Practice: A vertical spring is originally 60 cm long. When you attach a 5 kg object to it, the spring stretches to 70 cm. (a) Find the force constant on the spring. (b) You now attach an additional 10 kg to the spring. Find its new length.

Concept #1: Review: Intro to Springs

Practice: A vertical spring is originally 60 cm long. When you attach a 5 kg object to it, the spring stretches to 70 cm. (a) Find the force constant on the spring. (b) You now attach an additional 10 kg to the spring. Find its new length.

Concept #2: Work by Variable Forces & Springs

Example #1: Work by Variable Forces

Example #2: Work by Variable Forces

Practice: It takes 200 J of energy to compress a 1.0 m-long spring to 70 cm. How much work would it take to compress this same spring from 70 cm to 50 cm?

A 17000-kg jet takes off from an aircraft carrier via a catapult. The gases thrust out from the jets engines exert a constant force of 180 kN on the jet; the force exerted on the jet by the catapult is plotted in the figure b. Determine the work done on the jet by the gases expelled by its engines during launch of the jet.Determine the work done on the jet by the catapult during launch of the jet.

The net force exerted on a particle acts in the positive exttip{x}{x} direction. Its magnitude increases linearly from zero at x = 0, to 410 N at exttip{x}{x_2} = 3.1 m . It remains constant at 410 N from exttip{x}{x_2} = 3.1 m to exttip{x}{x_3} = 7.1 m , and then decreases linearly to zero at exttip{x}{x_4} = 13.1 m .Determine the work done to move the particle from x = 0 to exttip{x}{x_4} = 13.1 m graphically, by determining the area under the Fx versus exttip{x}{x} graph.

The force on a particle, acting along the x axis, varies as shown in the figure .Determine the work done by this force to move the particle along the x axis: from x = 0.0 to x = 10.0m.Determine the work done by this force to move the particle along the x axis: from x = 0.0 to x = 15.0m.

The figure is the force-versus-position graph for a particle moving along the
x-axis. Determine the work done on the particle during each of the three intervals 0-1 m,
1-2 m, and 2-3 m.

A child applies a force F parallel to the
x
-axis to a 10.0-kg sled moving on the frozen surface of a small pond. As the child controls the speed of the sled, the
x
-component of the force she applies varies with the
x
-coordinate of the sled as shown in the figure .Calculate the work done by the force F when the sled moves from x=0 to x=8.0 m.Calculate the work done by the force F when the sled moves from x=8.0 m to x =12.0 m.Calculate the work done by the force F when the sled moves from x=0 to x =12.0 m.
.

A force F is applied to a 2.0-kg radio-controlled model car parallel to the
x-axis as it moves along a straight track. The
x-component of the force varies with the
x-coordinate of the car as shown in the figure .Calculate the work done by the force F when the car moves from x=0 to x=3.0 m.Calculate the work done by the force F when the car moves from x=3.0 m to x=4.0 m.Calculate the work done by the force F when the car moves from x=4.0 m to x=7.0 m.Calculate the work done by the force F when the car moves from x=0 to x=7.0 m.Calculate the work done by the force F when the car moves from x=7.0 m to x=2.0 m.

A force is applied to a 3.5-kg radio-controlled model car parallel to the x-axis as it moves along a straight track. The x-component of the force varies with the x-coordinate of the car as shown in figure .Suppose the model car is initially at rest at x=0 and F is the net force acting on it.Use the work-energy theorem to find the speed of the car at x=3.0 m.Use the work-energy theorem to find the speed of the car at x=4.0 m.Use the work-energy theorem to find the speed of the car at x=7.0 m.

A child applies a force F
parallel to the
x-axis to a
8.00-kg
sled moving on the frozen surface of a small pond. As the child controls the speed of the sled, the
x-component of the force she applies varies with the
x-coordinate of the sled as shown in figure
. Suppose the sled is initially at rest at
x=0. You can ignore friction between the sled and the surface of the pond.You may want to review (Pages 183 - 189).
For related problemsolving tips and strategies, you may want to view a Video Tutor Solution of Motion on a curved path.Use the work-energy theorem to find the speed of the sled at
5.0
m.Use the work-energy theorem to find the speed of the sled at
11.0
m.

