You are trying to loosen a stuck bolt on your RV using a big
wrench that is 50 cm long. If you hang from the wrench, and your
mass is 55 kg, what is the maximum torque you can exert on the
A physical therapist wants her patient to rehabilitate his injured
elbow by laying his arm flat on a table, and then lifting a 2.1 kg
mass by bending his elbow. In this situation, the weight is 33 cm
from his elbow. He calls her back, complaining that it hurts him
to grasp the weight. He asks if he can strap a bigger weight onto
his arm, only 17 cm from his elbow. How much mass should she
tell him to use so that he will be exerting the same torque? (He is
raising his forearm itself, as well as the weight.)
An object thrown straight up in the air is momentarily at rest
when it reaches the top of its motion. Does that mean that it is in
equilibrium at that point? Explain.
An object is observed to have constant angular momentum.
Can you conclude that no torques are acting on it? Explain.
on a problem by Serway and Faughn.]
A person of weight W stands on the ball of one foot. Find the
tension in the calf muscle and the force exerted by the shinbones
on the bones of the foot, in terms of W, a, and b. For simplicity,
assume that all the forces are at 90-degree angles to the foot, i.e.,
neglect the angle between the foot and the floor.
Two objects have the same momentum vector. Assume that
they are not spinning; they only have angular momentum due to
their motion through space. Can you conclude that their angular
momenta are the same? Explain.
[Based on a problem by Serway
The sun turns on its axis once every 26.0 days. Its mass is
2.0 × 1030 kg and its radius is 7.0 × 108 m. Assume it is a rigid
sphere of uniform density.
(a) What is the sun's angular momentum?
In a few billion years, astrophysicists predict that the sun will use
up all its sources of nuclear energy, and will collapse into a ball of
exotic, dense matter known as a white dwarf. Assume that its radius
becomes 5.8 × 106 m (similar to the size of the Earth.) Assume it
does not lose any mass between now and then. (Don't be fooled by
the photo, which makes it look like nearly all of the star was thrown
off by the explosion. The visually prominent gas cloud is actually
thinner than the best laboratory vacuum every produced on earth.
Certainly a little bit of mass is actually lost, but it is not at all
unreasonable to make an approximation of zero loss of mass as we
(b) What will its angular momentum be?
(c) How long will it take to turn once on its axis?
A uniform ladder of mass m and length L leans against a
smooth wall, making an angle q with respect to the ground. The dirt
exerts a normal force and a frictional force on the ladder, producing
a force vector with magnitude F1 at an angle Φ with respect to the
ground. Since the wall is smooth, it exerts only a normal force on
the ladder; let its magnitude be F2.
Problems 8 and 9
(a) Explain why Φ must be greater than θ. No math is needed.
(b) Choose any numerical values you like for m and L, and show
that the ladder can be in equilibrium (zero torque and zero total
force vector) for θ = 45.00 ° and Φ = 63.43 °.
Continuing the previous problem, find an equation for Φ in
terms of θ, and show that m and L do not enter into the equation.
Do not assume any numerical values for any of the variables. You
will need the trig identity sin(a - b) = sin a cos b - sin b cos a. (As
a numerical check on your result, you may wish to check that the
angles given in part b of the previous problem satisfy your equation.)
(a) Find the minimum horizontal force which, applied at the
axle, will pull a wheel over a step. Invent algebra symbols for whatever
quantities you find to be relevant, and give your answer in
symbolic form. [Hints: There are four forces on the wheel at first,
but only three when it lifts off. Normal forces are always perpendicular
to the surface of contact. Note that the corner of the step
cannot be perfectly sharp, so the surface of contact for this force
really coincides with the surface of the wheel.]
(b) Under what circumstances does your result become infinite?
Give a physical interpretation.
A yo-yo of total mass m consists of two solid cylinders of radius
R, connected by a small spindle of negligible mass and radius r. The
top of the string is held motionless while the string unrolls from the
spindle. Show that the acceleration of the yo-yo is g/(1 + R2/2r2).
[Hint: The acceleration and the tension in the string are unknown.
Use τ = ΔL/Δt and F = ma to determine these two unknowns.]
