Analysis of forces
5.6 Simple Machines: the pulley
Even the most complex machines, such as cars or pianos, are built out of certain basic units called simple machines. The following are some of the main functions of simple machines:
transmitting a force: The chain on a bicycle transmits a force from the crank set to the rear wheel.
changing the direction of a force: If you push down on a see-saw, the other end goes up.
changing the speed and precision of motion: When you make the “come here” motion, your biceps only moves a couple of centimeters where it attaches to your forearm, but your arm moves much farther and more rapidly.
changing the amount of force: A lever or pulley can be used to increase or decrease the amount of force.
You are now prepared to understand one-dimensional simple ma-chines, of which the pulley is the main example.
z/Example 10.
A pulley example 10
. Farmer Bill says this pulley arrangement doubles the force of his tractor. Is he just a dumb hayseed, or does he know what he’s doing?
aa/Example 11.
.To use Newton’s first law, we need to pick an object and con-sider the sum of the forces on it. Since our goal is to relate the tension in the part of the cable attached to the stump to the ten-sion in the part attached to the tractor, we should pick an object to which both those cables are attached, i.e., the pulley itself. The tension in a string or cable remains approximately constant as it passes around an idealized pulley.1 There are therefore two left-ward forces acting on the pulley, each equal to the force exerted by the tractor. Since the acceleration of the pulley is essentially zero, the forces on it must be canceling out, so the rightward force of the pulley-stump cable on the pulley must be double the force exerted by the tractor. Yes, Farmer Bill knows what he’s talking about.
More complicated pulley systems can be constructed to give greater amplification of forces or to redirect forces in different direc-tions. For an idealized system,2 the fundamental principles are:
1. The total force acting on any pulley is zero.3
2. The tension in any given piece of rope is constant throughout its length.
3. The length of every piece of rope remains the same.
A compound pulley example 11
.Find the mechanical advantage T5/F of the pulley system. The bar is massless.
.By rule 2, T1= T2, and by rule 1, F = T1+ T2, so T1= T2= F /2.
Similarly, T3 = T4 = F /4. Since the bar is massless, the same reasoning that led to rule 1 applies to the bar as well, and T5 = T1+ T3. The mechanical advantage is T5/F = 3/4, i.e., this pulley system reduces the input force.
How far does the tractor go compared to the stump? example 12 .To move the stump in figure z by 1 cm, how far must the tractor move?
.Applying rule 3 to the the right-hand piece of rope, we find that the pulley moves 1 cm. The upper leg of the U-shaped rope there-fore shortens by 1 cm, so the lower leg must lengthen by 1 cm.
Since the pulley moves 1 cm to the left, and the lower leg extend-ing from it also lengths by 1 cm, the tractor must move 2 cm.
1This was asserted in section 5.4 without proof. Essentially it holds because of symmetry. E.g., if the U-shaped piece of rope in figure z had unequal tension in its two legs, then this would have to be caused by some asymmetry between clockwise and counterclockwise rotation. But such an asymmetry can only be caused by friction or inertia, which we assume don’t exist.
2In such a system: (1) The ropes and pulleys have negligible mass. (2) Friction in the pulleys’ bearings is negligible. (3) The ropes don’t stretch.
3F = ma, and m = 0 since the pulley’s mass is assumed to be negligible.
Section 5.6 Simple Machines: the pulley 189
ab/The black box marked with an X is a machine that doubles force while leaving the amount of motion unchanged.
If 1 cm of rope is pulled out through the input on the bottom at tension T , the amount of rope consumed at tension 2T on top is not 1/2 cm, as we would normally expect, but 1 cm. This machine is impossible.
Examples 10 and 12 showed that the pulley system in figure z amplifies the force by a factor of 2, but it reduces the motion by 1/2. This is an example of a more general inverse proportionality for all such systems. Superficially, it follows from rules 1-3 above.
If, for example, we try to construct a pulley system that doubles the force while keeping the motion the same, we will find that the rules seem to mysteriously conspire against us, and every attempt ends in failure. We could in fact prove as a mathematical theorem that the inverse proportionality always holds if we assume these rules.
