Physics 8 Wednesday, November 9, 2011

Transcription

1 Physics 8 Wednesday, November 9, 2011 For HW8 problems to sketch out in class, I got 5 requests for # 9 3 requests for # 3 2 requests for # 14 1 request each for # 2, 7, 10, 11, 15, 19 I will be in DRL 3W2 at 7pm both Wed and Thu this week. Next week Zoey will do both days.

2 People also asked about rolling motion in ch12 A 215 g can of soup is 10.8 cm tall and has a radius of 3.19 cm. (a) Calculate its theoretical rotational inertia, assuming it to be a solid cylinder. (b) When it is released from rest at the top of a ramp that is 3.00 m long and makes an angle of 25 with the horizontal, it reaches the bottom in 1.40 s. What is the experimental rotational inertia? (c) Compare the experimental and theoretical rotational inertias and suggest possible sources of the difference.

3 People also asked about rolling motion in ch12 A 215 g can of soup is 10.8 cm tall and has a radius of 3.19 cm. (a) Calculate its theoretical rotational inertia, assuming it to be a solid cylinder. (b) When it is released from rest at the top of a ramp that is 3.00 m long and makes an angle of 25 with the horizontal, it reaches the bottom in 1.40 s. What is the experimental rotational inertia? (c) Compare the experimental and theoretical rotational inertias and suggest possible sources of the difference. One constant-acceleration trick that you may not have seen: if you solve x f = 1 2 at2 for a and then plug a = 2x f into v 2 t 2 f = 2ax f, you get (for v i = 0, x i = 0, a = constant): ( ) vf 2 2xf = 2 t 2 x f = 4x f 2 t 2 Now use energy conservation to relate I to v 2 f...

4 9. Archimedes screw, one of the first mechanical devices for lifting water, consists of a very large screw surrounded by a hollow, tight-fitting shaft (shown below). The bottom end of the device is placed in a pool of water. As the screw is turned, water is carried up along its ridges and comes out the top of the shaft and into a storage tank. As the handle is turned, work done by the torque exerted on the handle is converted to gravitational potential energy of the water-earth system. Let s say you want to take a shower using this device. You figure your shower will consume 44 liters of water, and so you have to raise this amount to the storage tank 2.5 m above the pool, so it can fall down on you. When you turn the handle, you apply a torque of 12 N m. How many times must you turn the handle?

5 Problem 3. A 35 kg child stands on the edge of a 400 kg playground merry-go-round that is turning at the rate of 1 rev every 2.2 s. She then walks to the center of the platform. If the radius of the platform is 1.5 m, what is its rotational speed once the child arrives at the center?

6 I got two requests for #14 A square clock of inertia M = 5.0 kg is hung on a nail driven in a wall, as shown at right. The length of each side of the square is l = 20 cm, the thickness is w = 10 cm, and the clock is a distance d = 0.1 cm from the wall at the top, where it hangs on the nail. Assume that the surface of the wall is very smooth and that the center of mass of the clock corresponds to its geometric center. What is the magnitude of the normal force exerted by the wall on the clock? Make simplifying approximations any answer within 10% of the correct answer gets full credit.

7 Chapter 13 gravity If the density of all the planets were the same, how would the surface acceleration of gravity on each planet depend on its radius? (A) (B) a = GM R 2 so the acceleration would be proportional to 1/R 2 a = G(4 3 πr3 ρ) R 2 so the acceleration would be proportional to R

8 g = (4π/3)(density)G R The surface gravity of a planet or with a given mass will be approximately inversely proportional to the square of its radius, and the surface gravity of a planet or star with a given average density will be approximately proportional to its radius. Acceleration due to gravity is shown through the equation g = GM/R 2. On different scales the mass is proportional to its radius cubed. When this is plugged back into the first equation, g R. Surface acceleration of gravity and the planet s radius are directly related (for constant density). Given that volume is proportional to the radius cubed, which implies that mass is proportional to the radius cubed if the density is uniform, then the acceleration of gravity will be given by g R directly proportional to the radius of the planet.

9 Chapter 13 gravity An astronaut on the international space station gently releases a satellite that has a gravitational mass much less than that of the station. Describe the behavior of the satellite after release. (A) The satellite goes into orbit around the space station (B) The satellite will float along with the astronaut and everything else in the space station (C) The satellite is pulled back toward the station s center of mass by the small force of gravity between the station and the satellite (D) There would be a small gravitational pull between the station and the satellite, but this pull is so small that even if the astronaut is pretty careful, the relative speed of the satellite and station will still exceed the escape velocity, and the satellite will slowly drift away.

10 Chapter 13 gravity The satellite is only subject to the force of Earth s gravity, so it experiences weightlessness and just floats. It would tend to follow in the same trajectory of the space station and continue to orbit. The satellite will float along with the astronaut and everything else in the space station. The object is essentially in free fall, experiencing weightlessness. If the satellite with much less gravitational mass is released outside of the international space station, I believe it would begin to orbit the planet it was near, if any. Because every piece of matter attracts another piece of matter according to Newton s law of gravity, the satellite would have to interact with the gravitational pulls of the nearest planet. The satellite will orbit around the space station in an elliptical manner. The space station will be at one of the foci of this ellipse.

11 Chapter 13 gravity The International Space Station is in low-earth orbit, about 400 km above the earth s surface. Earth s radius is about 6400 km. The gravitational acceleration toward the center of the earth at this altitude is G M earth (6800 km) 2 = ( N m2 )( kg) kg 2 ( m) 2 = 8.7 m s 2 By comparison, gravitational acceleration toward (500,000 kg) ISS of a nearby satellite (say 10 m away) is ( N m2 kg 2 )( kg) (10 m) 2 = m s 2 So attraction between satellite and ISS is more than 10 million times smaller than attraction between satellite and earth.

12 Plugging in numbers, it looks as if satellite would take about 2 hours to fall 10 m back to ISS, or if released at the right relative velocity, it could orbit the ISS with a period of about 10 hours. So on a time scale of minutes, the satellite would just sit there next to the space station. On a time scale of hours, it looks as if it could be very slowly accelerated toward the space station s center of mass. (I wouldn t have guessed this.) If you gave it precisely the right push w.r.t. the space station, it may be conceivable that the satellite would very slowly orbit the space station, going around a couple of times a day. But I think in reality it would be very difficult not to give the satellite some nonzero initial velocity w.r.t. the space station, e.g. at least m/s, so it would drift out of reach of the space station s gravity.

13 Note that the time that it takes both the space station and the satellite (in an orbit 400 km above earth s surface) to go around the earth is only about R 2π = 1.5 hours a

14 What is the acceleration due to gravity at a distance of one earth radius above earth s surface? (A) g/4 (B) g/2 (C) g (D) 2g (E) 4g where g is our old friend 9.8 m/s 2.

15 What is the acceleration due to gravity at a distance of one earth radius above earth s surface? a = GM earth (2R earth ) 2 = ( N m2 kg 2 )( kg) ( m) 2 = 2.44 m s 2 or more simply a = g/4 = 2.45 m/s 2