Physics Exam 1 Formulas
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1 INSTRUCTIONS: Write your NAME on the front of the blue exam booklet. The exam is closed book, and you may have only pens/pencils and a calculator (no stored equations or programs and no graphing). Show all of your work in the blue book. For problems II-V, an answer alone is worth very little credit, even if it is correct so show how you get it. Do not assume ANYTHING is obvious if it s not clear, please ask. It s hard to write these questions, and we often forget things. Suggestions: Draw a diagram when possible, circle or box your final answers, and cross out parts which you do not want us to consider. Physics Exam 1 Formulas g = 9.8 m/s 2 Vectors : B = B 2 x + B 2 y + B 2 z î = ˆx; ĵ = ŷ; ˆk = ẑ va,c = v A,B + v B,C Kinematics : Constant a x : x = x 0 + v 0,x t a xt 2 v x = v 0,x + a x t v 2 x = v 2 0,x + 2a x (x x 0 ) 3D : v = d r = v xî + v yĵ + v zˆk = dx î + dy ĵ + dz ˆk at 2 + bt + c = 0 t = b ± b 2 4ac ; Newton : 2a a = d v = a xî +... = dv x î +... = d2 x 2 î +... F = Ftot = m a Work : W on object = Fon object d r U = W K = W U gravity = mg h Power = F v = τ ω = W/ t = dw/ F x, spring = kx U spring = 1 2 kx2 F centrip = mv2 r Conservation of Energy : K f + U f = K i + U i + W into system K lin = 1 2 mv2 F kinetic friction = µ k F normal F static friction µ s F normal p = m v J = p = pf p i Systems : M tot x c.m. = i m i x i p tot = M tot v c.m. v c.m. = d r c.m. U grav = M tot gy c.m. Rotation : α = const. ω = ω 0 + αt ω 2 = ω α(θ θ 0 ) θ = θ 0 + ω 0 t αt2 1 rev. = 2π rad τ = Iα vt = rω a t = rα a r = rω 2 I = i m i r 2 i K rot = 1 2 Iω2 L = r p = I ω τ = r F = r F sin φ = r F I parallel = I c.m. + Mh 2 F net = d P c.m. τ net = d L c.m. Elastic : v i + v f = V i + V f f = ω/2π f = 1/T F grav = GM 1M 2 r 2 U grav = GM 1M 2 r (U = 0 at r = ) G = Nm 2 /kg 2 Fluid Statics : p 2 = p 1 + ρg h Pascal : p 1 = p 2 F 1 A 1 = F 2 A 2 Archimedes : F b = ρ W gv displ Some Constants: Sphere : Area = 4πr 2 Volume = 4 3 πr3 Density : dm = ρdv Mass of the Earth kg Radius of Earth m Mass of the Sun kg Radius of Earth s Orbit m Density of Water 1000 kg/m 3 Page 1 of 5
2 I. Multiple Choice Questions (4 points each) Please write the letter corresponding to your answer for each question in the grid stamped on the first inside page of your blue book. No partial credit is given for these questions. 1. In the figure below there are several possible elliptical orbits of a satellite around a planet. Which orbit has the highest total energy? (c) (d) (b) (a) 2. Bucket A contains only water; an identical bucket B contains water and a solid object in the water. If the object floats in bucket B, and the buckets have the same water level, which bucket has the greater total weight? A B (a) Bucket A (b) Bucket B (c) The two buckets have the same weight (d) The answer cannot be determined from the given information 3. The magnitude of the gravitational force between two objects is given by F 0. If the mass of each object is doubled but the distance between them is halved, then the new force of gravity between the objects will be (a) 16 F 0 (b) 4 F 0 (c) F 0 (d) F 0 /2 (e) F 0 /4 Page 2 of 5
3 4. Consider a spherically-symmetric planet with a mass density that varies as a function of distance from the center of the planet. The magnitude of the gravitational potential energy of the system consisting of the planet and a test mass placed at some distance r measured from the planet s center would be (a) Zero at the center, and the maximum value would occur with the test mass at the surface of the planet. (b) Non-zero at the center, and the maximum value would occur with the test mass at the surface of the planet. (c) Non-zero at the center, but the maximum value would occur at with the test mass some point beneath the surface, but not at the center. (d) Non-zero at the center, and the maximum value would occur with the test mass at the center. (e) Zero at the center, and the potential energy is zero everywhere. 