A) Consider a driven oscillator with dissipation in its steady state, similar to that discussed on page 314 of the textbook. But assume the motor is connected directly to the mass, as sketched above. The maximum force exerted by the motor is F0, which replaces ksD in the textbook. Using the expression for x(t) in the textbook, with D replaced by F0/ks, calculate the sum of the kinetic plus spring potential energy, K+U=(1/2)mv2+(1/2)ksx2. Explain why K+U is not generally constant, unlike the case of an ideal non-driven oscillator with no dissipation. Where does the energy come from and go to? b) At what times during one period is this energy a maximum? Express your answer in tems of ωD and ϕ. What are the values of x and v at those times? HINT: Your answer will depend on whether the driving frequency is greater or less than the resonant frequency (labelled by subscripts D and F in the textbook).

c) Now consider an oscillator with dissipation but no driving force, so that the total force is -ksx(t)-cv(t) where v is the velocity. Derive a formula for the rate of change of (K+U), d(K+U)/dt as a function of t. Express your answer in terms of v(t) and c.

Acoal-fired plant generates 600 mw of electric power. the plant uses 4.8 x 10^6 kg of coal each day, and the heat of combustion of coal is 3.3x 10^7 j/kg. the steam that drives the turbines is at a temperature of 300° c, and the exhaust water is at 37° c. (a) what is the overall efficiency of the plant for generating electric power? (b) how much thermal energy is exhausted each day? (c) what is the change in entropy of the universe for this heat engine? (d) using the same heat reservoirs, what is the maximum possible (carnot) efficiency for a heat engine?

Chapter 23, problem 075 the figure shows a geiger counter, a device used to detect ionizing radiation (radiation that causes ionization of atoms). the counter consists of a thin, positively charged central wire surrounded by a concentric, circular, conducting cylindrical shell with an equal negative charge. thus, a strong radial electric field is set up inside the shell. the shell contains a low-pressure inert gas. a particle of radiation entering the device through the shell wall ionizes a few of the gas atoms. the resulting free electrons (e) are drawn to the positive wire. however, the electric field is so intense that, between collisions with gas atoms, the free electrons gain energy sufficient to ionize these atoms also. more free electrons are thereby created, and the process is repeated until the electrons reach the wire. the resulting "avalanche" of electrons is collected by the wire, generating a signal that is used to record the passage of the original particle of radiation. suppose the radius of the central wire is 24 âµm, the inner radius of the shell 2.3 cm, and the length of the shell 14 cm. if the electric field at the shell's inner wall is 2.8 ă— 104 n/c, what is the total positive charge on the central wire?

Consider a system to be two train cars traveling toward each other. what is the total momentum of the system before the train cars collide?   kg • what must the total momentum of the system be after the train cars collide?   kg •