Mechanics Problems - Force Problems
Note: Each problem begins with a list of forces
necessary to solve the context-rich problem. These are for the
benefit of the instructor. Delete the list before using the
problems in your class.
- Tension, WeightSOLUTION :FOR THE FOLLOWING PLAN
THE PROBLEM. An artist friend of yours wants your opinion
of his idea for a new kinetic sculpture. The basic
concept is to balance a heavy object with two lighter
objects using two very light pulleys, which are
essentially frictionless, and lots of string. The
sculpture has one pulley hanging from the ceiling by a
string attached to its center. Another string passes over
this pulley. One end of this string is attached to a 25
lb object while the other supports another pulley at its
center. This second pulley also has a string passing over
it with one end attached to a 10 lb object and the other
to a 15 lb object. Your friend hasn't quite figured out
the rest of the sculpture but wants to know if, ignoring
the mass of the pulley and string, the 25 lb object will
remain stationary during the time that the 10 and 15 lb
objects are accelerating. DO NOT SOLVE THE PROBLEM.
- Weight, Normal: You have always been impressed by the
speed of the elevators in the IDS building in Minneapolis
(especially compared to the one in the Physics building).
You wonder about the maximum acceleration for these
elevators during normal operation, so you decide to
measure it by using your bathroom scale. While the
elevator is at rest on the ground floor, you get in, put
down your scale, and stand on it. The scale reads 130
lbs. You continue standing on the scale when the elevator
goes up, carefully watching the reading. During the trip
to the 50th floor, the greatest scale reading was 180
lbs.
- You are designing a lamp for the interior of a special
executive express elevator in a new office building. The lamp has two
sections that hang one directly below the other. The bottom section is
attached to the top one by a single thin wire and the upper section is
attached to the ceiling by another single thin wire. Because the idea is
to make each section appear to be floating without support, you want to
use the thinnest (and thus weakest) wire possible. You decide to
calculate the force each wire must exert on the lamp sections in case of
an emergency stop. The elevator has all the latest safety features and
will stop with an acceleration of g/3 in any emergency. Each section of
the lamp weighs 7.0 N.
- You are investigating an elevator accident which happened
in a tall building. An elevator in this building is
attached to a strong cable which runs over a pulley
attached to a steel support in the roof. The other end of
the cable is attached to a block of metal called a
counterweight which hangs freely. An electric motor on
the side of the elevator drives the elevator up or down
by exerting a force on the side of the elevator shaft.
You suspect that when the elevator was fully loaded,
there was too large a force on the motor . A fully loaded
elevator at maximum capacity weighs 2400 lbs. The
counterweight weighs 1000 lbs. The elevator always starts
from rest at its maximum acceleration of g/4 whether it
is going up or down. (a) What force does the wall of the
elevator shaft exert on the motor if the elevator starts
from rest and goes up? (b) What force does the wall of
the elevator shaft exert on the motor if the elevator
starts from rest and goes down?
- Tension, Weight: An artist friend of yours wants your
opinion of his idea for a new kinetic sculpture. The
basic concept is to balance a heavy object with two
lighter objects using two very light pulleys, which are
essentially frictionless, and lots of string. The
sculpture has one pulley hanging from the ceiling by a
string attached to its center. Another string passes over
this pulley. One end of this string is attached to a
25-lb object while the other supports another pulley at
its center. This second pulley also has a string passing
over it with one end attached to a 10-lb object and the
other to a 15-lb object. Your friend hasn't quite figured
out the rest of the sculpture but wants to know if,
ignoring the mass of the pulley and string, the 25-lb
object will remain stationary during the time that the
10-lb and 15-lb objects are accelerating. DO ONLY THE
PROBLEM SOLVING STEPS NECESSARY TO FOCUS THE PROBLEM,
DESCRIBE THE PHYSICS OF THE PROBLEM, AND PLAN A SOLUTION.
DO NOT SOLVE THIS PROBLEM.