But these rules are only an idealized mathematical model of a specific type of simple machine. What about other machines built out of other parts such as levers, screws, or gears? Through trial and error we will find that the inverse proportionality holds for them as well, so there must be some more fundamental principles involved.
These principles, which we won’t discuss formally until ch. 11 and 13, are conservation of energy and the equation for mechanical work.
Informally, imagine that we had a machine that violated this rule.
We could then insert it into a setup like the one in figure ab. When we release the single weight at the top, it drops to the ground while lifting the pan, which holds double the weight, all the way to the top. This is the ultimate free lunch. Once the pair of weights is up at the top, we can use them to hoist four more, then 8, 16, and so on. This is known as a perpetual motion machine.
If this seems to be too good to be true, it is. Just as small machines can be put together to make bigger ones, any machine can also be broken down into smaller and smaller ones. This process can be continued until we get down to the level of atoms. The law of conservation of energy essentially says that atoms don’t act like perpetual motion machines, and therefore any machine built out of atoms also fails to be a perpetual motion machine.
ac/Ernst Mach (1838-1916) is mainly known for having pro-posed a radical extension of the principle of inertia to state that all motion, not just constant-velocity motion, was relative. His ideas strongly influenced Einstein. The Mach factor (used, e.g., when we describe a jet as traveling at
“Mach 2”) is named after him.
5.7 ? Does Newton’s third law mean anything, and if so, is it true?
This section discusses Newton’s third law in the same spirit as sec-tion 4.7 on the first and second laws.
Ernst Mach gave a cogent critique of the third laws’s logical assumptions in his book The Science of Mechanics. The book is available online for free at archive.org, and is very readable. To understand Mach’s criticism, consider the experiment illustrated in figure a on p. 166, in which a large magnet and a small magnet are found to exert equal forces on one another. I use this as a student lab, and I find that most students are surprised by the result. Nevertheless, the lab can be considered a swindle, for the following reason. If we wanted to, we could cut the large magnet apart into smaller pieces, each of which was the same size as the small magnet. In fact, the large magnets I use for this lab were constructed simply by taking six small ones, stacking them together, and wrapping them in plastic. To represent this symbolically, let the small magnet be [A] and the large one [BCDEFG]. Since A and B are identical, and they are oriented in the same way, it follows simply by symmetry that A’s force on B and B’s on A obey the third law.
The same holds for A on C and C on A, and so on. Since Newton claims that forces combine by addition, it follows that the result of the experiment must be in accord with the third law, despite the superficial asymmetry.
Now suppose that material objects 1 and 2 have the same chem-ical composition. By a similar argument it seems likely that F12and F21 obey Newton’s third law.
This argument shows how pointless it can be to attempt to test a scientific theory unless you have in your possession a sensible al-ternative theory that predicts something different. One could spend decades doing experiments of the kind described above without real-izing that the tests were all trivially guaranteed to give null results, even if nature was really described by a theory that violated New-ton’s third law.
Here is an example of a fairly sane theory that could violate Newton’s third law. Einstein’s famous E = mc2states that a certain amount of energy E is equivalent to a certain amount of mass m, with c being the speed of light. (We won’t formally encounter energy until ch. 11, or the reasons for E = mc2 until section 12.5, but for now just think of energy as the kind of thing you intuitively associate with food calories or a tank full of gasoline, and take E = mc2 for granted.) Einstein claimed that this would hold for three different kinds of mass: the mass measured by an object’s inertia, the “active”
gravitational mass ma that determines the gravitational forces it makes on other objects, and the “passive” gravitational mass mp
Section 5.7 ?Does Newton’s third law mean anything, and if so, is it true? 191
ad/1. A balance that mea-sures the gravitational attraction between massesM and m. (See section 10.5 for a more detailed description.) When the two masses M are inserted, the fiber twists. 2. A simplified diagram of Kreuzer’s modification. The moving teflon mass is submerged in a liquid with nearly the same density. 3. Kreuzer’s actual apparatus.
that measures how strongly it feels gravity. Einstein’s reason for predicting similar behavior for ma and mp was that anything else would have violated Newton’s third law for gravitational forces.
