1. Horizontal frictionless road of length Lo is connected to a long and frictionless slope of angle a. A cart equipped with a rocket has mass M. The rocket can produce a constant force F for duration to. Ignore the mass of fuel and calculate the maximum height up the slope the cart can reach as a function of Lo.
a) Give a cogent argument for the physical reasons that the value of Co affects the final height that the cart can reach.
b) Solve the motion and find the value of Lo that will allow the cart to reach the maximum height. What is the value of Lo for the minimum height?
2. Two massless sticks of length 2r, each with a mass m fixed at its center, are hinged together at the end. One stands on top of the other. The bottom end of the lower stick is hinged at the ground. They are held at rest such that the lower stick is vertical and upper one is tilted at a small angle 6.
a) Use Lagrangian method to calculate their initial angular accelerations.
b) Use Lagrangian multiplier to calculate the horizontal and vertical forces provided by the hinge at the initial moment.
3. A point mass is placed on the tip of a semi-spherical dome of radius R=1, with kinetic friction coefficient p.k=0.3 between the point mass and the dome surface. The particle is given an initial horizontal speed vo.
a) Use Lagrangian method and find the Euler-Lagrange equation of motion.
b) Change variables to obtain a 1st order differential equation for v2 as a function of A (angle with respect to vertical), and solve it.
c) Try different numerical values of the initial speed and find the minimum vo that will cause the particle to fall off the dome.
d) At such a minimum speed as determined in (c), calculate the normal force of the dome. Determine the position where the normal force is the largest, and the position where particle loses contact with the dome.
4. A mass M is connected to a smaller mass m by a string that hangs over two frictionless pulleys of negligible size at the same height. Mass M hangs vertically while mass m is initially held at the same height of the two pulleys with r=ro (r being the distance to the closest pulley).
a) Rewrite the Lagrangian in dimensionless quantities (x=r/ro etc).
b) Find the equation of motion for r/ro and A (angle between r and the horizontal) as pair of 2nd order differential equations.
c) Use the 1st integral to obtain the equation of motion as one rd order and one 1st order differential equations.
d) Change independent variable from t to 0 and further reduce the equation of motion to a single 1st order differential equation (plus one algebraic equation).
e) (bonus) Solve numerically the equation of motion and find the minimum value of x for m/M=0.1.