- Weight, Normal, Friction: Because of your physics background,
you have been asked to check the feasibility of a action movie stunt. In
the script, the hero and villain are fighting on top of a locomotive
heading down a straight track at 25 mph. Having jumped on the train as
it passed over a lake, the hero is dressed in a rubber wet suit. During
the fight, the hero slips off and barely hangs on over the top edge of
the front of the locomotive, which is essentially a vertical smooth
steel face. The villain stomps on the hero's fingers to cause the hero
to let go and be crushed under the train. Meanwhile, the hero's partner
has been trying to stop the train, whose brakes have been locked by the
villain. Seeing the hero’s fingers give way, the partner immediately
opens the throttle, causing the train to accelerate forward and the hero
to stay on the front face of the locomotive without slipping down until
the brakes can be unlocked. The movie company wants to know what minimum
acceleration is necessary to perform this stunt. The hero weighs 180
lbs. and the locomotive weighs 100 tons. Looking in a book giving the
properties of materials, you find that for rubber on steel, the
coefficient of kinetic friction is 0.50 and the coefficient of static
friction is 0.60.
- Weight, Normal, Friction: While working in a mechanical
structures laboratory, your boss assigns you to test the
strength of ropes under different conditions. Your test
set-up consists of two ropes attached to a 30 kg block
which slides on a 5.0 m long horizontal table top. Two
low friction, light weight pulleys are mounted at
opposite ends of the table. One rope is attached to each
end of the 30 kg block. Each of these ropes runs
horizontally over a different pulley. The other end of
one of the ropes is attached to a 12 kg block which hangs
straight down. The other end of the second rope is
attached to a 20 kg block also hanging straight down. The
coefficient of kinetic friction between the block on the
table and the table's surface is 0.08. The 30 kg block is
initially held in place by a mechanism that is released
when the test begins so, that the block is accelerating
during the test. During this test, what is the force
exerted on the rope supporting the 12 kg block?
Note: Each problem begins with a list of forces
necessary to solve the context-rich problem. These are for the
benefit of the instructor. Delete the list before using the
problems in your class.
- Human, Weight, Normal: You are taking care of two small
children, Sarah and Rachel, who are twins. On a nice
cold, clear day you decide to take them ice skating on
Lake of the Isles. To travel across the frozen lake you
have Sarah hold your hand and Rachel's hand. The three of
you form a straight line as you skate, and the two
children just glide. Sarah must reach up at an angle of
60 degrees to grasp your hand, but she grabs Rachel's
hand horizontally. Since the children are twins, they are
the same height and the same weight, 50 lbs. To get
started you accelerate at 2.0 m/s2. You are concerned
about the force on the children's arms which might cause
shoulder damage. So you calculate the force Sarah exerts
on Rachel's arm, and the force you exert on Sarah's other
arm. You assume that the frictional forces of the ice
surface on the skates are negligible.
- Tension, Weight, Normal, and Friction: You are planning to
build a log cabin and will need to pull the logs up a hill to the
building site by means of a rope attached to a winch. In order to buy
the rope, you need to know how strong the rope must be and decide to do
a quick calculation for this. The logs weigh 500 lbs. at most and the
hill is at an angle of 30° with respect to the horizontal. You estimate
that the coefficient of kinetic friction between a log and the hill is
0.90. When pulling a log up the hill, you will make sure that the rope
stays parallel to the surface of the hill and the acceleration of the
log is never more than 0.80 m/s^2.
- Tension, Weight, Normal, Friction: At your job at a
warehouse, you have designed a method to help get heavy packages up a
15º ramp. The package is attached to a rope that runs parallel to the
ramp and passes over a pulley at the top of the ramp. The other end of
the rope is attached to a counterweight that hangs straight down. The
mass of the counterweight is always adjusted to be twice the mass of the
package. However, your boss is worried that the acceleration of the
package will make it too difficult to handle at the top of the ramp and
tells you to calculate it. To determine the influence of friction on the
package by the ramp, you run some tests and find that using a horizontal
force of 250 Newtons, you can push a 50 kg package at a constant speed
along a level floor made of the same material as the ramp.
