Download Skill Sheet 5.1 Isaac Newton

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Skill Sheet 5.1
Isaac Newton
Isaac Newton is one of the most brilliant figures in scientific history. His three laws of motion are probably
the most important natural laws in all of science. He also made vital contributions to the fields of optics,
calculus, and astronomy.
Isaac Newton was born in 1642 in Lincolnshire, England. His childhood years were
difficult. His father died just before he was born, and when he was 3, his mother
remarried and left her son to live with his grandparents. Newton bitterly resented his
stepfather throughout his life.
An uncle helped Newton remain in school and in 1661, he entered Trinity College at
Cambridge University. He earned his bachelor’s degree in 1665.
Ironically, it was the closing of the university due to the bubonic plague in 1665 that
ushered in the blossoming of Newton’s genius. He returned to Lincolnshire and spent
the next two years in solitary academic pursuit. During this period, he made significant
advances in calculus, worked on a revolutionary theory of the nature of light and color,
developed early versions of his three laws of motion, and gained new insights into the nature of planetary motion.
When Cambridge reopened in 1667, Newton was given a minor position at Trinity and began his academic
career. His studies in optics led to his invention of the reflecting telescope in the early 1670s. In 1672, his first
public paper was presented, on the nature of light and color. Newton longed for public recognition of his work
but dreaded criticism. When another bright young scientist, Robert Hooke, challenged some of his points,
Newton was incensed. An angry exchange of words left Newton reluctant to make public more of his work.
In the 1680s, Newton turned his attention to forces and motion. He worked on applying his three laws of motion
to orbiting bodies, projectiles, pendulums, and free-fall situations. This work led him to formulate his famous
universal law of gravitation.
This concept was truly revolutionary. Less than 50 years earlier, it was commonly believed that some sort of
invisible shield held the planets in orbit. Newton’s law explained that it was the gravitational force between the
sun and the planets that is responsible.
In 1687, Newton published his ideas in a famous work known as the Principia. He jealously guarded the work as
entirely his. He bitterly resented the suggestion that he should acknowledge the exchange of ideas with other
scientists (especially Hooke) as he worked on his treatise.
Newton left Cambridge to take a government position in London in 1696. His years of active scientific research
were over. However, almost three centuries after his death in 1727, Newton remains one of the most important
contributors to our understanding of how the universe works.
Questions
1.
2.
Research the legend of Newton’s apple. Which of Newton’s laws does it help explain?
Newton was outraged when, in 1684, German mathematician Wilhelm Leibniz published a calculus book.
Find out why, and describe how the issue is generally resolved today.
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Skill Sheet 5.2
Newton's Second Law
As you work through the problems here, you will develop your understanding of Newton’s second law of
motion and how it relates to Newton’s first law of motion. The second law states that the acceleration of an
object is directly proportional to the force acting on the object and indirectly proportional to the mass of the
object.
1. Newton’s first law of motion
Newton's first law of motion (the law of inertia) states that the motion of an object will continue until an outside
force changes it. The amount of force needed to change the motion of an object depends on the amount of inertia
the object has. The inertia of an object is related to its mass. You need more force to move or stop an object with
a lot of mass or inertia than you need for an object with less mass or inertia.
In Newton's second law, the acceleration of an object is directly related to the force on it, and inversely related to
the mass of the object. This is shown in the formula below:
acceleration = force
-----------mass
Units for acceleration are m/sec2. Units for force are newtons (N). One newton is equivalent to 1 kg-m/sec2.
Units for mass are kilograms (kg). The equation for acceleration illustrates that acceleration is equal to the ratio
of force to mass. This means that the force on an object causes it to accelerate, but the object’s mass is a measure
of how much it will resist acceleration.
2. Three ways to write Newton’s second law of motion
In the formula for the second law of motion, acceleration equals force divided by mass. What does mass equal?
What does force equal? Rearrange the equation to solve for mass. Then rearrange the equation to solve for force.
What do you want to know?
What do you know?
acceleration (a)
force (F) and mass (m)
mass (m)
acceleration (a) and force (F)
force (F)
acceleration (a) and mass (m)
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The formula you will use
acceleration = force
-----------mass
Skill Sheet 5.2 Newton's Second Law
3. Solving problems using Newton’s second law
Solve the following problems using Newton’s second law. The first two problems are done for you.
1.
How much force is needed to accelerate a truck with a mass of 2,000 kilograms at a rate of 3m/sec2?
3m
m
F = m × a = 2,000 kg × ---------2- = 6,000 kg ⋅ ---------2- = 6,000 N
sec
sec
2.
