I.9 You are riding a bike and have just reached the top of a hill.
Before the descent you would like to know what is better: to
pedal initially say half the distance, or to pedal all the
distance with half that force (thus spending the same energy).
Of course the air resistance is important. Make reasonable
assumptions about all forces, the slope angle and so on, and
come up with a suggestion based on a few different cases. Of
course the word "better" has to be given a precise meaning.
I.20 There is a theory that meteor grains have fallen down on the
surface of the earth carrying organic molecules, which then
have started life on earth (possibly more than once, because
of extinctions due to comets or large meteors hitting earth).
Assume an atmosphere similar to that today, with negligable
temperature differences, and assume that the grain falls
vertically. Find the speed relative to the atmosphere, and
how it depends on grain size and density. Also find the maximal
power of the friction force. If the power is low enough, the
molecules will not be destroyed by heat -- setting a limit
is not necessary in this exercise, though).
For simplicity assume that the grains have the same velocity
as earth when they are well above the atmosphere, say at a
height of 500 km. What difference does it make if the grain
initially has a velocity of typically, say, 10 km/s?
I.21 A rocket is fired vertically from ground level, and should
reach the upper atmosphere. Try to chose realistic dimensions,
masses and exhaust velocity to achieve the task. How big a
payload can the rocket carry?
I.26 A person stands at rest on a raft, which is also at rest on
the water. When she starts walking, we all know that the raft
starts moving. Examine the motion of the raft, taking into
account the friction from the water. In particular, what is
the position of the raft after a long time (i.e., a long time
after the person has come to rest at the other end of the raft).
Use different models of the water resistance.
II.22 A small ball moves between two points A and B located at the
same height. It can either roll on a horizontal track, or on
a curved track which goes down (giving the particle an acceleration
due to gravity), and then up again to B. Choose a simple shape
for the curved track, and examine under what circumstances
the curved one provides the fastest transport. Friction
can be neglected.
II.29 A wood model of a seagull consists (essentially) of two rods
(the wings) and a central body. Two strings are attached to the
wings and to the ceiling. Select lengths and masses, and arrange
the strings suitably so that the gull can perform a beautiful
motion similar to that of a real bird.
III.15 The support of a pendulum has an oscillatory horizontal motion.
Set up its equation of motion, and analyze the motion for different
parameter values. Determine how the behavior varies with the
frequency of the horizontal motion? For example, is there chaotic
motion?
III.25 A person is trying to balance a stick by moving the supported
end horisontally. Make a model for how the information about
the inclination of the stick should result in a motion of the
hand.
III.31 A sphere is elastically suspended by two springs going straight
up and straight down, and is free to swing horizontally
under the influence of a constant wind. Show that an oscillation may
build up if the size and wind speed are appropriately chosen.
For experimental results for the drag coefficient of a sphere,
see Homework 1.
To simplify the equation, you may assume that the restoring force
from the springs depends linearly on the displacement. What is
the lesson to be learnt?
IV.21 When the spring of a simple music box is wound up, it produces a
large torque, that then decreases, and the speed of the mechanical
parts will also decrease. Consequently the pitch of the music (really?! Mats),
as well as the tempo, will change annoyingly. A simple stabilizer
of the speed consists of an air-brake. It is simply a rectangular
plate on one of the rotating axes in the clockwork. Evaluate the
efficiency of such a brake, making all necessary assumptions about
the parameters of a simplified music box.