HW 6 - Solutions Due November 20, 2017

Transcription

1 Conteporary Physics I HW 6 HW 6 - Solutions Due Noveber 20, A 4 kg block is attached to a spring with a spring constant k 200N/, and is stretched an aount 0.2 [5 pts each]. (a) Sketch the potential energy curve for the spring. Make sure that the xaxis goes fro at least -0.2 to 0.2. (b) How uch potential energy is stored in the spring when it is stretched? U 1 2 kx2 1 2 (200N/)(0.2)2 U 4J (c) The block is then released fro rest. Where (what value of x) will the block have its axiu kinetic energy? How fast will it go at that oent? The block will have axiu kinetic energy right before it is pulled back in the other direction, at 0.2.

2 U i + K i + W U f + K f 1 2 kx2 1 2 v2 k v x v 1.41/s 200N/ (0.2) 4kg 2. A very coplicated olecule has an energy diagra as shown below: where the x-axis represents the separation between the two atos. Your answers to the following will be approxiate, but in each case, I ve tried to ake it so that correct answer(s) will be integers or half-integers. [3 pts each] (a) What separations are the equilibria of the syste? Are they stable or unstable? Equilibria points occur at extrea of the syste, axius (unstable) and inius (stable). Fro the iage it s clear at which values of r we have extrea. r 1Å(stable) r 3Å(unstable) r 4Å(stable)

3 (b) Suppose the atos are separated by 2Å and are at rest, describe what will happen to the olecule. The separation will oscillate fro 2Å to 0.5 Å, eventually settling to 1 Å. (c) Under those circustances, what are the range of otion of the atos? By investigating the diagra we can see that at 2.5eV, it can ove fro 0.5 Å to 2.5 Å (d) How uch energy is required to disassociate the olecule (provided it started at rest separated by 2Å)? Dissociating a olecule eans that r, looking at the graph is shows that as r increases then U x 0, Since we are at 2.5eV we would then need 2.5eV to reach 0 (e) Estiating fro the curve at r 2Å, what is the approxiate force on an ato? (Hint: 1eV/Å N). F du dx ( 2.5eV ( 5)eV ) 2Å 1Å ( 2.5eV/Å ) ( N/(eV/Å)) F N 3. I going to ake soe not entirely accurate stateents about air. In order to avoid confusion, I ll indicate true stateents with a T. The other stateents should be treated as true for the purposes of this proble. (T) Air is ade up predoinantly of olecular nitrogen, which has a olecular ass of approxiately 28. (28 the ass of a proton). (T) At sea level, typical air density is approxiately 1.3kg/ 3. (T) Roo teperature is about 300K. (T) An ideal onatoic gas has an energy of: E 3 2 k bt per ato and an equation of state (pressure relation) P V Nk b T where N is the nuber of atos and V is the volue of the gas. You ay treat air as being ade entirely of olecular nitrogen, and you should treat the olecular nitrogen as a onatoic gas. [3 pts each] (a) What is the nuber density of air olecules? n ρ 1.3kg/ 3 28 ( kg) n

4 (b) What is the typical kinetic energy of an air olecule? E K 3 2 k BT 3 2 ( J/K)(300K) E K J (c) How fast is a typical air olecule oving? K 1 2 v2 2K v 2 ( J) 28 ( kg) v /s (d) Incidentally, it can be shown that the sound speed of a onatoic gas is: kb T c s where is the ass of an ato (olecule). What is the sound speed of air, and how does that copare to the result fro the previous part? kb T c s ( J/K)(300K) 28 ( kg) c s /s

5 (e) What is air pressure at roo teperature? P V Nk B T P N V k BT nk B T ( )( J/K)(300K) P N/ P.62) Calculate the speed of a satellite in a circular orbit near the Earth (just above the atosphere). If the ass of the satellite is 200kg, what is the iniu energy required to ove the satellite fro this near-earth orbit to very far away fro the Earth? [15 pts] Let s assue the distance of the satellite fro the centre of Earth is approxiately the sae as the radius of Earth. We can balance the graviational force with the centripetal force. GM E RE 2 v2 R E GME v R E ( /kg/s 2 )( kg) v /s W E W (K f K i ) + (U f U i ) 1 2 ( GME R E GM E 2R E ) + GM E R E ( /kg/s 2 )( kg)(200kg) 2( ) W J

6 5. 6.P.69 (a,d only): A pendulu consists of a very light but stiff rod of length L hanging fro a nearly frictionless axle, with a ass at the end of the rod. (a) Calculate the gravitational potential energy as a function of the angle θ, easured fro the vertical. [10 pts] cos θ L h L L cos θ L h h L(1 cos θ) U gh U gl(1 cos θ) (d) Suppose that you hit the stationary hanging ass so it has an initial speed v i. What is the iniu initial speed needed for the pendulu to go over the top (theta 180 )? [5 pts] Fro conservation of energy... K U 1 2 v2 gl(1 cos θ) v 2gL(1 cos θ) 2gL(1 cos(180 )) 4gL v 2 gl

7 6. 7.P.23) A relaxed spring of length 0.15 stands vertically on the floor; its stiffness is 1000 N/. You release a block of ass 0.4 kg fro rest, with the botto of the block 0.8 above the floor and straight above the spring. How long is the spring when the block coes oentarily to rest on the copressed spring? [15 pts] We can reason that any height the brick will fall, both before it hits the spring (D) and after it is contact with the spring (s, which also is the aount the spring has been copressed), will contribute to it s potential energy fro gravity. All of this energy will turn into spring potential energy. D H L i U gd + gs 1 2 k(s)2 0 1/2ks 2 gs gd s g ± 2 g kgD 2.5 k (.4kg) (9.8/s) ± s (.4kg 2 ) (9.8/s 2 ) (1000N/)(.4kg)(9.8/s 2 )(.65) 1000N/ s.754 Since this is how uch the string has copressed, L f P.36) During three hours one winter afternoon, when the outside teperature was 0 C (32 F), a house heated by electricity was kept at 20 C (68 F) with the expenditure of 45 kwh (killowatthours) of electric energy. What was the average energy leakage in joules per second through the walls of the house to the environent (the outside air and ground)? The rate at which energy is transferred between two systes is often proportional to their teperature difference. Assuing this to hold in this case, if the house teperature had been kept at 25 C (77 F), how any kwh of electricity would have been consued? [10 pts] Energy leakage 45kW h 3h 15kW J/s E 2 ( T2 T 1 ) E 1 ( 25 ) C 20 45kW h C E kW h