- Tension, Weight, Normal, Friction: After graduating you
get a job in Northern California. To move there, you rent
a truck for all of your possessions. You also decide to
take your car with you by towing it behind the truck. The
instructions you get with the truck tells you that the
maximum truck weight when fully loaded is 20,000 lbs and
that the towing hitch that you rented has a maximum
strength of 1000 lbs. Just before you leave, you weigh
the fully loaded truck and find it to be 15,000 lbs. At
the same time you weigh your car and find it to weigh
3000 lbs. You begin to worry if the hitch is strong
enough. Then you remember that you can push your car and
can easily keep it moving at a constant velocity. You
know that air resistance will increase as the car goes
faster but from your experience you estimate that the sum
of the forces due to air resistance and friction on the
car is not more than 300 lbs. If the largest hill you
have to go up is sloped at 10o from the horizontal, what
is the maximum acceleration you can safely have on that
hill? DO ONLY THE PROBLEM SOLVING STEPS NECESSARY TO
FOCUS THE PROBLEM, DESCRIBE THE PHYSICS OF THE PROBLEM,
AND PLAN A SOLUTION. DO NOT SOLVE THIS PROBLEM.
- Weight, Normal, Friction: Because of your physics
background, you have been able to get a job with a
company devising stunts for an upcoming adventure movie
being shot in Minnesota. In the script, the hero has been
fighting the villain on the top of the locomotive of a
train going down a straight horizontal track at 20 mph.
He has just snuck on the train as it passed over a lake
so he is wearing his rubber wet suit. During the fight,
the hero slips and hangs by his fingers on the top edge
of the front of the locomotive. The locomotive has a
smooth steel front face sloped at 20o from the vertical
so that the bottom of the front is more forward that the
top. Now the villain stomps on the hero's fingers so he
will be forced to let go and slip down the front of the
locomotive and be crushed under its wheels. Meanwhile,
the hero's partner is at the controls of the locomotive
trying to stop the train. To add to the suspense, the
brakes have been locked by the villain. It will take her
10 seconds to open the lock. To her horror, she sees the
hero's fingers give way before she can get the lock off.
Since she is the brains of the outfit, she immediately
opens the throttle causing the train to accelerate
forward. This causes the hero to stay on the front face
of the locomotive without slipping down giving her time
to save the hero's life. The movie company wants to know
what minimum acceleration is necessary to perform this
stunt. The hero weighs 180 lbs in his wet suit. The
locomotive weighs 100 tons. You look in a book giving the
properties of materials and find that the coefficient of
kinetic friction for rubber on steel is 0.50 and its
coefficient of static friction is 0.60.
- Gravitational: You have been hired as a consultant for
the new Star Trek TV series to make sure that any science
on the show is correct. In this episode, the crew of the
Enterprise discovers an abandoned space station in deep
space far from any stars. This station is obviously the
work of an advanced race and consists of four identical 3
x 1020 kg asteroids configured so that each is at the
corner of a square with 200 km sides. According to the
tricorder, the station has been abandoned for at least
two centuries. You know that such a configuration is
unstable and worry whether there would be observable
motion of the asteroids after two hundred years so you
calculate the acceleration of one of the asteroids in the
proposed configuration. Make sure you give both the
magnitude and the direction of the acceleration.
- Gravitational: Because the movie industry is trying to
make the technical details of movies as correct as
possible, you have been made a member of a panel
reviewing the details of a new science fiction script.
Although neither astronomy nor navigation is your field,
you are disturbed by one scene in which a space ship
which is low on fuel is attempting to land on the Earth.
As the ship approaches, it is heading straight for the
center of the Earth. The commander cuts off the ship's
engines so that it will be pulled in by the Earth's
gravitational force. As the commander looks in the
viewer, she sees the Earth straight ahead and the Moon
off to the left at an angle of 30o. The line between the
centers of the Moon and Earth is at right angles to the
initial path of the space ship. Under these conditions
you don't think the ship will continue heading toward the
Earth, so you calculate the component of its acceleration
which is perpendicular to the initial path of the ship.
First you look up the distance between the Earth and the
Moon (3.8 x 105 km), the mass of the Earth (6.0 x 1024
kg), the mass of the Moon (7.3 x 1022 kg), the radius of
the Earth (6.4 x 103 km), the radius of the Moon (1.7 x
103 km), and the universal gravitational constant (6.7 x
10-11 N m2/kg2). As a first approximation, you decide to
neglect the effect of the Sun and the other planets in
the solar system. You guess that a space ship such as
described in the script might have a mass of about
100,000 kg.