5. The pressure in a fluid as a function of depth is plotted to the right. What is the density of the liquid? (a) 1.76 g/cm 3 (b) 1.36 g/cm 3 (c) g/cm 3 (d) 1.70 g/cm 3 (e) 3.27 g/cm 3 p (kpa) Problems (20 points each) depth (m) II. Two cylinders are connected by a small tube at their bases. The left cylinder has cross-sectional area A, the right has cross-sectional area 3A. The cylinders are originally filled with a liquid of density 0. A cantaloupe of mass m and density C = is placed into the right cylinder. (a) Will the cantaloupe float? Why or why not? Make sure your argument references Archimedes. (b) Find the buoyancy force on the cantaloupe. (Note that there is a buoyancy force regardless of whether or not it floats.) (c) How much water is displaced by the cantaloupe? (d) Find the change in elevation of the water level in the two cylinders. A 3A (e) Assuming the cantaloupe fits, would your answer for (d) change if the cantaloupe was placed in the left cylinder? Why or why not? Page 3 of 5
4 III. A satellite of mass 220 kg travels in an approximately circular orbit 640 km above the surface of the earth. (The earth s radius is m.) (a) Determine its speed. (b) What is the total mechanical energy of this orbit? The satellite develops a small leak in a pressurized thruster pack. The leak points directly forward, along the path of the orbit, causing the satellite to lose kinetic energy very slowly at the average rate of J per orbital revolution. Make the approximation that the trajectory is a circle of slowly diminishing radius with constant energy loss per revolution. (c) Find the distance from the surface of the earth and the speed of the satellite at the end of its 1500 th revolution. (d) Find the average value of the force generated by the leaky thruster pack. IV. A sketch of a hydraulic lift is shown in the figure below. You wish to lift an M1 Abrams tank, mass 61,000 kg. Your tank of desire will be placed on a platform of area 32 m 2. Just for amusement, you decide that you want to design a hydraulic lift such that you, alone, can support the tank. Assume your mass is 70kg, just so we have a round number. Assume that the fluid levels in both sides of the lift are equal, and that you and the tank step onto your respective platforms simultaneously. tank you 32 m 2 (a) Find the fluid pressure necessary to hold up the tank. (b) Assume the platform on which you stand is square. What is the length of a side of the square if you are to provide exactly the pressure necessary to hold up the tank? (c) Now, you wish to raise the tank 1 centimeter. How far down will you need to push your platform in order to do so? (d) In doing this, you find that you need to add some additional force to compensate for the fact that the column height of fluid is now larger under the tank than it is beneath your platform because the tank moves up, while your platform moves down. How much extra force is required to hold the tank up at its new height? Assume the fluid has a density of 1500 kg/m 3. Page 4 of 5
5 V. Most planets are not uniform spheres. Instead, the density of the material varies with radius. We can still use the available tools to calculate the gravitational properties of such a planet, however. Consider a planet made of some material that has a density (r) = 0 /r, so the density varies inversely with the radius. In order to use our gravitational expressions, we need to find the mass enclosed by a sphere of a given radius. (a) Consider a thin spherical shell of thickness dr and radius r. How much mass dm is there just included in the shell itself, not in the entire volume enclosed by the sphere? (b) Use your expression from part (a) to find (by integration) the total mass enclosed by a sphere of arbitrary radius r. (c) Find the magnitude of the gravitational force on a test mass m for r < R, where R is the radius of the planet. (d) If the planet has radius R, what is its total mass in terms of the given constants? (e) Find the magnitude of the gravitational force on a test mass m for r > R, everywhere outside the planet. (Use M tot for the planet s mass if you didn t get an answer for (d)). (f) Bonus: Find the change in gravitational potential energy to move a test mass m from the surface of the planet to the planet s center. Page 5 of 5
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