Exercise:1. A particle moving along x axis starts from x 0 with initial velocityv 0. Its acceleration can be expressed in a =-kv 2 where k is a knownconstant. Find its velocity function v =v (x ) with the coordinate x asvariable.2. A particle moves in xy plane with the motion function asj t i t t r )3sin 5()3cos 5()(+=(all in SI). Find (a) its velocity )(t v and (b)acceleration )(t a in the unit-vector notation. (c) Show that v r⊥.3. A bullet of mass m is shot into a sand hill along a horizontalpath, assume that the drag of the sand is kv f -=, find the velocityfunction v(t) if 0)0(v v = and the gravitation of the bullet can beignored.4. what work is done by a conservative force j i x f 32+= thatmoves a particle in xy plane from the initial position j i r i 32+= to the final position j i r f 34--=. All quantities are in SI.5. The angular position of a point on the rim of a rotating wheel isgiven by 320.30.4t t t +-=θ, where θ is in radians and t is inseconds. Find (a) its angular velocities at t=0s and t =4.0s? (b)Calculate its angular acceleration at t =2.0s. (c) Is its angularacceleration constant?6. A uniform thin rod of mass M and length L can rotate freelyabout a horizontal axis passing through its top end o (231ML I =). Abullet of mass m penetrates the rod passing its center of masswhen the rod is in vertical stationary. If the path of the bullet ishorizontal with an initial speed v o before penetration and 20v after penetration . Show that (a) the angular velocity of the rod just after the penetration is MLmv 430=ω. (b) Find the maximum angular max θ the rod will swing upward after penetration.7. A 1.0g bullet is fired into a block (M=0.50kg) that is mountedon the end of a rod (L=0.60m). The rotational inertia of the rodalone about A is 206.0m kg ⋅. The block-rod-bullet system thenrotates about a fixed axis at point A. Assume the block is smallenough to treat as a particle on the end of the rod. Question: (a)What is the rotational inertia of the block-rod-bullet system aboutA? (b) If the angular speed of the system about A just after thebullet ’s impact is 4.5rad/s , What is the speed of the bullet justbefore the impact?8. A clock moves along the x axis at a speed of 0.800c and readszero as it passes the origin. (a) Calculate the Lorentz factor γbetween the rest frame S and the frame S* in which the clock isrest. (b) what time does the clock read as it passes x =180m ?9. What must be the momentum of a particle with mass m sothat its total energy is 3 times rest energy?10. Ideal gas within a closed chamber undergoes the cycle shownthe Fig. Calculate Q net the net energy added to the gas as heatduring one complete cycle.11. One mole of a monatomic ideal gas undergoes the cycleshown in the Fig. temperature at state A is 300K.(a). calculate the temperature of state B and C.(b). what is the change in internal energy of the gas between stateA and state B? (int E )(c). the work done by the gas of the whole cycle .(d). the net heat added to the gas during one complete cycle.12. The motion of the electrons in metals is similar to the motionof molecules in the ideal gases. Its distribution function of speedis not Maxwell ’s curve but given by.⎩⎨⎧=0)(2Av v pthe possible maximum speed v F is called Fermi speed. (a)plot the distribution curve qualitatively. (b) Express the coefficientA in terms of v F . (c) Find its average speed v avg .13. Two containers are at the same temperature. The firstcontains gas with pressure 1p , molecular mass 1m , and rmsspeed 1rms v . The second contains gas with pressure 12p , molecularmass 2m , and average speed 122rm s avg v v =. Find the mass ratio21m m .14. In a quasi-static process of the ideal gas, dW =PdV andd E int =nC v dT . From the 1st law of thermodynamics show that thechange of entropy i f v i fT T nC V V nR S ln ln +=∆ .Where n is the numberof moles, C v is the molar specific heat of the gas at constantvolume, R is the ideal gas constant, (V i , T i ) and (V f , T f ) . are theinitial and final volumes and temperatures respectively.15. It is found experimentally that the electric field in a certainregion of Earth ’s atmosphere is directed vertically down. At analtitude of 300m the field is 60.0 N /C ; at an altitude of 200m , thefield is 100N /C . Find the net charge contained in a cube 100m onedge, with horizontal faces at altitudes of 200m and 300m .Neglect the curvature of Earth.16. An isolated sphere conductor of radius R with charge Q . (a)Find the energy U stored in the electric field in the vacuum outsidethe conductor. (b) If the space is filled with a uniform dielectrics ofknown r ε what is U * stored in the field outside the conductorthen?17. Charge is distributed uniformly throughout the volume of aninfinitely long cylinder of radius R. (a) show that, at a distance rfrom the cylinder axis (r<R), r E 02ερ=, where ρis the volume charge density. (b) write the expression for E when r>R .18. A non-uniform but spherically symmetric distribution ofcharge has a volume density given as follow:⎩⎨⎧-=0)/1()(0R r r ρρwhere 0ρ is a positive constant, r is the distance to the symmetric center O and R is theradius of the charge distribution. Within the charge distribution (r <R ), show that (a) the charge contained in the co-center sphere ofradius r is )34(31)(430r Rr r q -=πρ, (b) Find the magnitude of electricfield E (r ) within the charge (r < R ). (c) Find the maximum field E max =E (r *) and the value of r *.19. In some region of space, the electric potential is the followingfunction of x,y and z: xy x V 22+=, where the potential is measuredin volts and the distance in meter . Find the electric field at thepoint x=2m, y=2m . (express your answer in vector form)20. The Fig. shows a cross section of an isolated spherical metalshell of inner radius R 1 and outer radius R 2. A point charge q is located at a distance 21R from the center of the shell. If the shell is electrically neutral, (a) what are the induced charges (Q in , Q out )on both surfaces of the shell? (b) Find the electric potential V(0) atthe center O assume V (∞)=0.21. Two large metal plates of equal areaare parallel and closedto each other with charges Q A , Q B respectively. Ignore the fringingeffects, find (a) the surface charge density on each side of bothplates, (b) the electric field atp 1, p 2 . (c) the electric potentialA and B)22．In a certain region of space, the electric potential is ()2=-+where A,B,C are positive constant. The ,,,V x y z Axy Bx Cyelectric field is ; at which point is the electric field equal to zero .23. A 9.60-μC point charge is at the center of a cube with sides of length 0.500m. The electric flux through one of the six faces of the cube is ; the answer would be if the sides were of length 0.250m.。