A ball is connected by a string to a vertical post. The ball is
set in horizontal motion so that it starts winding the string around
the post. Assume that the motion is confined to a horizontal plane,
i.e., ignore gravity. Michelle and Astrid are trying to predict the
final velocity of the ball when it reaches the post. Michelle says
that according to conservation of angular momentum, the ball has
to speed up as it approaches the post. Astrid says that according to
conservation of energy, the ball has to keep a constant speed. Who
is right? [Hint: How is this different from the case where you whirl
a rock in a circle on a string and gradually pull in the string?]
In the 1950's, serious articles began appearing in magazines
like Life predicting that world domination would be achieved by the
nation that could put nuclear bombs in orbiting space stations, from
which they could be dropped at will. In fact it can be quite difficult
to get an orbiting object to come down. Let the object have energy
E = KE +PE and angular momentum L. Assume that the energy
is negative, i.e., the object is moving at less than escape velocity.
Show that it can never reach a radius less than
[Note that both factors are negative, giving a positive result.]
Two bars of length L are connected with a hinge and placed on
a frictionless cylinder of radius r.
(a) Show that the angle θ shown
in the figure is related to the unitless ratio r/L by the equation
(b) Discuss the physical behavior of this equation for very large and
very small values of r/L.
You wish to determine the mass of a ship in a bottle without
taking it out. Show that this can be done with the setup shown in
the figure, with a scale supporting the bottle at one end, provided
that it is possible to take readings with the ship slid to two different
Two atoms will interact via electrical forces between their protons
and electrons. One fairly good approximation to the potential
energy is the Lennard-Jones potential,
where r is the center-to-center distance between the atoms.
Show that (a) there is an equilibrium point at r = a, (b) the equilibrium
is stable, and (c) the energy required to bring the atoms
from their equilibrium separation to infinity is k. [Hints: The first
two parts can be done more easily by setting a = 1, since the value
of a only changes the distance scale. One way to do part b is by
Suppose that we lived in a universe in which Newton's law
of gravity gave forces proportional to r-7 rather than r-2. Which,
if any, of Kepler's laws would still be true? Which would be completely
false? Which would be different, but in a way that could be
calculated with straightforward algebra?
The figure shows scale drawing of a pair of pliers being used to
crack a nut, with an appropriately reduced centimeter grid. Warning:
do not attempt this at home; it is bad manners. If the force
required to crack the nut is 300 N, estimate the force required of
the person's hand.
Solution, p. 161
Show that a sphere of radius R that is rolling without slipping
has angular momentum and momentum in the ratio L/p = (2/5)R.
Suppose a bowling ball is initially thrown so that it has no
angular momentum at all, i.e., it is initially just sliding down the
lane. Eventually kinetic friction will bring its angular velocity up to
the point where it is rolling without slipping. Show that the final
velocity of the ball equals 5/7 of its initial velocity. [Hint: You'll
need the result of problem 21.]
The rod in the figure is supported by the finger and the string.
(a) Find the tension, T, in the string, and the force, F, from the
finger, in terms of m, b,L, and g.
(b) Comment on the cases b = L and b = L/2.
(c) Are any values of b unphysical?
Two horizontal tree branches on the same tree have equal
diameters, but one branch is twice as long as the other. Give a
quantitative comparison of the torques where the branches join the
trunk. [Thanks to Bong Kang.]
(a) Alice says Cathy's body has zero momentum, but Bob
says Cathy's momentum is nonzero. Nobody is lying or making a
mistake. How is this possible? Give a concrete example.
(b) Alice and Bob agree that Dong's body has nonzero momentum,
but disagree about Dong's angular momentum, which Alice says is
zero, and Bob says is nonzero. Explain.
Penguins are playful animals. Tux the Penguin invents a new
game using a natural circular depression in the ice. He waddles at
top speed toward the crater, aiming off to the side, and then hops
into the air and lands on his belly just inside its lip. He then bellysurfs,
moving in a circle around the rim. The ice is frictionless, so
his speed is constant. Is Tux's angular momentum zero, or nonzero?
What about the total torque acting on him? Take the center of the
crater to be the axis. Explain your answers.
Make a rough estimate of the mechanical advantage of the
lever shown in the figure. In other words, for a given amount of
force applied on the handle, how many times greater is the resulting
force on the cork?