Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information....

A little old lady and a pro football player collide head-on.
Compare their forces on each other, and compare their accelerations.
Explain.

2

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?

3

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.

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 participates.

4

A magnet is stuck underneath a parked car. (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. Analyze
the forces in which the farmer participates. (See instructions
above.)

8

A propeller plane is cruising east at constant speed and altitude.
(See instructions above.)

9

Today's tallest buildings are really not that much taller than the
tallest buildings of the 1940s. One big problem with making an even
taller skyscraper is that every elevator needs its own shaft running
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 impossible
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
µ_{s}, 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 µ_{s}. 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?

10

Unequal masses M and m are suspended from a pulley as
shown in the figure.

(a) Analyze the forces in which mass m participates, using a table
the format shown in section 5.3. [The forces in which the other mass
participates will of course be similar, but not numerically the same.]

(b) Find the magnitude of the accelerations of the two masses.
[Hints: (1) Pick a coordinate system, and use positive and negative
signs consistently to indicate the directions of the forces and
accelerations. (2) The two accelerations of the two masses have to
be equal in magnitude but of opposite signs, since one side eats up
rope at the same rate at which the other side pays it out. (3) You
need to apply Newton's second law twice, once to each mass, and
then solve the two equations for the unknowns: the acceleration, a,
and the tension in the rope, T.]

(c) Many people expect that in the special case of M = m, the two
masses will naturally settle down to an equilibrium position side by
side. Based on your answer from part b, is this correct?

(d) Find the tension in the rope, T.

(e) Interpret your equation from part d in the special case where one
of the masses is zero. Here "interpret" means to figure out what happens
mathematically, figure out what should happen physically, and
connect the two.

11

A tugboat of mass m pulls a ship of mass M, accelerating it.
The speeds are low enough that you can ignore fluid friction acting
on their hulls, although there will of course need to be fluid friction
acting on the tug's propellers.

(a) Analyze the forces in which the tugboat participates, using a
table in the format shown in section 5.3. Don't worry about vertical
forces.

(b) Do the same for the ship.

(c) Assume now that water friction on the two vessels' hulls is negligible.
If the force acting on the tug's propeller is F, what is the
tension, T, in the cable connecting the two ships? [Hint: Write
down two equations, one for Newton's second law applied to each
object. Solve these for the two unknowns T and a.]

(d) Interpret your answer in the special cases of M = 0 and M = ∞.

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12

Explain why it wouldn't make sense to have kinetic friction
be stronger than static friction.

13

In the system shown in the figure, the pulleys on the left and
right are fixed, but the pulley in the center can move to the left or
right. The two masses are identical. Show that the mass on the left
will have an upward acceleration equal to g/5. Assume all the ropes
and pulleys are massless and rictionless.

14

The figure shows two different ways of combining a pair of
identical springs, each with spring constant k. We refer to the top
setup as parallel, and the bottom one as a series arrangement.

(a) For the parallel arrangement, analyze the forces acting on the
connector piece on the left, and then use this analysis to determine
the equivalent spring constant of the whole setup. Explain whether
the combined spring constant should be interpreted as being stiffer
or less stiff.

(b) For the series arrangement, analyze the forces acting on each
spring and figure out the same things.

Solution, p. 279

15

Generalize the results of problem 14 to the case where the
two spring constants are unequal.

16

(a) Using the solution of problem 14, which is given in the
back of the book, predict how the spring constant of a fiber will
depend on its length and cross-sectional area.

(b) The constant of proportionality is called the Young's modulus,
E, and typical values of the Young's modulus are about 10^{10} to
10^{11}. What units would the Young's modulus have in the SI (meterkilogram-
second) system?

Solution, p. 280

17

This problem depends on the results of problems 14 and 16,
whose solutions are in the back of the book When atoms form chemical
bonds, it makes sense to talk about the spring constant of the
bond as a measure of how "stiff" it is. Of course, there aren't really
little springs - this is just a mechanical model. The purpose of this
problem is to estimate the spring constant, k, for a single bond in
a typical piece of solid matter. Suppose we have a fiber, like a hair
or a piece of fishing line, and imagine for simplicity that it is made
of atoms of a single element stacked in a cubical manner, as shown
in the figure, with a center-to-center spacing b. A typical value for
b would be about 10^{-10} m.

(a) Find an equation for k in terms of b, and in terms of the Young's
modulus, E, defined in problem 16 and its solution.

(b) Estimate k using the numerical data given in problem 16.

(c) Suppose you could grab one of the atoms in a diatomic molecule
like H_{2} or O_{2}, and let the other atom hang vertically below it. Does
the bond stretch by any appreciable fraction due to gravity?

18

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. 280

19

Ginny has a plan. She is going to ride her sled while her
dog Foo pulls her. However, Ginny hasn't taken physics, so there
may be a problem: she may slide right off the sled when Foo starts
pulling.

(a) Analyze all the forces in which Ginny participates, making a
table as in section 5.3.

(b) Analyze all the forces in which the sled participates.

(c) The sled has mass m, and Ginny has mass M. The coefficient
of static friction between the sled and the snow is µ_{1}, and µ_{2} is
the corresponding quantity for static friction between the sled and
her snow pants. Ginny must have a certain minimum mass so that
she will not slip off the sled. Find this in terms of the other three
variables.

(d) Under what conditions will there be no solution for M?

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20

Example 2 on page 154 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. 280

21

Example 5 on page 170 describes a force-doubling setup involving
a pulley. Make up a more complicated arrangement, using
more than one pulley, that would multiply the force by a factor
greater than two.

22

Pick up a heavy object such as a backpack or a chair, and
stand on a bathroom scale. Shake the object up and down. What
do you observe? Interpret your observations in terms of Newton's
third law.

23

A cop investigating the scene of an accident measures the
length L of a car's skid marks in order to find out its speed v at
the beginning of the skid. Express v in terms of L and any other
relevant variables.

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24

The following reasoning leads to an apparent paradox; explain
what's wrong with the logic. A baseball player hits a ball. The ball
and the bat spend a fraction of a second in contact. During that
time they're moving together, so their accelerations must be equal.
176 Chapter 5 Analysis of Forces
Newton's third law says that their forces on each other are also
equal. But a = F/m, so how can this be, since their masses are
unequal? (Note that the paradox isn't resolved by considering the
force of the batter's hands on the bat. Not only is this force very
small compared to the ball-bat force, but the batter could have just
thrown the bat at the ball.)

25

Driving down a hill inclined at an angle θ with respect to
horizontal, you slam on the brakes to keep from hitting a deer.

(a) Analyze the forces. (Ignore rolling resistance and air friction.)

(b) Find the car's maximum possible deceleration, a (expressed as
a positive number), in terms of g, θ, and the relevant coefficient of
friction.

(c) Explain physically why the car's mass has no effect on your
answer.

(d) Discuss the mathematical behavior and physical interpretation
of your result for negative values of θ.

(e) Do the same for very large positive values of θ.

26

(a) Compare the mass of a one-liter water bottle on earth, on
the moon, and in interstellar space.
(b) Do the same for its weight.