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6. Below is a picture of an inverted pendulum of mass m and length L. Its angular position with respect to the ground is given by 0. (10 pts) m L a) Based on the figure above, derive expressions for the x and y positions of tilfe mass, m, as a function of the length of the limb, L, and specified rotation angle, 9 b) Derive expressions for the vertical and horizontal ground reaction forces as a function of the length of the limb, L, and rotation angle, 0. (Hint: assume the presence of gravity that is directed in the -y direction. Use this term if you need it) c) Given the following: L = 2 m m = 2 kg O'=-2.5rad/s 0"-0 rad/s2 Plot the vertical position of the mass as 0 varies over a range from 135° to 60° (37/4 to 7/3 rads). Briefly discuss how this plot relates to the motion paths of the center of mass in walking. d) For the same constants given in part (c), plot the vertical ground reaction force as 0 varies over a range from 135° to 60° (37/4 to 7/3 rads). Briefly discuss how your plot compares to the plot of vertical ground reaction force.
e) Plot the vertical ground reaction force for (not zero) and the initial conditions shown below (over the same range as before). Briefly discuss how your plot compares to the plot of vertical ground reaction force in part (d). 0-2.5rad/s: 0*= 2 rad/s2; 0o = 135 degree; Qüvat= 60 degree