# Take-Home Midterm Exam #2, Part A

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2 F T1 F T2 F T3 M 3. (5 pts.) The mass M = 125 kg hangs at rest as shown above, suspended by three massless cables. The cables are joined at the center, and cable #3 hangs straight down. a. What is the magnitude of tension F T1? b. What is the magnitude of tension F T2? c. What is the magnitude of tension F T3? F T1 = 375 N 12.0 m F T3 = 430. N F T2 = 380. N 4. (6 pts.) A heavy rail car (m = 1500 kg) rolls on the ground on rails, and is constrained so that it can only move to the right or left, as shown in the top-view above. Three people supply constant forces by pulling on cables attached to the car s center. (Assume that the three forces do NOT change magnitude or direction as the car moves.) The car starts from rest, and there is no friction. When the car has moved 12.0 m to the right, what is a. the work done on the car by F T1? b. the work done on the car by F T2? c. the work done on the car by F T3? d. the final speed of the car?

3 5. Mid-2005 saw the announcement of the discovery of a so-called 10 th planet, now officially named Eris (but at the time nicknamed Xena ). Eris was measured to Sun be slightly larger in physical size than Pluto, so this led to a debate in 2006 about what should qualify as a planet. (As you probably know, it was decided to demote Pluto from its planet status, and now Pluto, Eris, and other small bodies are called dwarf planets. ) This diagram schematically shows the orbits of Eris, Pluto, and some of the planets (Jupiter through Neptune) about the Sun. (Throughout this problem, when considering a particular planet s orbit, you may consider only the gravitational effect of the Sun, and ignore the gravitational effect of other planets on it.) Pluto s orbit Eris s orbit a. (1 pt.) What is the name of the mathematical curve that describes the shape of Eris s orbit? b. (1 pt.) Label the points on Eris s orbit (above) where its linear speed along its orbit is at its fastest and its slowest. c. (1 pt.) Based on the diagram, which of these two bodies has the greater average linear speed along its orbit? A. Pluto B. Eris d. (1 pt.) Based on the diagram, which of these two bodies has the longer orbital period? A. Pluto B. Eris e. (2 pts.) Eris has a physical radius of 1200 km and a mass of kg. If you were to visit the surface of Eris, how many gees of surface gravity would you experience? gees (Recall: 1 gee = 9.80 m/s 2.) We can only use the equation U gr = mgy to calculate gravitational potential energy of a small body m that is in a region of uniform gravitational acceleration g (such as near the surface of the Earth). However, for astronomical situations like this one, the value of g is NOT uniform throughout space. So, to find U gr for a body like Eris relative to the Sun, we must use the more general equation instead: U gr = " Gm 1m 2 r where m 1 = M Sun, m 2 = m Eris, and r is the distance between their centers. (Note the minus sign! Also, U gr = 0 when r =.) f. (2 pts.) At Eris s farthest distance from the Sun, r = 97.6 AU, it has a linear speed of 2.57 km/s. (1 AU = average Earth Sun distance = km.) Eris experiences no friction in orbit. Use conservation of energy to determine Eris s linear speed (in km/s) when it is at its closest approach to the Sun, 37.8 AU: km/s Note: Does your answer here agree with your answer to part (b) above?

4 Physics 151 Spring 2007 Take-Home Midterm Exam #2, Part B Roster No.: Score: m v 0 α µ k h 1. A mass m is launched with an initial speed v 0 up an incline with a coefficient of kinetic friction µ k. a. (3 pts.) Find an algebraic expression for the maximum height h reached by the mass when there is NO friction (µ k = 0). SIMPLIFY your final answer algebraically, and express it ONLY in terms of m, v 0, g, α, and any numerical constants. You may use whatever method you wish, but you must show your work. b. (6 pts.) Now, find a new algebraic expression for the maximum height h reached by the mass when there IS kinetic friction present. SIMPLIFY your final answer algebraically, and express it ONLY in terms of µ k, m, v 0, g, α, and any numerical constants. You may use whatever method you wish, but you must show your work.

5 1. continued: c. (1 pt.) Show that, as µ k 0, your answer to part (b) simplifies to the same expression as your answer to part (a). m v f α µ k h When the mass reaches its maximum height, it pauses momentarily, and then it slides back down the incline. When the mass reaches its original position, it has a final speed v f, as shown above. Kinetic friction µ k acts on the mass during both ascent and descent. (Assume that the coefficient of static friction is sufficiently small that the mass does not remain stationary when it pauses at maximum height.) d. (1 pt.) The mass s final speed v f is its initial speed v 0. A. greater than B. equal to C. less than D. cannot determine with information given e. (1 pt.) The mass s time of ascent is its time of descent. A. greater than B. equal to C. less than D. cannot determine with information given

6 2. A small satellite (m = 100. kg) orbits the Earth at a distance greater than our Moon. Assume that both the Moon and the satellite have exactly circular orbits. When all three bodies are in a straight line, as shown at right, the distances between the centers of Earth, Moon, and satellite are: r M = 384,000 km r L = 61,500 km Masses of the Earth and Moon can be found inside the rear cover of your textbook. Earth r M Moon satellite r L a. (4 pts.) If the Moon exerted NO gravitational pull on the satellite (i.e., only the Earth pulls gravitationally on the satellite), calculate what the orbital period of the satellite would be, in days. Show your work. Express your final answer to 3 significant figures. b. (5 pts.) If BOTH the Moon AND Earth exert gravitational forces on the satellite, calculate the magnitude of the net (total) gravitational force on the satellite, at the moment shown in the diagram above. Show your work.

7 2. continued: c. (3 pts.) If the net gravitational force on the satellite calculated in part (b) is the actual centripetal force acting on the satellite, calculate the satellite s actual period (in days) of its uniform circular motion about the Earth. (Assume that the Earth, Moon, and satellite remain aligned as they orbit, so that the magnitude of the centripetal force on the satellite remains constant.) Show your work. Express your final answer to 3 significant figures. Earth Moon F net satellite Note: Your answer to part (c), for the satellite s orbital period, should be quite close (to within ±1 day) of the Moon s own orbital period! This special location in space is called the L2 Lagrangian Point; there are a total of five such points in space around the Earth Moon system where a satellite can be placed so that it orbits around the Earth with approximately the same period as the Moon does. There are many advantages to placing a satellite at the L2 point; for example, it would be especially useful for telecommunications to the far side of the Moon someday.

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