Question

The cart below caries a pendulum with the characteristics shown. If the cart is subjected to acceleration "a", the pendulum settles to an angle θ that varies as "a" varies. Using a programming language on your choice present a graph that shows the angle θ for different values of acceleration when (Oka<20g where g is gravitational acceleration) Prepare a report and communicate the results in a clear well-explained way. Present your Free-body diagram and all the details as you see fit. Present your code as an appendix to the report. 1 kg k-1000 N/m 0.5 m

Answer 1

From the reference frame co-moving with the car, the free body diagram of the pendulum will look like, Initial length of the spring l_0 = squareroot (0.5)^2 + (0.5)^2 m = squareroot 2 times 0.5 m = 0.71 m length at the new position l = squareroot (0.5)^2 + (0.5)^2 - 2 times (0.5)^2 times cos(pi/2 + theta) = (0.5) squareroot 2 + 2 sin theta = squareroot 2 times 0.5 (1 + sin theta)^1/2 = l_0 (1 + sin theta)^1/2 Extension in the spring is delta l = l - l_0 [(1 + sin theta)^1/2 - 1)] - (i) \r\n From the above figure we have theta + 2 phi = pi/2 phi = (pi/4 - theta/2) - (ii) Now considering equilibrium along x direction we have (M + m)a = (k delta l) cos phi (M + m)a = kl_0 [(1 + sin theta)^1/2 - 1] cos (pi/4 - theta/2) [squareroot (1 + sin theta) - 1] cos (pi/4 - theta/2) = (M + m)a/v l_0 squareroot 1 + sin theta - 1 =