What is the mass of an object that requires 15 N to accelerate it at a rate of 1.5 m/sec2?
15 kg-m
-------------------2
sec - = 10 kg
15 N = ------------------m = F
--- = ------------a
1.5 m1.5
m----------------------2
2
sec
sec
3.
What is the rate of acceleration of a 2,000.-kilogram truck if a force of 4,200 N is used to make it start
moving forward?
4.
What is the acceleration of a 0.30 kilogram ball that is hit with a force of 25 N?
5.
How much force is needed to accelerate a 68 kilogram-skier at a rate of 1.2 m/sec2?
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Skill Sheet 5.2 Newton's Second Law
6.
What is the mass of an object that requires a force of 30 N to accelerate at a rate of 5 m/sec2?
7.
What is the force on a 1,000 kilogram-elevator that is falling freely under the acceleration of gravity only?
8.
What is the mass of an object that needs a force of 4,500 N to accelerate it at a rate of 5 m/sec2?
9.
What is the acceleration of a 6.4 kilogram bowling ball if a force of 12 N is applied to it?
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Skill Sheet 6.1
Mass and Weight
The mass of an object is a characteristic property and doesn't change unless the object changes. Weight is the
result of gravitational attraction and depends on the strength of that attraction. You can calculate changes in
weight based on changes in the force of gravity.
1. Mass, weight, and gravity
The key to these problems is to remember that gravity is a force that acts on mass. The result of that interaction is
a force that we call weight. Mass is always measured in kilograms, but weight is measured in force units, or
newtons.
This is yet another area of confusion within the English system of measurement because pounds are a force unit.
Here is the conversion for pounds force to newtons.
1 pound = 4.45 newtons
The problem is that we also convert pounds into kilograms, the unit of mass. Here is the conversion for pounds
weight to kilograms. You can see that the conversion constants are very similar numerically.
1 pound = 0.454 kilograms
It’s essential to keep these units clear. For that reason, these skill problems will be done in International System
units, newtons and kilograms. Weight in newtons is calculated by multiplying mass (in kilograms) by the
acceleration of gravity. On Earth, the acceleration of gravity is 9.8 meters/second/second. On other planets, the
acceleration of gravity is different.
Example:
What is the weight of a girl with a mass of 50 kilograms in a space station with an artificial gravity of 7 N/kg?
F w = mg
F w = 50 kg × 7 N/kg
F w = 350 N = 78.6 pounds (force units)
2. Mass and weight practice
1.
Science fiction movies depict astronauts landing on large asteroids. The gravitational force of such bodies is
typically less that 0.1 N/kg. Given an astronaut with a mass of 77.3 kilograms:
a. What would his weight be on Earth (g = 9.8 N/kg) in newtons and pounds (force)?
b. What would his weight be on an asteroid (g = 0.08 N/kg) in newtons and pounds (force)?
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Skill Sheet 6.1 Mass and Weight
2.
If you could embark on Jules Verne’s Journey to the Center of the Earth, you would discover that the force
of gravity would at first increase, as you got closer to the main mass of Earth. But then it would decrease as
your progress left more mass behind you and less in front of you. Imagine now that you are very close to the
center of Earth and the force of gravity has decreased to 0.001 N/kg. What would you weigh in newtons and
pounds (force) if your mass were 52.0 kilograms?
3.
You are a typical young male with a mass of 68.4 kilograms. On Earth, you weigh 670 newtons, or
150.5 pounds (force). If you could survive, what would you weigh on the other planets and the sun? Express
your answer in newtons and pounds (force).
Force of gravity
Sun
newtons
274.4 N/kg
Mercury
3.7 N/kg
Venus
8.9 N/kg
Mars
3.7 N/kg
Saturn
10.4 N/kg
Uranus
8.8 N/kg
Neptune
10.7 N/kg
Pluto
0.7 N/kg
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pounds
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Skill Sheet 6.2
Friction
Friction is a force that resists motion in all real-world mechanics. This skill sheet will provide you with the
opportunity to practice solving problems that involve static friction and sliding friction. Static friction is the
friction between two surfaces that are not moving. Sliding friction is the friction between two moving
surfaces.
1. Calculating friction
In the illustration below, a force (F) is applied to an object. This force is resisted by
friction (Ff), also a force.
Note that the normal force, (Fn), includes all forces pressing the moving surfaces
together.
Here is the equation for finding friction:
In this equation:
•
Ff = force of friction.
•
µ = coefficient of friction.
