Exercise:1. A particle moving along x axis starts from x 0 with initialvelocity v 0. Its acceleration can be expressed in a =-kv 2where k isa known constant. Find its velocity function v =v (x ) with the coordinate x as variable.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 horizontal path, assume that the drag of the sand is kv f -=, find the velocity function 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+= that movesa particle in xy plane from the initial position j i r i 32+= to the final position j i r f34--=. All quantities are in SI.5. The angular position of a point on the rim of a rotating wheel is given by 320.30.4t t t +-=θ, where θ is in radians and t is in seconds. Find (a) its angular velocities at t=0s and t = (b) Calculate its angular acceleration at t =. (c) Is its angular acceleration constant6. A uniform thin rod of mass M and length L can rotate freely about a horizontal axis passing through its top end o (231ML I =). A bulletof mass m penetrates the rod passing its center of mass when the rod is in vertical stationary. If the path of the bullet is horizontal 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 mounted on the end of a rod (L=0.60m). The rotational inertia of the rod alone about A is 206.0m kg ⋅. The block-rod-bullet system then rotates about a fixed axis at point A. Assume the block is small enough to treat as a particle on the end of the rod. Question: (a) What is the rotational inertia of the block-rod-bullet system about A (b) If the angular speed of the system about A just after the bullet ’s impact is s , What is the speed of the bullet just before the impactzero as it passes the origin. (a) Calculate the Lorentz factor γ between the rest frame S and the frame S* in which the clock is rest. (b) what time does the clock read as it passes x =180m9. What must be the momentum of a particle with mass m so that itstotal energy is 3 times rest energychamber undergoes the cycle shown in the Fig. Calculate Q net the net energy added to the gas as heat during one complete cycle.11. One mole of a monatomic ideal gas undergoes the cycle shown 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 stateB (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 motion of 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) plotthe 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 first contains gas with pressure 1p , molecular mass 1m , and rms speed 1rms v . The second contains gas with pressure 12p , molecular mass 2m , and average speed 122rms avg v v =. Find the mass ratio 21m m .14. In a quasi-static process of the ideal gas, dW =PdV and d E int =nC v dT . From the 1st law of thermodynamics show that the change of entropy i f v i fT T nC V V nR S ln ln +=∆ .Where n is the number of moles,C v is the molar specific heat of the gas at constant volume, R is the ideal gas constant, (V i , T i ) and (V f , T f ) . are the initial and final volumes and temperatures respectively.15. It is found experimentally that the electric field in a certain region of Earth ’s atmosphere is directed vertically down. At an altitude of 300m the field is N /C ; at an altitude of 200m , the field is 100N /C . Find the net charge contained in a cube 100m on edge, 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 outside the conductor. (b) If the space is filled with a uniform dielectrics of known r ε what is U * stored in the field outside the conductorthen17. Charge is distributed uniformly throughout the volume of an infinitely long cylinder of radius R. (a) show that, at a distance r from 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 of charge has a volume density given as follow:⎩⎨⎧-=0)/1()(0R r r ρρ0ρ is a positive constant, r is the distance to the symmetric center O and R is the radius of the charge distribution. Within the charge distribution (r < R ), show that (a) the charge contained in the co-center sphere of radius r is )34(31)(430r Rr r q -=πρ, (b) Find the magnitude of electric field 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 following function of x,y and z: xy x V 22+=, where the potential is measured in 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 metal shell of inner radius R 1 and outer radius R 2. A point charge q islocated 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) at the center O assume V (∞)=0.21. Two large metal plates of equal areaare parallel and closed to each other with charges Q A , Q B respectively. Ignore the fringing effects, find (a) the surface charge density on each side of both plates,(b) the electric field at p 1, p 2 . (c) the electric potentialA and B)。