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The Laws of Motion…or, Newtonian mechanics

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The Laws of Motion…or, Newtonian mechanics

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Isaac Newton

Newton believed that mathematics can describe nature, accurately.He solved the premier scientific problem of his day – to explain the motion of the planets.

Where is this sculpture from?

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1. The law of inertia. An object in motion remains in motion with constant velocity if the net force on the object is 0.

2. Force and acceleration. If the net force acting on an object of mass m is F, then the acceleration of the object is a = F/m. Or, F = ma.

3. Action and reaction. For every action there is an equal but opposite reaction.

Newton’s Laws of Motion

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Atwood machine with equal weights

Net force = 0 impliesconstant velocity.

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Newton’s second law of motion

F = m a• F = force acting on the moving object, a vector

• m = mass of the moving object

• a = acceleration of the moving object, a vectorIf the force is known, Newton’s second law states how the object (on which the force is acting) accelerates:

m

Fa

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Understanding vectors

A vector is a mathematical quantity that has both a magnitude and a direction.

F = m a an equation of vectors

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Three kinds of acceleration

Example: Circular motion has centripetal acceleration and centripetal force; F = ma.

m slows down

m speeds up

m changes direction

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Quantitative applications of F = ma

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Mass is a property of matter = quantitative measure of inertia.

The standard unit of mass is the kilogram (kg).A gram is 0.001 kg.

This picture shows the international prototype 1 kg mass standard. It is a platinum-iridium cylinder kept in Sevres, France.

Any physical object has a mass m, which could be measured against a standard, e.g., using a balance.

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Force = a push or a pull exerted on one object by another object.

The standard unit of force is the newton (N).The newton is defined in terms of more fundamental units by

1 N = 1 kg m/s2

(consistent with F = m a)

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Acceleration and Force

Given the mass of the object, and given the force that is acting on it, the acceleration is

mF

a

If necessary, identify a and F as vectors and take into account their direction (both in the same direction!)

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Example. Playing catch with a softball

The trajectory has 3 parts.

the throw the catch

free fall

What is the force acting on the ball, during each part of the trajectory?

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Example ProblemA dragster accelerates from 0 to 60 mi/hr in 6 seconds. Calculate the force on the car if the mass is 103 kg.

Answer: 4.47 x 103 N

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Two examples of forces

• weight (the force of gravity on an object)

• string tension (the force exerted by a taut string)

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Gravity

What is the force acting on the mass m due to the Earth’s gravity?

downward gaIf released, the acceleration of m would be … By Newton’s second law the force on m must be …

downward mgF

By the way, what is the reaction force?

Solution

That is, the magnitude, or strength, is mg and the direction is downward.

g = 9.81 m/s2

F

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Weight = the force of gravity

The weight of an object is, by definition, the strength of the force of gravity pulling the object downward.

force of gravity

W = m g

newtons

kg

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Example: What is the weight of a 1 kg mass near the Earth’s surface?

W = mg = 9.81 N

Or, W = 2.21 pounds

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Mass and Weight different!

► Mass is an intrinsic property of an object. It is completely determined by the number and type of atoms that make up the object. It does not depend on the environment in which the object is located.

► But weight is different. Weight depends on both the object itself, and on some other object that exerts the gravitational force.So, for example, the mass of an object

would be the same on the moon or the Earth; but the weight would be different.

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An astronaut would not need a car jack to change a flat tire on the Moon Buggy.

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Analyze free fall including air friction

The aerodynamic drag force (C) depends on the size, shape and surface roughness; it is about the same for both balls.

• The gravitational acceleration (g) is independent of mass.• Effect of air is inversely proportional to mass:

heavy --- small effect of airlight --- large effect of air

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String tension

Example. Suppose a string can withstand string tension 500 N without breaking. What is the maximum mass M that it can hold suspended in Earth’s gravity? Answer: 51.0 kg

The force exerted by the string, at either end:• direction is parallel to the string• magnitude (same at both ends) is called the tension

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Analyze the motion of a pendulum.

Fa

TFF

gravity

m

Solve by calculus.

period gL

2

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More example problems

3. A gymnast weighs 100 pounds.(a) What is her weight in newtons?(b) What is her mass?

4. A car of 2000 pounds moving 30 mi/hr crashes into a brick wall. The collision lasts 0.3 seconds. Calculate the force acting on the car during the collision. Express the answer in both newtons and pounds.

45.3 kg

445 N

4.05 x 104 N or 9,100 pounds

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Newton’s second law and calculus

. as

ison accelerati ousinstantane

; or

0ttv

a

mF

amaF

mF

dtdv

differential equation

Isaac Newton invented calculus to solve the equations of motion; i.e., to calculate motion for the force that is acting. Generally, calculus is the mathematics that describes continuous change.

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Quiz Question

A car accelerates away from a stop sign with acceleration 0.1 g ( = 0.981 m/s2). The mass of the driver is 50 kg. What is the force on the driver? (Be sure to include the unit of measurement!)