Physics 101 Lawrence Weinstein Homework set 6 Solutions Project: -------- 8.4: The average woman is able to stand closer to the wall while touching her toes than the average man. This is because the average woman has a greater fraction of her mass in her hips and less in her shoulders than the average man. When you touch your toes, your shoulders go forwards of your feet; therefore your hips need to move backwards so that your center of gravity remains over your feet. The average woman's hips do not need to move as far back to balance their less massive shoulders. The average woman has a lower center of gravity relative to her height than the average man. One Steps: ---------- 8.2: Torque = lever arm * force = 0.5 m * 50 N = 25 Nm 8.4: The force of friction has to provide the centripetal force to keep the person moving in a circle. F = mv^2/R = 75 kg * (3 m/s)^2 / 2 m = 3.4*10^2 N Exercises: ---------- 7.60: The work done by you on the handles equals the work done on the paper or metal by the blades. In the case of the bolt cutters it takes a very large force to cut the bolts. Therefore you apply a medium force over a large distance and the blades apply a LARGE force over a small distance. In the case of scissors or metal shears (which are used to cut very thin metal), you do not a large force to cut the material. Therefore, you apply a small to medium force over a small distance, and the scissors apply a smaller force over a larger distance (so you can cut more quickly). 7.68: Energy is conserved. If you use less energy than you consume, your body will store the remainder (as fat). If you use more energy than you consume, your body will metabolize (ie: consume) itself to provide the difference. Initially, it will metabolize the stored fat. But when it runs out of fat, it will metabolize muscle tissue. An undernourished person cannot perform extra work without extra food if they do not have any fat or 'extra' muscle tissue left to metabolize. 7.extra: Let's choose our mass to be equal to 2 (in some arbitary units) so that KE = (1/2)mv^2 = v^2. Then we have: v KE ---------- 15 225 30 900 45 2025 Thus, it takes 900 - 225 = 675 units of energy to increase your speed from 15 to 30 mph, but it takes 2025 - 900 = 1125 units of energy to increase yoru speed from 30 to 45 mph. Thus it takes more energy to increase your speed from 30 to 45 mph than from 15 to 30 mph. 8.2: Since the two wheels are connected by a belt, points along the rims of the two wheels all have the same tangential speed. This means that the smaller wheel makes two rotations for every one rotation of the larger wheel. Thus the rotational speed of the small wheel is twice that of the large wheel. 8.4: This problem is exactly the same as the previous one. Points along the rim of both wheels have exactly the same tangential speed since both bikes are travelling at the same linear speed. This means that the smaller wheels have a larger rotational speed (since tangential speed = rotational speed times radius). 8.12: The mass of the object has no effect on its acceleration (as we showed for two objects that are just dropped). Therefore, the only thing that matters is the distribution of the mass. The volleyball has all of its mass on the periphery (ie: the outside) and the bowling ball has its mass distributed uniformly. Since the bowling ball has its mass closer to the center of the ball, it will have a greater acceleration rolling down an incline. 8.18: The lever arm does not change when someone sits on, stands on, or hangs from the end of the seesaw. The force is still directed vertically downward at the same point. Therefore the net torque is not changed. We demonstrated this in class. 8.26: Because it lowers the center of mass of the tightrope walker and therefore makes him more stable. (He uses a long pole in order to increase him rotational inertia. However, that does not explain why the long pole works better if it droops.) Problems: --------- 8.2: The passenger covers a distance of 2*pi* 10 m = 62 m in 30 seconds. Therefore her speed is v = 62 m / 30 s = 2.1 m/s. Answers should not have more than 3 significant figures. 8.extra: the 3 kg weight hangs at the point 0 cm. The 6 kg weight hangs at the point 100 cm. We will call the balance point x. Then, the torque exerted by the 3 kg weight is 3 kg * 10 m/s^2 * x = 30 N * x. The torque exerted by the 6 kg weight is 60 N * (100 cm - x) in the opposite direction. These two torques must be equal so that 30 x = 6000 - 60 x or x = 6000 / 90 = 66.6 cm Therefore the balance point is 66.6 cm from the 3 kg weight and 33.3 cm from the 6 kg weight. Estimation: ----------- The mass of an 18-wheeler is about 40 tons = 4*10^4 kg so that the mass of 6 cars is 2.4*10^5 kg. The train travels at about 40 mph or 20 m/s. Therefore, the kinetic energy is KE = (1/2) (2.4*10^5 kg) (20 m/s)^2 = 5*10^7 J Spiderman has to do 5*10^7 J of work to stop the train. Work = F * d so that F = W / d and F = 5*10^7 J / 10^3 m = 5*10^4 N. This is about the weight of 5 tons (since 1 ton = 1000 kg has a weight of 10,000 N)! A superhero can do that, but I cannot. Answers within a factor of 10 should get full credit.