Suppose instead that an object’s energy content contributes only to mp, not to ma. Atomic nuclei get something like 1% of their mass from the energy of the electric fields inside their nuclei, but this percentage varies with the number of protons, so if we have objects m and M with different chemical compositions, it follows that in this theory mp/ma will not be the same as Mp/Ma, and in this non-Einsteinian version of relativity, Newton’s third law is violated.
This was tested in a Princeton PhD-thesis experiment by Kreuzer4 in 1966. Kreuzer carried out an experiment, figure ad, using masses made of two different substances. The first substance was teflon.
The second substance was a mixture of the liquids trichloroethy-lene and dibromoethane, with the proportions chosen so as to give a passive-mass density as close as possible to that of teflon, as de-termined by the neutral buoyancy of the teflon masses suspended inside the liquid. If the active-mass densities of these substances are not strictly proportional to their passive-mass densities, then mov-ing the chunk of teflon back and forth in figure ad/2 would change the gravitational force acting on the nearby small sphere. No such change was observed, and the results verified mp/ma = Mp/Ma to within one part in 106, in agreement with Einstein and Newton. If electrical energy had not contributed at all to active mass, then a violation of the third law would have been detected at the level of about one part in 102.
The Kreuzer result was improved in 1986 by Bartlett and van Buren5 using lunar laser ranging data similar to those described in section 4.7. Since the moon has an asymmetrical distribution of iron and aluminum, a theory with mp/ma 6= Mp/Ma would cause it to have an anomalous acceleration along a certain line. The lack of any such observed acceleration limits violations of Newton’s third law to about one part in 1010.
4Kreuzer, Phys. Rev. 169 (1968) 1007
5Phys. Rev. Lett. 57 (1986) 21
Summary
Selected vocabulary
repulsive . . . describes a force that tends to push the two participating objects apart
attractive . . . . describes a force that tends to pull the two participating objects together
oblique . . . describes a force that acts at some other angle, one that is not a direct repulsion or attraction normal force . . . the force that keeps two objects from
occupy-ing the same space
static friction . . a friction force between surfaces that are not slipping past each other
kinetic friction . a friction force between surfaces that are slip-ping past each other
fluid . . . a gas or a liquid
fluid friction . . . a friction force in which at least one of the object is is a fluid
spring constant . the constant of proportionality between force and elongation of a spring or other object un-der strain
Notation
FN . . . a normal force
Fs . . . a static frictional force Fk . . . a kinetic frictional force
µs . . . the coefficient of static friction; the constant of proportionality between the maximum static frictional force and the normal force; depends on what types of surfaces are involved
µk . . . the coefficient of kinetic friction; the constant of proportionality between the kinetic fric-tional force and the normal force; depends on what types of surfaces are involved
k . . . the spring constant; the constant of propor-tionality between the force exerted on an ob-ject and the amount by which the obob-ject is lengthened or compressed
Summary
Newton’s third law states that forces occur in equal and opposite pairs. If object A exerts a force on object B, then object B must simultaneously be exerting an equal and opposite force on object A.
Each instance of Newton’s third law involves exactly two objects, and exactly two forces, which are of the same type.
There are two systems for classifying forces. We are presently using the more practical but less fundamental one. In this system, forces are classified by whether they are repulsive, attractive, or oblique; whether they are contact or noncontact forces; and whether
Summary 193
the two objects involved are solids or fluids.
Static friction adjusts itself to match the force that is trying to make the surfaces slide past each other, until the maximum value is reached,
Fs,max= µsFN.
Once this force is exceeded, the surfaces slip past one another, and kinetic friction applies,
Fk= µkFN.
Both types of frictional force are nearly independent of surface area, and kinetic friction is usually approximately independent of the speed at which the surfaces are slipping. The direction of the force is in the direction that would tend to stop or prevent slipping.
A good first step in applying Newton’s laws of motion to any physical situation is to pick an object of interest, and then to list all the forces acting on that object. We classify each force by its type, and find its Newton’s-third-law partner, which is exerted by the object on some other object.
When two objects are connected by a third low-mass object, their forces are transmitted to each other nearly unchanged.