•
Fn = normal force (force pressing together).
Example: A cinder block sitting on a sidewalk weighs 90 newtons. The coefficient of friction is 0.4. How much
force is required to start the block sliding?
Ff = µ × F n
Ff = 0.4 × 90 N
Ff = 36 N
To slide the block, F must be greater than 36 N.
2. Static friction
Solve these problems and state your answers in the following way: “More than _____ newtons of force is
needed.” Fill in the blank with the correct number of newtons.
1.
A huge pile of leaves was wrapped in a tarp in the middle of a lawn. The wrapped leaves weigh
580 newtons. The coefficient of friction for the lawn is 0.55. How much force is required to start sliding the
wrapped leaves?
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Skill Sheet 6.2 Friction
2.
Although the collie Lassie could easily pull the 110-newton sled empty, she could not even budge it with
380-newton Timmy aboard. How much force would Lassie have to apply to slide the sled with Timmy
aboard? Assume a coefficient of friction of 0.15.
3.
The plastic wading pool weighs 50 newtons. But the water in it weighs 4,000 newtons. How much force is
required to slide the filled pool if the coefficient of friction is 0.22?
4.
The boys pushed and pushed. At an applied force of 804 newtons, they were just able to move the
1,340-newton car. What was the coefficient of friction?
5.
At an applied force of 530 newtons, the boulder just began to slide. Assuming a coefficient of friction of
0.65, how much did the boulder weigh?
3. Sliding friction
Sliding friction is usually less than static friction. Each of the problems below is related to those in part 2.
1.
Although the pile of leaves in the tarp still weighed 580 newtons, once the family that had raked and
wrapped them up got the huge bundle moving, it was easy to keep it moving. The family estimated that they
were applying 93 newtons of force. What was the coefficient of sliding friction?
2.
Aboard the sled, Timmy kicked off with his feet, and although it wasn't easy, Lassie was able to pull the
combined 490-newton weight of sled and Timmy. Assuming that the sliding coefficient of friction was 0.05,
how much force was Lassie applying to the sled?
3.
The children had been splashing in the pool all day. Their father was surprised that he could drag the plastic
wading pool over the wet grass if he gave it a jerk to get it started. Assume the coefficient of sliding friction
is now 0.08. How much force must he apply to the 4,050-newton pool (and water) once it is moving?
4.
The boys were relieved to find that the 1,340-newton car was easier to push once it got going. Just how hard
did they have to push, assuming a coefficient of friction of 0.4?
5.
Try to do this one logically in your head. The boulder in problem 5 in Part 2 required 530 newtons of force to
just begin to slide at a coefficient of static friction of 0.65. What is the coefficient of sliding friction if the
force required to keep it moving is 265 newtons, half of the 530 newtons? Check your work using the
boulder’s weight you calculated in the earlier problem.
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Skill Sheet 6.3
Equilibrium
When all forces acting on a body are balanced, the forces are in equilibrium. Here are free-body diagrams for
you to use for practice working with equilibrium.
1. Equilibrium
Remember that an unopposed force results in acceleration. Therefore, the forces acting
on a body that is at rest or moving at a constant velocity must be at equilibrium.
Conceptually, free-body equilibrium problems are identical to solving equations.
600 N = 400 N + ? N
200 N = ? N
You can see that the free-body diagram is conceptually interchangeable with the
equation. However, it is important to keep in mind that equilibrium means that forces
are in balance; the condition tells us nothing about the magnitude of those forces.
2. Achieving equilibrium
For each free-body diagram, supply the force or forces necessary to achieve equilibrium.
1.
Draw a force arrow, and write in the force.
2.
Supply the missing force.
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Skill Sheet 6.3 Equilibrium
3.
Distribute the unknown forces evenly to prevent rotation.
4.
Supply the missing force.
3. Think about these
1.
Here is the classic “asteroid destroys Earth” scenario. The momentum of the
asteroid is beyond the forces that even thermonuclear bombs might apply to
stop its approach. Assuming that you can apply only modest forces, where
might they best be applied to result in a new acceleration that will, as they
say, “save the world”? Draw an arrow to show the best location on the
asteroid to apply force so that it avoids hitting Earth.
2.
Helium balloons stay the same size as you hold them, but swell and burst
as they rise to high altitudes when you let them go. Draw and label force
arrows inside and/or outside the balloons on the graphic at right to show
why the near Earth balloon does not burst, but the high altitude balloon
does eventually burst. Hint: What are the forces on the inside of the
balloon? What are the forces on the outside of the balloons?
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