Physics 218 (Mechanics)Exam 3November 18, 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] This is a one-dimensional problem. There are two masses m1and m2on a frictionlesssurface without any external force. The distance between m1and the center of mass (CM)of the system is r1and r2is the distance between m2and the CM of the system.Figure for (a) and (b)(a) [6 points] Write the ratio of r1/r2in terms of m1and m2.(b) [8 points] Suppose that m1= 1 kg, m2= 3 kg, and the distance between the two is12 m. The internal force between the two is repulsive and the two masses move awayfrom the center of mass of the system. If the displacement of m1from its originalposition is 3 m, what is the distance between m1and m2?(c) [6 points] This is a two-dimensional problem. Obtain the center of mass position ofthe system in the figure below, where m1= 3 kg, m2= 2 kg, and m3= 1 kg. Thepositions of the bodies are (x1= 0, y1= 1) for m1, (x2= 3, y2= 0) for m2, and(x3= 2, y3= 3) for m3. (Do NOT use decimal numbers for your answer.)Figure for (c)1[2] A spool of thread falls and unwinds under the gravitational force. The spool is a thin-walledhollow cylinder of mass M and radius R. The moment of inertia of the spool in this motionis I = MR2. Neglect the mass of the thread. The gravitational acceleration is g.(a) [10 points] Find the downward acceleration acmof the spool.(b) [10 points] Use energy consideration to find the speed vcmof the spool after it hasunwound a length h of the thread.2[3] A ladder of mass mLand length L leans against a frictionless wall. There are two firefighterson the ladder. Firefighter A of mass mAstops a third of the way up the ladder (point A inthe figure) and firefighter B of mass mBstops 2/3 of the way up the ladder (point B inthe figure). The ladder makes an angle θ with the horizontal and the coefficient of frictionbetween the ladder and the floor is µ. The gravitational acceleration is g and the ladder is inequilibrium.(a) [6 points] Find the normal force on the ladder at its base.(b) [7 points] Find the friction force on the ladder at its base.(c) [7 points] If mA= mB= 3mL, what is the coefficient of static friction µ?3[4] There is a slender rod of mass M and length L on a frictionless surface. The mass densityof the rod is uniform. When the axis of rotation passes its center of mass, the moment ofinertia of the rod is I =112ML2. But one end of the rod is hinged so that the rod can rotateabout a vertical axis as shown in the figure. A ball of mass m with velocity v hits the exactcenter of the rod. After the collision, the ball sticks to the rod.(a) [10 points] Find the moment of inertia of the ball+rod system after the collision withrespect to the vertical axis along the hinge.(b) [10 points] Find the angular speed of the ball+rod system just after the collision.4[5] (a) [6 points] A block of mass m = 4.0 kg, moving on a frictionless surface with a speedvi= 3.0 m/s, makes a perfectly elastic collision with a block of mass M at rest. Afterthe collision, the block of mass m recoils with a speed of vf= 1.0 m/s. What is thespeed of the block of mass M after the collision?(b) [7 points] A block of mass m = 2 kg has a speed V and is behind a block of massM = 50 kg that has a speed of 0.5 m/s. The surface is frictionless. The blocks collideand couple. After the collision, the blocks have a common speed of 1.0 m/s. What isthe magnitude of the impulse on the block of mass m due to the collision?(c) [7 points] The masses of the blocks and their velocities before and after collisionare given below. Determine whether this collision is elastic, partially inelastic, com-pletely inelastic, or not possible. (Merely writing down the answer will get NO credit.Explain or prove your answer mathematically.)(Use the back side of this paper for this problem.)5Useful information• Center of mass~rcm=Pimi~riPimi.• Parallel axis theorem for moment of inertiaIP= Icm+ Md2,where M is the mass of the rigid body and d is the distance between the two axes of rotation.• Work-Energy theoremW = ∆K = K2− K1• Energy ConservationE1+ Wother= E2,where Ei= Ki+ Uiand Wotheris the work done by nonconservative forces.• Torque and Angular momentum~τ = ~r ×~F → τ = rF sin φXτ = Iα.~L = ~r × ~p → L = rp sin φL = Iω.• Trigonometric Functionssin(90◦+θ) = cos θ, cos(90◦+θ) = −sin θ, sin(90◦−θ) = cos θ, cos(90◦−θ) = sin θ.sin(180◦+θ) = −sin θ, cos(180◦+θ) = −cos θ, sin(180◦−θ) = sin θ, cos(180◦−θ) = −cos θ.sin 30◦= 1/2, cos 30◦=√3/2, sin 60◦=√3/2, cos 60◦= 1/2.tan 30◦= 1/√3, tan 60◦=√3.• Impulse and momentum~J = ~p2− ~p1=Zt2t1X~F dt.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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