Physics 218 (Mechanics)Exam 2October 21, 2008Please print neatly in ink.Family Name:First Name:Student ID Number:Your Section Number:Signature:IMPORTANT• You have 60 minutes to complete 5 problems totaling 100 points. Make sure you have allthe pages. Work each problem on the page the problem is on. You may use the back of thesheet it is written on, if needed, but indicate it in the front side.• Formulas are provided on a separate sheet at the end of this exam paper. You may use acalculator.• Mark your answers clearly by drawing boxes around them.• Write the procedure to solve the problem. You cannot have full credit for merely writingdown the answer. Generous partial credits will be given if you show that you are on the righttrack. Therefore, write neatly and in a well-organized way.Do not write below.STOP• You may use a pencil from this page.[1] Two masses m1and m2are connected by a massless nonstretching string. The mass m1moves along the inclined surface which makes an angle θ from the horizontal line and m2moves vertically. (See the figure.) The gravitational acceleration constant is g. Assume thatthere is no friction force for questions (a) and (b). Ignore the mass of the pulley as well.(a) [ 8 points]Obtain the acceleration of the masses. (You should define your coordinate system.)(b) [6 points] If the system is moving with a constant speed, what is the ratio m2/m1?(This does not depend on the direction of the motion.)(c) [6 points] Assume that the inclined surface is rough and the objects are not moving. Ifm2= 3m1, obtain the coefficient of static friction µ in terms of m1and θ.1[2] An object with mass m = 40 kg is attached to the ceiling by a rope. Use the gravitationalacceleration constant g = 10 m/s2.(a) [6 points] If the rope is massless, what is the tension of the rope at the top of the rope?[See Fig. (a)](b) [7 points] Consider the case when the rope is not massless. If the rope exerts the tensionforce T = 500 N on the ceiling, what is the mass of the rope? [See Fig. (a)](c) [7 points] Assume that the rope is massless. There is a constant force F exerting onthe object as shown in Fig. (b). If θ = 30◦, what is the magnitude of F ? (Use thatsin 30◦= 1/2, cos 30◦=√3/2. sin 60◦=√3/2, cos 60◦= 1/2. Do NOT use adecimal number for your answer.)2[3] A massless rod with length R is loosely pinned to a frictionless pivot at O. A ball withmass m is attached to the other end of the rod. The ball is held at A, where the rod makesa 30◦angle above the horizontal, and is released. The ball-rod assembly then swings freelyin a vertical circle between A and D. In the figure, the ball passes through C, where therod makes an angle of 30◦below the horizontal. The gravitational acceleration constant isg. (Use that sin 30◦= 1/2, cos 30◦=√3/2. sin 60◦=√3/2, cos 60◦= 1/2. Do NOT usedecimal numbers for your answers.)(a) [6 points] What is the speed of the ball as it passes through C?(b) [7 points] What is the tension of the rod as it passes through B?(c) [7 points] What is the work done by the gravity on the ball from B to D?3[4] This is a two-dimensional problem. A particle is moving with a potential energy,U(r) = αr,where α is a constant and r is the distance from the origin, r =√x2+ y2.(You may use the formulas given in the formula sheet.)(a) [7 points] Obtain the x and y components of the force, Fxand Fy, arising from thispotential energy.(b) [7 points] The particle is moving from r = 1 m to r = 5 m. If α = 10 J/m, what is∆K, the change in its kinetic energy? Ignore all the other forces such as gravity.(c) [6 points] At what value of x is the point of stable equilibrium located? Describe themotion of the particle around the stable equilibrium point. If the particle has a verylarge but finite energy, can this particle escape from this potential?4[5] A box with mass m is compressing a spring by a distance a. The spring constant is k.Assume that the spring is ideal and you can ignore its mass. At x = 0 the spring is notstretched or compressed. After it is released, the box will pass x = 0 point and then movesa distance of d on a horizontal surface and then climbs up a frictionless surface as shown inthe figure. The horizontal surface is frictionless except the region between x = 0 and x = d.The gravitational acceleration constant is g.(a) [6 points] There is no friction between x = −a and x = 0. What is the speed of thebox when it passes x = 0?(b) [7 points] Between x = 0 and x = d, the surface is rough and µ is the coefficient offriction. How fast will the box at x = d?(c) [7 points] There is no friction between x = d and x = L. How fast will it be going atx = L where the height of the hill is H? Assume that k is large enough to allow thismotion.5Useful information• Integration & Differentiationddxxn= nxn−1,Zdxxn=1n + 1xn+1+ Constant,Zx2x1dxxn=1n + 1xn+12− xn+11,for an integer n (6= −1).Partial derivative∂∂xr =∂∂xqx2+ y2=x√x2+ y2=xr,∂∂yr =yr,where r =√x2+ y2.• Work-Energy theoremW = ∆K = K2− K1• Energy ConservationE1+ Wother= E2,where Ei= Ki+ Uiand Wotheris the work done by nonconservative forces.• Force and Potential Energy~F = −~∇U,which meansFx= −∂∂xU, Fy= −∂∂yU.• Trigonometric Functionssin(90◦+ θ) = cos θ, cos(90◦+ θ) = −sin θ,sin(90◦− θ) = cos θ, cos(90◦− θ) = sin θ.sin 30◦= 1/2, cos 30◦=√3/2, sin 60◦=√3/2, cos 60◦= 1/2.tan 30◦= 1/√3, tan 60◦=√3.You may remove this page. If you do remove this page, do not turn it in!6This is a scratch paper. Remove this page and do not turn it
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