Chapter 13: Newton’s Theory of Gravity Fg = GMm r2 E = K + Ug Ug = −GMm r Circular motion: a =v2r(centripetal acceleration) Circumference of a circle = 2πr Chapter 14: Fluids p =FAρ =mass p = p0 + ρgh volume FB = (ρf luid)(Vdisplaced)g [Buoyant force = weight of fluid displaced] Continuity equation: v1A1 = v2A2 Bernoulli’s equation: p +ρv2 Volume of a sphere = 43πr3 Chapter 15: Oscillations Hooke’s law: F = −kx ω = Simple pendulum: F = −mg qk m 2 + ρgy = constant q g Ls ω = L Simple harmonic motion: x(t) = A cos(ωt + φ0) or x(t) = A sin(ωt + φ0) Period: T =2πω Frequency (in cycles per unit time): f =1T =ω2π E = K + Uspring Uspring =kx2 2 Some physical constants: Gravitational acceleration on the surface of the Earth: g = 9.8m/s2 Universal gravitational constant: G = 6.67 × 10−11Nm2/kg2 Mass of the Earth: ME = 5.97 × 1024kg Average Earth radius: RE = 6.37 × 106m Mass of the Sun: MS = 1.989 × 1030kg Average distance between Earth and Moon: rM = 3.84 × 108m Average distance between Earth and Sun: rE = 1.50 × 1011m Orbital period of Moon: TM = 27.3 days Atmospheric pressure at sea level: p0 = 1.013 × 105 Pa Mass density of pure water: ρwater = 1.00 × 103kg/m3 Note: It is not enough to simply write the final answers. Justify your answers by writing the equations you are using and intermediate steps in the solution. You are allowed to use your calculator for numerical calculations. Make sure all numerical answers are provided with proper units.Name: Student ID: Problem 1: The asteroid Gaspra has a mass of 1.0 × 1016 kg and a mean diameter of about 15 km. (a) What is the speed v of a man-made satellite orbiting around the asteroid Gaspra at a constant distance of 20 km from its center? (b) When the astronauts on the surface of Gaspra want to leave Gaspra, they fire their rockets briefly to make the radial component of their velocity vr just sufficient to escape to an indefinitely large distance. What is vr?Name: Student ID: Problem 2: A spring with spring constant k = 10 N/m hangs from a rigid support. A ball is hung from the spring and allowed to come to rest. The ball is then pulled down 0.1 m and released from rest. The ball now is oscillating at a frequency of 1 Hz. (a) What is the mass of the ball? (b) What is the maximum speed of the ball? (c) What is the velocity at time t = 1 s?Name: Student ID: Problem 3: A device for measuring specific gravities of liquids is a floating hollow cylinder of cross sectional area A = 1 cm2that is weighted at the sealed bottom by a mass to keep it upright in the liquid. The total mass of the cylinder (including the mass at the bottom of the cylinder) is m = 0.01 kg. (a) What is the equilibrium depth h (below the surface of the liquid) of the bottom of the cylinder floating in pure water contained in a large diameter beaker, as shown in the Figure? (b) If the cylinder is pushed downward from its equilibrium position by a small distance z, what is the magnitude and direction (up or down) of the change in the buoyant force F(z) on the cylinder? (c) If the cylinder is pushed downward from its equilibrium position by a small distance z, what is its oscillation frequency f after it is released? (Neglect the small perturbation to the oscillation frequency caused by the small motions of the water in the beaker.) (d) What mass of water must be added inside the cylinder for its equilibrium depth to change from h to 3h/2?
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