Note: Each problem begins with a list of forces
necessary to solve the context-rich problem. These are for the
benefit of the instructor. Delete the list before using the
problems in your class.
- Weight - Buoyancy, Normal, Friction, Electric: The
quarter is almost over so you decide to have a party. To
add atmosphere to your otherwise drab apartment, you
decide to decorate with balloons. You buy about fifty and
blow them up so that they are all sitting on your carpet.
After putting most of them up, you decide to play with
the few balloons left on the floor. You rub one on your
sweater and find that it will "stick" to a
wall. Ah ha, you know immediately that you are observing
the electric force in action. Since it will be some time
before you guests arrive and you have already made the
onion dip, you decide to calculate the minimum electric
force of the wall on the balloon. You know that the air
exerts a net upward force (the "buoyant" force)
on the balloon which makes it almost float. You measure
that the weight of the balloon minus the buoyant force of
the air on the balloon is 0.05 lb. By reading your
physics book, you estimate that the coefficient of static
friction between the wall and the balloon (rubber and
concrete) is 0.80.
- Tension, Weight, Electric: While working in a
University research laboratory you are given the job of
testing a new device for precisely measuring the weight
of small objects. The device consists of two very light
strings attached at one end to a support. An object is
attached to the other end of each string. The strings are
far enough apart so that objects hanging on them don't
touch. One of the objects has a very accurately known
weight while the other object is the unknown. A power
supply is slowly turned on to give each object an
electric charge which causes the objects to slowly move
away from each other (repel) because of the electric
force. When the power supply is kept at its operating
value, the objects come to rest at the same horizontal
level. At that point, each of the strings supporting them
makes a different angle with the vertical and that angle
is measured. To test the device, you want to calculate
the weight of an unknown sphere from the measured angles
and the weight of a known sphere. You use a standard
sphere with a known weight of 2.000 N supported by a
string which makes an angle of 10.0º with the vertical.
The unknown sphere's string makes an angle of 20.0º with
the vertical.
- Gravitational: You are writing a short science fiction
story for your English class. You get your idea from the
fact that when people cross the Earth's equator for the
first time, they are awarded a certificate to commemorate
the experience. In your story it is the 21st Century and
you are the tour director for a trip to the moon.
Transplanetary Tours promises tour participants a
certificate to commemorate their passage from the
stronger influence of the Earth's gravitational pull to
the stronger gravitational pull of the moon. To finish
the story, you need to figure out where on the trip you
should award the certificate. In your physics book you
look up the distance between the Earth and the Moon (3.8
x 105 km), the mass of the Earth (6.0 x 1024 kg), the
mass of the Moon (7.3 x 1022 kg), the radius of the Earth
(6.4 x 103 km), the radius of the Moon (1.7 x 103 km),
and the universal gravitational constant (6.7 x 10-11 N
m2/kg2).
- Gravitational: You have been hired as a consultant for
the new Star Trek TV series to make sure that the science
in the show is correct. In this episode, the crew of the
Enterprise goes into standard orbit around a newly
discovered planet. The plot requires that the planet is
hollow and contains the underground cities of a lost
civilization. From orbit the science officer determines
that the radius of the planer is 1/4 (one-fourth) that of
Earth. The first officer beams down to the surface of the
planet and measures that his weight is only 1/2
(one-half) of his weight on Earth. How does the mass of
this planet compare with the mass of the Earth? If it
were hollow, its density would be less than Earth. Are
the measurements consistent with a hollow planet?
- Gravitational, Electric: You and a friend are reading a
newspaper article about nuclear fusion energy generation
in stars. The article describes the helium nucleus, made
up of two protons and two neutrons, as very stable so it
doesn't decay. You immediately realize that you don't
understand why the helium nucleus is stable. You know
that the proton has the same charge as the electron
except that the proton charge is positive. Neutrons you
know are neutral. Why, you ask your friend, don't the
protons simply repel each other causing the helium
nucleus to fly apart? Your friend says she knows why the
helium nucleus does not just fly apart. The gravitational
force keeps it together, she says. Her model is that the
two neutrons sit in the center of the nucleus and
gravitationally attract the two protons. Since the
protons have the same charge, they are always as far
apart as possible on opposite sides of the neutrons. What
mass would the neutron have if this model of the helium
nucleus works? Is that a reasonable mass? Looking in your
physics book, you find that the mass of a neutron is
about the same as the mass of a proton and that the
diameter of a helium nucleus is 3.0 x 10-13 cm.