Objects under strain always obey Hooke’s law to a good approx-imation, as long as the force is small. Hooke’s law states that the stretching or compression of the object is proportional to the force exerted on it,
F ≈ k(x − xo).
Problem 6.
Problem 8.
Problem 7.
Problems
Key√
A computerized answer check is available online.
R A problem that requires calculus.
? A difficult problem.
1 In each case, identify the force that causes the acceleration, and give its Newton’s-third-law partner. Describe the effect of the partner force. (a) A swimmer speeds up. (b) A golfer hits the ball off of the tee. (c) An archer fires an arrow. (d) A locomotive slows
down. . Solution, p. 548
2 Example 2 on page 169 involves a person pushing a box up a hill. The incorrect answer describes three forces. For each of these three forces, give the force that it is related to by Newton’s third law, and state the type of force. . Solution, p. 548 3 (a) Compare the mass of a one-liter water bottle on earth, on the moon, and in interstellar space. . Solution, p. 548 (b) Do the same for its weight.
In problems 4-8, analyze the forces using a table in the format shown in section 5.3. Analyze the forces in which the italicized object par-ticipates.
4 Some people put a spare car key in a little magnetic box that they stick under the chassis of their car. Let’s say that the box is stuck directly underneath a horizontal surface, and the car is parked.
(See instructions above.)
5 Analyze two examples of objects at rest relative to the earth that are being kept from falling by forces other than the normal force. Do not use objects in outer space, and do not duplicate problem 4 or 8. (See instructions above.)
6 A person is rowing a boat, with her feet braced. She is doing the part of the stroke that propels the boat, with the ends of the oars in the water (not the part where the oars are out of the water).
(See instructions above.)
7 A farmer is in a stall with a cow when the cow decides to press him against the wall, pinning him with his feet off the ground. An-alyze the forces in which the farmer participates. (See instructions above.)
8 A propeller plane is cruising east at constant speed and alti-tude. (See instructions above.)
Problems 195
Problem 9.
Problem 12.
9 A little old lady and a pro football player collide head-on.
Compare their forces on each other, and compare their accelerations.
Explain.
10 The earth is attracted to an object with a force equal and opposite to the force of the earth on the object. If this is true, why is it that when you drop an object, the earth does not have an acceleration equal and opposite to that of the object?
11 When you stand still, there are two forces acting on you, the force of gravity (your weight) and the normal force of the floor pushing up on your feet. Are these forces equal and opposite? Does Newton’s third law relate them to each other? Explain.
12 Today’s tallest buildings are really not that much taller than the tallest buildings of the 1940’s. One big problem with making an even taller skyscraper is that every elevator needs its own shaft run-ning the whole height of the building. So many elevators are needed to serve the building’s thousands of occupants that the elevator shafts start taking up too much of the space within the building.
An alternative is to have elevators that can move both horizontally and vertically: with such a design, many elevator cars can share a few shafts, and they don’t get in each other’s way too much because they can detour around each other. In this design, it becomes im-possible to hang the cars from cables, so they would instead have to ride on rails which they grab onto with wheels. Friction would keep them from slipping. The figure shows such a frictional elevator in its vertical travel mode. (The wheels on the bottom are for when it needs to switch to horizontal motion.)
(a) If the coefficient of static friction between rubber and steel is µs, and the maximum mass of the car plus its passengers is M , how much force must there be pressing each wheel against the rail in order to keep the car from slipping? (Assume the car is not
accelerating.) √
(b) Show that your result has physically reasonable behavior with respect to µs. In other words, if there was less friction, would the wheels need to be pressed more firmly or less firmly? Does your equation behave that way?
13 An ice skater builds up some speed, and then coasts across the ice passively in a straight line. (a) Analyze the forces, using a table in the format shown in section 5.3.
(b) If his initial speed is v, and the coefficient of kinetic friction is µk, find the maximum theoretical distance he can glide before coming
to a stop. Ignore air resistance. √
(c) Show that your answer to part b has the right units.
(d) Show that your answer to part b depends on the variables in a way that makes sense physically.
(e) Evaluate your answer numerically for µk= 0.0046, and a world-record speed of 14.58 m/s. (The coefficient of friction was measured by De Koning et al., using special skates worn by real speed skaters.)√