1.

Write down the lagrangian for a mass m on a frictionless table attached to a spring with spring constant k and relaxed length L. The other end of the spring is fixed to the table so the mass can move in two dimensions around it. Write down the two corresponding Euler-Lagrange Equations. Using Python Numerically solve them, plot the (x,y) location of the mass for 10 seconds if: L = 0.1 meters , m = 0.5 kg, k = 20 newton/meters, and initial conditions: x = /15 meters, y = 0, the derivative of x = 0, and the derivative of y = 1 meters per second.

2. The measured orbital period of an exoplanet is 20 days. The dip in luminosity from the reflecting planet transiting behind the star is one fifth of the duration of the dip from the planet transiting in front of the star. The line of sight is along the major axis of the orbit. Assume that the system is so far away that the transit distances in front and behind are both approximately 2 Rs. you can estimate that the star (relative to the sun) has a mass of 10 and radius Rs = 5.6R

What is the ratio of the perihelion to the aphelion? Which (perihelion or aphelion) is between us and the star, and which is behind the star? What is the value of the semi-major axis 'a' of the orbit in meters? Therefore, what are the aphelion and perihelion radii, the eccentricity, and the equation of the ellipse The minimum detectable wobble requires a displacement of the star a distance of one tenth Rs where the planet is a distance c from the center of mass as drawn in taylor figure 8.10. Therefore , what is the maximum mass of the planet? What is the percent difference between this mass and the corresponding reduced mass, and the percent difference between the star's mass and the total mass?

3. One a windless day, you fire a spherical flare of mass 5 grams and radium 1.25 cm straight up from the summit of mount everest(latitude: 28 degrees North and radius from the earths center: 6,382 kilometers) with muzzle velocity 90 meters per second. Where is the flare relative to you when it returns to your height? Take into account the Earth's gravity, Drag forces, and the inertial centrifugal and centripetal forces. Use python. For drag forces beta = 3 *(10^-5) ((Newton*second)/Meters^2) and gamma = 0.08 ((Newton*second^2)/Meters^4) A uniform rod with mass M and moment of inertia I = (1/12)*M*L^2 is balanced on a pivot. Each side is attached to a spring and spring constant k connected to a mass m. Write down the Lagrangian for the system and solve the Euler-Lagrange equations in matrix form to find the normal modes of oscillation and the corresponding relative amplitudes fpr the equations of motion.