Note: Each problem begins with a list of forces
necessary to solve the context-rich problem. These are for the
benefit of the instructor. Delete the list before using the
problems in your class.
- Tension, Weight, Friction: You are taking
advantage of an early snow to go sledding. After a long
afternoon of going up and down hills with your sled, you
decide it is time to go home. You are thankful that you
can pull your sled without climbing any more hills. As
you are walking home, dragging the sled behind you by a
rope fastened to the front of the sled, you wonder what
the coefficient of friction of the snow on the sled is.
You estimate that you are pulling on the rope with a 2
pound force, that the sled weighs 10 pounds, and that the
rope makes an angle of 25 degrees to the level ground.
- Human, Weight, Normal, Friction: You are helping a friend
move into a new apartment. A box weighing 150 lbs needs
to be moved to make room for a couch.. You are taller
than the box, so you reach down to push it at an angle of
50 degrees from the horizontal. The coefficient of static
friction between the box and the floor is 0.50 and the
coefficient of kinetic friction between the box and the
floor is 0.30. (a) If you want to exert the minimum force
necessary, how hard would you push to keep the box moving
across the floor? (b) Suppose you bent your knees so that
your push were horizontal. How hard would you push to
keep the box moving across the floor?
- Human, Weight, Normal, Friction: You are helping an
investigation of back injuries in the construction
industry. Your assignment is to determine why there is a
correlation of the height of the worker to the likelihood
of back injury. You suspect that some back injuries are
related to the way people push heavy objects in order to
move them. When people push an object, such as a box,
across the floor they tend to lean down and push at an
angle to the horizontal. Taller people push at a larger
angle with respect to the horizontal than shorter people.
To present your ideas to the rest of the research team,
you decide to calculate the force a 200-lb box exerts on
a 150-lb person when they push it across a typical floor
at a constant velocity of 7.0 ft/s as a function of the
angle with respect to the horizontal at which the person
pushes the box. Once you have your function, you will use
angles of 0o, 10o, 20o, 30o, and 40o to make a graph of
the result for the presentation. One of your coworkers
tells you that a typical coefficient of static friction
between a box and a floor of 0.60 and while a typical
coefficient of kinetic friction between a box and a floor
is 0.50. (Don't forget to make the graph).
- Tension, Weight: Your are part of a team to help design
the atrium of a new building. Your boss, the manager of
the project, wants to suspend a 20-lb sculpture high over
the room by hanging it from the ceiling using thin, clear
fishing line (string) so that it will be difficult to see
how the sculpture is held up. The only place to fasten
the fishing line is to a wooden beam which runs around
the edge of the room at the ceiling. The fishing line
that she wants to use will hold 20 lbs (20-lb test) so
she suggests attaching two lines to the sculpture to be
safe. Each line would come from the opposite side of the
ceiling to attach to the hanging sculpture. Her initial
design has one line making an angle of 20o with the
ceiling and the other line making an angle of 40o with
the ceiling. She knows you took physics, so she asks you
if her design can work.
- Electric, Weight, Tension: While working in a University
research laboratory you are given the job of testing a
new device, called an electrostatic scale, for precisely
measuring the weight of small objects. The device is
quite simple. It consists of two very light but strong
strings attached to a support so that they hang straight
down. An object is attached to the other end of each
string. One of the objects has a very accurately known
weight while the other object is the unknown. A power
supply is slowly turned on to give each object an
electric charge which causes the objects to slowly move
away from each other (repel) because of the electric
force. When the power supply is kept at its operating
value, the objects come to rest at the same horizontal
level. At that point, each of the strings supporting them
makes a different angle with the vertical and that angle
is measured. To test the device, you want to calculate
the weight of an unknown sphere from the measured angles
and the weight of a known sphere. You use a standard
sphere with a known weight of 2.00000 N supported by a
string which makes an angle of 10.00o with the vertical.
The unknown sphere's string makes an angle of 20.00o with
the vertical.