One end of a horizontal spring with force constant 130.0 N/m
is attached to a vertical wall. A
5.00-kg
block sitting on the floor is placed against the spring. The coefficient of kinetic friction between the block and the floor is
k = 0.400. You apply a constant force vec{F} to the block. vec{F} has magnitude
86.0 N
and is directed toward the wall. The spring is compressed 80.0 cm.What is the speed of the block?What is the magnitude of the blocks acceleration?What is the direction of the blocks acceleration?

Spiderman uses his spider webs to save a runaway train (see the figure). His web stretches a few city blocks before the 2.5×104 kg train comes to a stop.Assuming the web acts like a spring, estimate the spring constant. Assume the train is moving 25 m/s , and that the distance of "a few city blocks" is perhaps about 850 meters.

A mass m is attached to a spring which is held stretched a distance exttip{x}{x} by a force F and then released. The spring compresses, pulling the mass. Assuming there is no friction, determine the speed of the mass m when the spring returns to its normal length ( x = 0 ).Determine the speed of the mass m when the spring returns to half its original extension (x/2).

Stretchable ropes are used to safely arrest the fall of rock climbers. Suppose one end of a rope with unstretched length l is anchored to a cliff and a climber of mass m is attached to the other end. When the climber is a height l above the anchor point, he slips and falls under the influence of gravity for a distance 2l, after which the rope becomes taut and stretches a distance exttip{x}{x} as it stops the climber (see the figure). Assume a stretchy rope behaves as a spring with spring constant exttip{k}{k}.Applying the work-energy principle, find x.Assuming exttip{m}{m_0} = 86 kg , exttip{l}{l_0} = 8.6 m and exttip{k}{k_0} = 870 N/m , determine x/l (the fractional stretch of the rope) at the moment the climbers fall has been stopped.Assuming exttip{m}{m_0} = 86 kg , exttip{l}{l_0} = 8.6 m and exttip{k}{k_0} = 870 N/m , determine kx/mg (the force that the rope exerts on the climber compared to his own weight) at the moment the climbers fall has been stopped.

Three identical
8.50-kg
masses are hung by three identical springs
. Each spring has a force constant of
7.60 kN/m
and was
14.0 cm
long before any masses were attached to it.How long is the bottom spring when hanging as shown? (Hint: isolate only the bottom mass.)How long is the middle spring when hanging as shown? (Hint: treat the bottom two masses as a system.)How long is the top spring when hanging as shown? (Hint: treat all three masses as a system.)Draw a free-body diagram for the top mass.Draw a free-body diagram for the middle mass.Draw a free-body diagram for the bottom mass.

To stretch a spring 8.00 cm from its unstretched length, 16.0 J of work must be done.What is the force constant of this spring?What magnitude force is needed to stretch the spring 8.00 cm from its unstretched length?How much work must be done to compress this spring 4.00 cm from its unstretched length?What force is needed to compress it this distance?

As part of your daily workout, you lie on your back and push with your feet against a platform attached to two stiff springs arranged side by side so that they are parallel to each other. When you push the platform, you compress the springs. You do an amount of work of 81.0 J when you compress the springs a distance of 0.210 m from their uncompressed length.What magnitude of force must you apply to hold the platform in this position?How much additional work must you do to move the platform a distance 0.210 m
farther?What maximum force must you apply to move the platform to the position in Part
B?

An air-track glider of mass 0.100 kg
is attached to the end of a horizontal air track by a spring with force constant 20.0 N/m. Initially the spring is unstreched and the glider is moving at 1.50 m/s
to the right. With the air track turned off, the coefficient of kinetic friction is
k=0.47. It can be shown that
with the air track turned off, the glider travels 8.6 cm
before it
stops instantaneously.How large would the coefficient of static friction s have to be to keep the glider from springing back to the left when it stops instantaneously?If the coefficient of static friction between the glider and the track is exttip{mu_{
m s}}{mu_s} = 0.75, what is the maximum initial speed v1 that the glider can be given and still remain at rest after it stops instantaneously?

