Question

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1. An electron decays from its first excited state to its ground state in a 1-dimensional infinite square well of width 5.0 nm, emitting a photon. Calculate the wavelength of the photon. 2. A water turbine with diameter 40 cm is driven by a flow with constant speed. The turbine is fully immersed in the flow, and the generator driven by the turbine con- verts the energy of the flow to electrical energy with 50% efficiency. The electrical power output of the generator is 2 kW, and the density of the water is 1000 kg m-3. Estimate the speed of the flow. 3. Natural uranium is 3% 235U and 97% 238 U by mass. Only 235U is fissile, and the average energy released from one fission of a 235U atom is 200 MeV. Calculate the period of time during in which 1 kg of uranium can provide the power to fully illuminate a 100 W lightbulb, assuming that the total energy is stored and later released at a steady rate. 4. A large sail made of a material which perfectly reflects radiation is used to propel a small spacecraft through the planetary system of a star. The total mass of the spacecraft and sail is 25 kg. The sail is a square whose sides have length 125 m. The total radiation power output of the star is 4.5 x 1026 W. The spacecraft is initially at rest and located at a distance of 2.5 x 10 km from the star. Estimate the minimum time required for the spacecraft to travel 50 m directly away from the star. You may neglect the gravitational pull of the star.

Answer 1

2ma² so 2 mar 2 ma? (1) Energy of noth excited state nh Where a= width of the box m = mass of the election Ist excited state energy is at no 2 E, 2²7t² 47²h2 ground stale energy is at n=1 Prot 2maz ho hc этъ ama² hcxama² 6.626x 10 x 23 x 108 2X 9-11x10 Ysxiongo 30 h 3x Tx (1-054X 70-34) = 2.75 38 16 m = 27.53 um then E, E. m 2. Electrical power output at generator = 2 kw = 2000 J/s Since efficiency is 50% then work done by the turbine per unit time is = 4000 J/s. let the speed of s' the flow is to m/s. diameter of the turbine =D=40cm = 0.4m area of the turbine - A+ Ti (P)² = 7 (0.23 m² work done by the water flow per Second is (4 9. vA) = (1000x + (0.2) * 6 x 9.8) 5/5 So, wi= 4000 1000x 9.8 XT1 (0.2) * 10 = 4000 9.8 x 71 (0-2) = 3.25 m/s 07

(3) E 200 mer energy released in one fussion of 0235 the light bulb total no 235 power = 100 Watt - 100 g/s. =P in one kg of uranium is 6-023x 16²3 NX 3 х 3 100 100 = 235 total energo in these 4235 E = nE atom produce is x 200x106) ( 3x6-023x10 23 en 23500 23 3x6-023x10 =1 23500 200x10 х 6-242x1018) J total time the ball can be use is E 3x6-023 x 10² 200X10 Р 23 500 X 100 6. 242 x 10 8 23 Х S ( 41) 4.5x1026 W/m² 26 15625X4.5 X10 w 6843.37 hour = 285- 14 days mass of space craft = m = 25 kg area of sail = A= (125x125) = 15625 m2 radiation power at the position of the space coaft is P = Р P A, 47R? 41 (2-5xv power vereine receved by the sail = Pi A- 40 (2.5x103)? radiation pressure on reflecting surface is P = 15 = 2I where I =P and c= 3x108 m/s Force on spacecoceft = F = 21A 2P, A 2x15625x 4.5 x 10 C acceleration - 2x15625x4,5 x 1626 4118.5x187) * 3x 103 un (2.5x167) K3 x 168 x 25 S = ut + £at, but here u=0,s=50m so, then, t= (25) 2 2X 50x 47 (2.5x157) * 31109225172 2x15625x4.5x1026 26 F m= now we know วน ขนาด 42514 0.02 S