Consider the well-known problem of a cart of mass m moving along the x axis attached to a spring (force constant k), whose other end is held xed. if we ignore the mass of the spring, then we know that the cart executes simple harmonic motion with angular frequency ! = p k=m. 1. assuming, that the spring has mass m, that it is uniform and stretches uniformly, show that its kinetic energy is 1 6mx_ 2. (as usual x is extension of the spring from its equilibrium length.) 2. write down the lagrangian for the system of cart plus spring. write down the lagrange equation of motion. find the frequency of the small oscillations.

4. now look at the green lines you created by connecting the three boiling point data points and the three melting point data points. for each of these lines, describe any trends you see. 5. locate the elements on your periodic table that you circled in green on your graph. what term or description would you use to identify these elements with respect to the periodic table? 7. using the room temperature line (orange line) and your periodic table, make lists that identify the state of matter (gas, liquid, or solid) in which each element you plotted exists at room temperature. explain your answers.

Avehicle accelerates from 0 to 30 m/s in 10 seconds on a straight road

Aflask with vinegar in it has a mass of 160 grams. a balloon with baking soda in it has a mass of 40 grams. the balloon is attached to the flask to seal the opening and the vinegar and baking soda mixes. the balloon inflates to a large volume. what will the total mass of the balloon and flask be after the balloon inflates? explain. a) less than 200 grams because the solid baking soda disappears. b) 200 grams, because all the atoms remain in the balloon or flask. c) more than 200 grams because the size of the balloon is so much larger. d) less than 200 grams because gases such as the one in the balloon are lighter than solids and liquids.