Physics 101 Lawrence Weinstein Homework set 7 Solutions 1-Steps: ----------------------- 8.6: Angular momentum = mvr. If v doubles then Angular Momentum will double. 9.5: F = GMm/d^2 = (6.7*10^(-11) N-m^2/kg^2) * (3 kg) * (6.4*10^23 kg) --------------------------------------------------- (5.6*10^10 m)^2 F = 4.1*10^(-8) N 9.6: F = GMm/d^2 = (6.7*10^(-11) N-m^2/kg^2) * (3 kg) * (80 kg) --------------------------------------------------- (0.5 m)^2 F = 6.4*10^(-8) N The obstetrician exerts about the same gravitational force on the baby as Mars! (The 10^(-8) is much more important than the 4 or 6 that multiplies it.) Note that your doctor will have more or less mass than mine, so your answer will be slightly different. Exercises: ------------------------- 8.38: The tangential speed of the Earth is greatest at the equator and zero at the poles. Therefore, a cannon ball fired south from Norfolk will have the same eastward velocity as you do. However, Florida has a larger eastward velocity than Norfolk. Therefore the cannon ball will land 'behind' or to the west of Florida. 8.44: Centripetal force does not do work on a circularly moving object because the force is radial and the motion is tangential. This means that the force is perpendicular to the motion so that zero work is done. 8.52: The distance from the axis of the Earth (the line passing through the North and South poles) to the surface is shortest at the poles (ie: it is zero there) and longest at the equator (where it equals the radius of the Earth). The Mississippi River moves mass from higher latitudes, where it is closer to the axis of rotation, to lower latitudes, where it is further from the axis of rotation. This increases the rotational inertia of the Earth, thereby decresing the rotational speed (since angular momentum is conserved). Thus, the day should get longer. 8.57: Angular momentum is conserved. The system starts with zero angular momentum. If the train moves clockwise (CW), then the wheel must move counterclockwise (CCW). If the train backs up (ie: moves CCW), then the wheel will go in the opposite direction (ie: CW). The total angular momentum of the train plus wheel system does not change. If the train were much more massive than the track, the track would move very quickly and the train would hardly move. If the track were much more massive than the train, then the train would move quickly and the track would move slowly. 8.58: The small tail rotor keeps the helicopter from spinning wildly. In order to keep the main rotor turning, the engine exerts a torque on it. Therefore, by Newton's 3rd Law, the rotor exerts a torque on the rest of the helicopter. This will make the helicopter spin faster and faster unless there is a countertorque to balance it. The small rotor exerts a sideways force on the air to provide the countertorque. If the small rotor fails in flight, then the helicopter will spin wildly and almost certainly crash. 9.2: If you double the mass of an object, it has twice the gravitational force acting on it (twice the weight). However, since it also has twice the inertia, its acceleration will be unchanged (since a = F/m). 9.10: There are two falacies here: 1) Astronauts in orbit are not beyond the pull of Earth's gravity. Earth's gravity keeps them in orbit (otherwise they would keep going in a straight line and leave the Earth behind forever). 2) They only feel weightless because they are in free fall and there is no normal force acting on them. 9.14: The force is the same. Newton's 3rd Law. Problems: ------------------------ 8.6: a) torque = F*L = 80 N * 0.25 m = 20 Nm b) to exert the same torque with L = 0.1 m, you need to exert a force F = 20 Nm / 0.1 m = 200 N. c) These answers assume that you are pushing perpendicular to the handle of the wrench. At other angles you will get less torque for the same force. 8.other: A 70 kg astronaut would have a weight W = 70 kg * 10 m/s^2 * (1 lb / 4.5 N) W = 154 lb. The normal force that the floor of the space station exerts on the atmosphere provides the 'apparent weight'. That force is F = mv^2/R. If you double v, then you quadruple v^2 and F quadruples. Therefore, the astronaut will now have an apparent weight of 4*154 lb = 616 lb. Too much for me! Estimation: ----------------------- Angular momentum is conserved. If rotational inertia increases, then rotational speed will decrease by the same amount. When the density of the Sun increases, its volume will decrease. The Sun's volume will thus decrease by a factor of 10^14. In general, volume ~ R^3. Therefore, if V decreases by 10^14, R will decrease by (10^14)^(1/3) or a bit less than 10^5. This means that R^2 will decrease by almost 10^10. Since rotational inertia = mR^2 and mass does not change, the rotational inertia will decrease by a factor of almost 10^10. This means that rotational speed will increase by a factor of 10^10. Now, the sun rotates once per month or once per 2.6*10^6 seconds (using the answer from the first problem set). Afterwards, it will rotate 10^10 times per 2.6*10^6 s or 4000 times per second. This means that it will take 0.25*10^(-3) seconds per rotation. Wow! That's fast! (This was a little harder than I intended. Congratulations to all who got it.)