Ratio of the large moment of inertia to that of the smaller moment of inertia

7 problems on rotational motion and two problems on density

1. Competitive divers pull their limbs in a curl up their bodies when they do flips. Just before entering the water, they fully extend their limbs to enter straight down. Explain the effect of both actions on their angular velocity. What is the effect of these actions on their angular momentum?

2. (A) Show that for a hoop of mass M rolling along level ground with a velocity v,the kinetic energy of the hoop is mv2. How is the kinetic energy "divided" between translational kinetic energy and rotational kinetic energy?

(B) Repeat this for a solid sphere of mass M. (Show that the kinetic energy is 0.7mv2 in this case.)

3. (A) What is the final velocity of a hoop that rolls without slipping down a 5.00 m high hill, starting from rest? Solve this using conservation of energy. [7.0 m/s].

(B) Repeat for a solid sphere instead of a hoop. [8.37 m/s].

4. A 30-kg child is riding on the edge of a 1.60-m-radius, 300 kg merry-go-round, spinning at 20.0 rpm. Assume that the merry-go-round experiences no friction: its bearings are perfect and we ignore air friction. Also, assume that you can find the rotational inertia of the merry-go-round by treating it like a disk. (You find the child's contribution by treating her as a point mass.)

(A) If the child moves to the center of the merry-go-round, what is the new angular velocity in rpm? (Hint: angular momentum is conserved in this case, since no external torque is applied to the system.)

(B) Calculate the kinetic energy of the system before and after the child moves to the center.

(C)You should have found that the kinetic energy is greater after the child moves to the center. Where did the "extra" energy come from?
[(A): 24 rpm, (B) 1010 J; 1210 J].

5. A ball slides up a ramp without friction. (If it helps, you could think of a square block of the same mass sliding up without friction. It would act the same.) The ball is then rolled without slipping up another ramp, with the same initial velocity as before. In which case does it reach a greater height, and why?

6. (a) Calculate the angular momentum of the earth in its orbit around the sun. Use SI units, of course.

(b) Compare this with the angular momentum of the earth on its axis. [The former is about 3.8 million times the latter.]

7. A yo-yo may be modeled as a small cylinder of mass 0.12 kg and radius 8.0 cm.

(A) What is the rotational inertia of such a yo-yo?

(B) As the yo-yo unwinds from the string, its angular speed goes up, from 0 to 40 radians per second in a period of 1.4 seconds. What is the net torque on the yo-yo?

(C) What is the yo-yo's angular 1 momentum at the end of the period described above?

(D) When the yo-yo is rotating at 40 radians per second, how many revolutions is it making each second?

8. In chapter 28 of Galileo's Daughter, there is a description of the opening of Galileo's last book, Two New Sciences. The subject is what we usually call the "laws of scaling." In view of this, answer the following questions, making your reasoning clear:

(a) Consider a cube whose sides are each of length L. If we made L 10 times larger, and kept the density of the material the same, by what factor would the mass of the cube increase? (Answer: 1000 times.) We can generalize this to any shape.

If we make each of the 3 dimensions 10 times as large, the mass increases by the same amount as our answer for the cube.

(b) The strength of a rope is proportional to its cross-sectional area. If it were pro portional to the volume, then making a rope twice as long would make it twice as strong, which you hopefully agree is not the case. Similarly, the strength of a muscle is proportional to its cross-sectional area, because this area is what determines the number of myofibrils which contract when the muscle applies a force.

Finally, the strength of a bone is also roughly proportional to cross-sectional area (though the reason is not so obvious as in muscles.) In light of this, consider an arm or a leg. If we make it 10 times as wide, 10 times as deep, and 10 times as long, by what factor would its strength increase?

(c) We are now in a position to answer the question: why don't giants exist? Consider a giant with 10 times the height, 10 times the width, and 10 times the depth (front to back) of a normal human. How would the giant's mass compare to that of a normal human? How would the giant's strength compare to that of a normal human? What would the giant's strength-to-mass ratio be, compared to a normal human? Do you see why such a giant would collapse under his own weight?

9. We could go the other way and ask why humans are not, say, 2 cm tall instead of our roughly 2 m height. What would be the problem with being small? Strength would not be a limiting factor, since if we were 2 cm tall, our strength-to-mass ratio would be a hundred times larger than it is now. What would some other limiting factors be? (Hint: why does a shrew have to eat half its weight in food each day?) If you are interested, there are some web sites which discuss the laws of scaling.

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