PHYSICS 218 Honors EXAM 2 Retest Choose 5 of the following 6 problems. Indicate which problem is not to be graded. 1. A rope is affixed at one end to the rim of a pulley, and wrapped five turns around the pulley. The pulley has a mass of 5 kg, a radius of 0.2 m, and its mass is distributed as a uniform disk. A mass of 10 kg is suspended from the loose end of the rope, so that it hangs 2 m from the floor. The mass is then released. What is the velocity of the mass just before it strikes the floor? Solution: The rope will unwind by an angle radmmRL 102.02/ ===θ We require Newton's law for force and for torque: (M=10 kg is mass of weight, m=5 kg is mass of pulley) 22/82/22smmMMgaMamRMgmRIRaITRMaTMg=+==−====−ααα Now solve the equation of motion: smaxatvaxtatx/6.52/2212=====2. A lawn roller in the form of a thin-walled hollow cylinder of mass M is pulled horizontally with a constant force F applied by a handle attached to its axle. If it rolls without slipping, find the acceleration and the friction force. Solution: There will be a friction force Ff opposing the external force F, so as to increase the angular velocity as the linear velocity increases (no slipping). Require Newton’s law for both forces and torques: 2/2//R) (radius surface on the mass all shell, lcylindrica(thin 2FMaFMFaMaMaFMaRIFMrIRaIRFMaFFfff====−=−====−ααα3. Consider a uniform solid disk of radius 25 cm, with a mass of 20 kg. Three 10 cm diameter holes are bored through the disk as shown. What is the moment of inertia of the holey disk, about its axis? Solution: Use the parallel axis theorem: The moment of inertia of the big disk with no holes, about its axis, is 22m kg 625.021== MRIdisk The moment of inertia of each small disk (radius r, distance from center of big disk d, mass m = Mr2/R2) that is removed, about the central axis, is =+=+=033.019.009.032.018.008.001.2122mdmrIhole So overall, ∑==−=31nholendiskIII 0.565 kg m24. Two astronauts must retrieve a satellite that has begun to tumble in orbit. The satellite is in the shape of a long cylinder, whose mass of 1,000 kg is uniformly distributed inside. The satellite is tumbling end over end, with a period of 5 minutes. The astronauts want to stop the tumbling by catching the satellite at its ends. If they each apply constant force over a maximum stroke of 0.3 m, how much force must they apply to stop the satellite? Solution: Call the overall length of the cylindrical satellite L. The moment of inertia of a rod rotating about its center in a tumbling motion is 2121MLI = The torque exerted by the two astronauts is FL)2(2=τ The angular acceleration is then MLFI /12/ ==τα The initial angular velocity is srad /021.min52==πω The angle through which the ship rotates before stopping is FLttt/018.2 when 02120020===−=−=αωθωαωωαωθ We require that the angle θ correspond to a stroke s = 0.3 m. 2203.3.0/009.)2(LFFLLs====θ5. Communications satellites are placed in geosynchronous orbit: the satellite orbit remains over a fixed location on the Earth’s surface. Calculate the radius of a geosynchronous orbit. What is the highest latitude on Earth’s surface that can receive line-of-sight signals from such satellites? The universal constant of gravitation is 6.67 x 10-11 N m2/kg2. The mass of the earth is 6 x 1024 kg. Hint: you don’t need these, only g and the radius of the Earth, 6400 km, and the length of a day! Solution: For geosynchronous orbit, we must have that the orbit period equals one day: srads/103.7)3600(2425−⋅==πω Apply Newton’s law to require that the acceleration from gravity equals that needed to keep it in orbit: ()()( )()mrmsmsmgRrrmrvmFrRmRGMrGMmFEEgE832425262223222222109.1105.7/103.7104.6/8.9//⋅=⋅=⋅⋅⋅======−ωω43421 The highest latitude that can receive line-of-sight signals is an angle β from the pole, where °==⋅⋅==2034.109.1104.6sin86radmmrREβ So the highest latitude is 90o-2o = 88o.6. A uniform pulley wheel 8 cm diameter, mass 1 kg, has a 5 m long cord wrapped around its periphery. Starting from rest, the wheel is given an angular acceleration of 1.47 rad/s2. A) through what angle must the wheel turn for the cord to unwind? B) how long does it take? C) what is the final angular momentum of the wheel? Solution: A) radmmRL5.6208.5/ ===θ B) stt2.9/2212===αθαθ C)
View Full Document
Unlocking...