Show that if the potential in the Lagrangian contains velocity-dependent terms, the canonical momentum corresponding to a coordinate of rotation θ of the entire system is no longer the mechanical angular momentum Lθ but is given by pθ = Lθ − i n ? ri 3∇viU, where ∇v is the gradient operator in which the derivatives are with respect to the velocity components and n is a unit vector in the direction of rotation. If the forces are electromagnetic in character, the canonical momentum is therefore pθ = Lθ + i n ? ri 3 qi c Ai .

Askydiver weighing 140 lb (including equipment) falls vertically downward from an altitude of 19,000 ft and opens the parachute after 18 s of free fall. assume that the force of air resistance, which is directed opposite to the velocity, is of magnitude 0.55|v| when the parachute is closed and is of magnitude 14|v| when the parachute is open, where the velocity v is measured in ft/s. assume that acceleration due to gravity has magnitude 32 ft/s/s; remember that weight is the product of mass and gravitational acceleration. (a) find the speed of the skydiver when the parachute opens. (b) find the distance fallen before the parachute opens. (c) what is the limiting velocity vl after the parachute opens? (d) determine how long the sky diver is in the air after the parachute opens. (e) plot the graph of velocity versus time from the beginning of the fall until the skydiver reaches the ground.

4r-134a enters the condenser of a residential heat pump at 800 kpa and 50°c at a rate of 0.022 kg/s and leaves at 750 kpa subcooled by 3°c. the refrigerant enters the compressor at 200 kpa superheated by 4°c determine (a) the isentropic efficiency of the compressor, (b) the rate of heat supplied to the heated room, and (c) the cop of the heat pump. also, determine (d) the cop and rate of heat supplied to the heated room if this heat pump operated on the ideal vapor-compression cycle between the pressure limits of 200 and 800 kpa. (0.757, 4.37 kw, 5.12, 6.18, 3.91 kw)

The flame produced by the burner of a gas (propane) grill is a pale blue color when enough air mixes with the propane (c3h8) to burn it completely. for every gram of propane that flows through the burner, what volume of air is needed to burn it completely? assume that the temperature of the burner is 195.0°c, the pressure is 1.05 atm, and the mole fraction of o2 in air is 0.210.