Asimple pendulum of mass m and length l hangs from a trolley of mass m running on smooth horizontal rails. the pendulum swings in a plane parallel to the rails. using the position x of the trolley and the angle of inclination θ of the pendulum as generalized co-ordinates, find the hamiltonian and show that x is ignorable.

Abag of potato chips contains 2.00 l of air when it is sealed at sea level at a pressure of 1.00 atm and a temperature of 20.0 deg c. what will be the volume of the air in the bag if you take it with you, still sealed, to the mountains where the temperature is 7.00 deg c and atmospheric pressure is 70.0 kpa? assume that the bag behaves like a balloon and that the air in the bag is in thermal equilibrium with the outside air.