The pendulum bob of mass m, shown in the figure is suspended by an inextensible string...

    

Transcribed Image Text:

The pendulum bob of mass m, shown in the figure is suspended by an inextensible string from the point p. This point is free to move along a straight horizontal line under the action of the springs, each having a constant k. Assume that the mass is displaced only slightly from the equilibrium position and released. Neglecting the mass of the springs, show that the pendulum oscillates with a period of P= 22kg mg+2 use a first-order Taylor series approximation for sin Solve for 8(t) if m=0.1 kg, r=1 m, 0(0)=pi/6 rad, and a. k=20 N/m b. k= N/m c. Plot the solutions of 0(t) for 2 periods on one figure 1-1 m-0.1 k=20 8-9.81 and cos 0 1 (0)=0 rad/s for 2 cases: P=2*p1/np.sqrt(2*k*g/(2*k*1+m*g)) t-np. linspace(0,2*P); #your work # your new solutions, convert rad to deg with 180/pi a_inf = t# create solution for k=infty a 20 t# create solution for k=20 N/m plt.plot(t,a inf) plt.plot(t,a_20) plt.xlabel('time (s)') plt.ylabel('angle (deg)') Text(0, 0.5, 'angle (deg)') Fam of Jame X e Problem 2 angle (deg) time (s) from scipy.linalg import from scipy.optimize inport Tsolve, root from scipy, integrate impart solve ivp teport the ordinury differential equution integrutor in Python Spring Compound pendulum The pendulum arm of mass m, shown in the figure above, is held in place by two springs. This point is free to move along a straight horizontal line under the action of the springs, each having a constant k. Assume that the mass is displaced only slightly from the equilibrium position and released. Neglecting the mass of the springs, solve for the nonlinear equations of motion and use the solve_ivp to determine (+) Solve for (8) il m=1 kg. L=1 m, 0(0)=pi/6 rad, and (0)=0 rad/s for k-20 N/m Plot the nonlinear solutions of 8(t) for 2 periods on one figure def sy ode(t,r.): ***Help documentation for "my_ode" input is time, t (s) and reposition p (a), angle (rad), velocity p (m/s), angle velocity (rad/s)] and time output is or [velocity p (m/s), angle velocity (rad/s), accel p (a/s/s), angle accel (rad/s/s)] at time, t the CDE is defined by: dr = f(t.r)*** 1-1 -0.1 K-20 angle (deg) dr np.zeros(np.size(r)) dr[@]-[2] dr[1]-[3] your work here dr[2]... #dr[3] .... return dr Pi2 pi/np.sqrt(2*k*/(2*k*1+m*g)) solve ivp(my_ode, [0,2P], [0, pi/6,0,0]); default="RX45 plt.plot(e.t,ry[1] 180/pi, o,label-'nonlinear') # The pendulum bob of mass m, shown in the figure is suspended by an inextensible string from the point p. This point is free to move along a straight horizontal line under the action of the springs, each having a constant k. Assume that the mass is displaced only slightly from the equilibrium position and released. Neglecting the mass of the springs, show that the pendulum oscillates with a period of P= 22kg mg+2 use a first-order Taylor series approximation for sin Solve for 8(t) if m=0.1 kg, r=1 m, 0(0)=pi/6 rad, and a. k=20 N/m b. k= N/m c. Plot the solutions of 0(t) for 2 periods on one figure 1-1 m-0.1 k=20 8-9.81 and cos 0 1 (0)=0 rad/s for 2 cases: P=2*p1/np.sqrt(2*k*g/(2*k*1+m*g)) t-np. linspace(0,2*P); #your work # your new solutions, convert rad to deg with 180/pi a_inf = t# create solution for k=infty a 20 t# create solution for k=20 N/m plt.plot(t,a inf) plt.plot(t,a_20) plt.xlabel('time (s)') plt.ylabel('angle (deg)') Text(0, 0.5, 'angle (deg)') Fam of Jame X e Problem 2 angle (deg) time (s) from scipy.linalg import from scipy.optimize inport Tsolve, root from scipy, integrate impart solve ivp teport the ordinury differential equution integrutor in Python Spring Compound pendulum The pendulum arm of mass m, shown in the figure above, is held in place by two springs. This point is free to move along a straight horizontal line under the action of the springs, each having a constant k. Assume that the mass is displaced only slightly from the equilibrium position and released. Neglecting the mass of the springs, solve for the nonlinear equations of motion and use the solve_ivp to determine (+) Solve for (8) il m=1 kg. L=1 m, 0(0)=pi/6 rad, and (0)=0 rad/s for k-20 N/m Plot the nonlinear solutions of 8(t) for 2 periods on one figure def sy ode(t,r.): ***Help documentation for "my_ode" input is time, t (s) and reposition p (a), angle (rad), velocity p (m/s), angle velocity (rad/s)] and time output is or [velocity p (m/s), angle velocity (rad/s), accel p (a/s/s), angle accel (rad/s/s)] at time, t the CDE is defined by: dr = f(t.r)*** 1-1 -0.1 K-20 angle (deg) dr np.zeros(np.size(r)) dr[@]-[2] dr[1]-[3] your work here dr[2]... #dr[3] .... return dr Pi2 pi/np.sqrt(2*k*/(2*k*1+m*g)) solve ivp(my_ode, [0,2P], [0, pi/6,0,0]); default="RX45 plt.plot(e.t,ry[1] 180/pi, o,label-'nonlinear') #

Answer rating: 100% (QA)

To solve for the period of the pendulum we can use the following equation T 2lg where T is the period of the pendulum l is the length of the pendulum ... View the full answer