The spring of a spring gun has force constant
k = 400 N/m
and negligible mass. The spring is compressed 6.00 cm
and a ball with mass 0.0300 kg
is placed in the horizontal barrel against the compressed spring. The spring is then released, and the ball is propelled out the barrel of the gun. The barrel is 6.00 cm
long, so the ball leaves the barrel at the same point that it loses contact with the spring. The gun is held so the barrel is horizontal.Calculate the speed with which the ball leaves the barrel if you can ignore friction.Calculate the speed of the ball as it leaves the barrel if a constant resisting force of 6.00 N
acts on the ball as it moves along the barrel.For the situation in part B, at what position along the barrel does the ball have the greatest speed? (In this case, the maximum speed does not occur at the end of the barrel.)What is that greatest speed?

A
11-cm-long spring is attached to the ceiling. When a 2.1 kg mass is hung from it, the spring stretches to a length of 16 cm
.
You may want to review (Pages 219 - 221).What is the spring constant k?How long is the spring when a 3.0 kg
mass is suspended from it?

A 61 kg
student is standing atop a spring in an elevator that is accelerating upward at 3.1 m/s2 . The spring constant is 2600 N/m
.
You may want to review (Pages 219 - 221).By how much is the spring compressed?

A 7.1 kg mass hanging from a spring scale is slowly lowered onto a vertical spring, as shown in the figure.
You may want to review (Pages 219 - 221).What does the spring scale read just before the mass touches the lower spring?The scale reads 22 N when the lower spring has been compressed by 2.1 cm . What is the value of the spring constant for the lower spring?At what compression length will the scale read zero?

Two identical horizontal springs are attached to opposite sides of a box that sits on a frictionless table. The outer ends of the springs are clamped while the springs are at their equilibrium lengths. Then a
2.6 N
force applied to the box, parallel to the springs, compresses one spring by
3.1 cm
while stretching the other by the same amount.What is the spring constant of the springs?

A horizontal spring with spring constant
650 N/m
is attached
to a wall. An athlete presses against the free end of the spring,
compressing it 7.0
cm.How hard is the athlete pushing?

The left end of a spring is attached to a wall. When Bob pulls on the right end with a 200 N
force, he stretches the spring by 20 cm. The same spring is then used for a tug-of-war between Bob and Carlos. Each pulls on his end of the spring with a 200 N
force.How far does the spring stretch?

How much work is done by the following force over the 12 m displacement?

If it requires 5.5 J of work to stretch a particular spring by 2.0 cm from its equilibrium length, how much more work will be required to stretch it an additional 4.1 cm?

When a 4.00-kg object is hung vertically on a certain light spring that obeys Hooke’s law, the spring stretches 2.50 cm. If the 4.00-kg object is removed, (a) how far will the spring stretch if a 1.50-kg block is hung on it? (b) How much work must an external agent do to stretch the same spring 4.00 cm from its unstretched position?

A 4.00-kg particle is subject to a net force that varies with position as shown in the figure. The particle starts from rest at x 5 0. What is its speed at (a) x = 5.00 m, (b) x = 10.0 m, and (c) x = 15.0 m?

A 38-cm-long vertical spring has one end fixed on the floor. Placing a 2.2 kg physics textbook on the spring compresses it to a length of 29 cm. What is the spring constant?

A spring has an unstretched length of 10 cm. It exerts a restoring force F when stretched to a length of 11 cm.(a) For what length of the spring is its restoring force 3F?(b) At what compressed length is the restoring force 2F?

The force acting on a particle varies as shown in Figure P7.14. Find the work done by the force on the particle as it moves (a) from x = 0 to x = 8.00 m, (b) from x = 8.00 m to x = 10.0 m, and (c) from x = 0 to x = 10.0 m.

A particle is subject to a force Fx that varies with position as shown in Figure P7.15. Find the work done by the force on the particle as it moves (a) from x = 0 to x = 5.00 m, (b) from x = 5.00 m to x = 10.0 m, and (c) from x = 10.0 m to x = 15.0 m. (d) What is the total work done by the force over the distance x = 0 to x = 15.0 m?

An archer pulls her bowstring back 0.400 m by exerting a force that increases uniformly from zero to 230 N. (a) What is the equivalent spring constant of the bow? (b) How much work does the archer do on the string in drawing the bow?