2[x_] := 
      If[(0 ≤ x && 
            x - t ≤ 
              0), (1/2) (f[x + t] - f[-x + t]) + r[t - x], 
        0]
g[x_] := u1[x] + u2[x]
\(a[x_] := 
      Plot[g[x], {x, 0, 15}, 
        AspectRatio -> Automatic, 
	
        PlotStyle -> {RGBColor[1, 0.7, 0], 
            Thickness[0.005]}, 
	
        PlotRange -> {{\(-1\), 15}, {\(-1.5\), 1.5}}, 
        DisplayFunction -> Identity]\)
b[x_] := 
      Show[Graphics[Disk[{0, r[t]}, 0.2], 
          AspectRatio -> 
            Automatic, 
       PlotRange -> {{-1, 15}, {-1.5, 1.5}}], 
        DisplayFunction -> Identity]
c[x_] := 
      Show[{a[x], b[x]}, 
        DisplayFunction -> $DisplayFunction]
mass = Table[c[x], {t, 0, 20, 0.1}]

Clear[u1, u2, g, f, r]
ρ := 1
m := 0.001
f[x_] := 
      Sin[x] /; 2  π ≤ x ≤ 
          3  π
f[x_] := 
      0 /; (x > 3  π || x < 2  π)
r[x_] := ((ρ (m Cos[x] - ρ Sin[x]) (UnitStep[-3 π + x] - UnitStep[-2 π + x]))/((m^2) + (ρ^2))) + 
              ((ρ (E^((((2 π - x) ρ)/m))) m ((E^(((π ρ)/m))) UnitStep[-3 π + x] + UnitStep[-2 π + x]))/((m^2) + (ρ^2)))
u1[x_] := 
      If[(0 ≤ x && 
            x - t ≥ 0), (1/2) (f[x + t] + f[x - t]), 
        0]
u2[x_] := 
      If[(0 ≤ x && 
            x - t ≤ 
              0), (1/2) (f[x + t] - f[-x + t]) + r[t - x], 
        0]
g[x_] := u1[x] + u2[x]
\(a[x_] := 
      Plot[g[x], {x, 0, 15}, 
        AspectRatio -> Automatic, 
	
        PlotStyle -> {RGBColor[1, 0.7, 0], 
            Thickness[0.005]}, 
	
        PlotRange -> {{\(-1\), 15}, {\(-1.5\), 1.5}}, 
        DisplayFunction -> Identity]\)
b[x_] := 
      Show[Graphics[Disk[{0, r[t]}, 0.2], 
          AspectRatio -> 
            Automatic, 
       PlotRange -> {{-1, 15}, {-1.5, 1.5}}], 
        DisplayFunction -> Identity]
c[x_] := 
      Show[{a[x], b[x]}, 
        DisplayFunction -> $DisplayFunction]
mass = Table[c[x], {t, 0, 20, 0.1}]

g1 = 
      Plot[x, {x, 0, 20}, AspectRatio -> Automatic, 
        PlotStyle -> {RGBColor[0, 0, 1], Thickness[0.01]}, 
        PlotRange -> {{-1, 15}, {-1.1, 15}}, 
        DisplayFunction -> Identity]
g2 = 
      Plot[Sin[x], {x, 2  π, 4  π}, 
        PlotStyle -> {RGBColor[1, 0.7, 0], Thickness[0.01]}, 
        DisplayFunction -> Identity]
g3 = 
      Graphics[Disk[{0, 0}, 0.3], 
        DisplayFunction -> Identity]
g4 = 
      Plot[0, {x, 0.3, 2  π}, 
        PlotStyle -> {RGBColor[1, 0.7, 0], Thickness[0.01]}, 
        PlotRange -> {{-1, 15}, {-1.1, 15}}, 
        DisplayFunction -> Identity]
g5 = 
      Plot[0, {x, 4  π, 6  π}, 
        PlotStyle -> {RGBColor[1, 0.7, 0], Thickness[0.01]}, 
        PlotRange -> {{-1, 15}, {-1.1, 15}}, 
        DisplayFunction -> Identity]
Show[{g1, g2, g3, g4, g5}, 
      DisplayFunction -> $DisplayFunction]