# Disk Radius R Mass M Mass Moment Inertia Io Point O Centroid Sits Inclined Surface Also Co Q17963072 A disk of radius R, mass m and mass moment of inertia I_o (with point o being the centroid) sits on an inclined surface; it is also connected to a spring, as shown. The disk rolls on the surface without slipping. Note that as the disk moves, its centroid translates (x) and the entire body rotates (0). In the state shown, the spring has already been elongated statically by x_st due to gravity. In other words, x is measured from the equilibrium position of the system. Consider displacing the system an arbitrary amount (x > 0 and theta > 0). Write down the total kinetic and potential energies at that instant. Write it so that the only unknown (dependent variable) that appears is x(t). i.e. eliminate theta(t). In class we showed that if energy is conserved (which is the case here), then Delta T + Delta v = 0 so that (T_f – T_i) + (V_f – V_i) = 0 T_f + v_f = T_i + V_i Since the final time is any arbitrary instant later, this suggests that the total energy at any instant is constant. So we can write: T_f + V_f = C. Substitute your kinetic and potential energies into Equation (1) and then differentiate the result with respect to time. This will zero out the right hand side (you don’t know that constant – so it’s better to just get rid of it). The result is your equation of motion. What is this equation?Show transcribed image text