Newton’s Laws of Motion Forces Some WRONG views about MotionSome WRONG views about Motion Galileo’s “Thought Experiments” Inertia Newton’s 1 st LawNewton’s

Newton’s Laws of Motion• Forces

• Some WRONG v ws out Mot on ie ab i

• Galileo’s “Thought Experiments”

• Inertia

• Newton’s 1st Law

• Newton’s 2nd Law

• Free Body Diagrams (FBDs)

• The 2 Viewpoints

• View-from-Above Problems

• Force of Gravity (Weight)

• Simple Problems • Wagon/Lawnmower-Style Problems

• Newton’s 3rd Law

• The Normal Force • Frictional Forces • Block-Pushing-Block Problems

• Train-Style Problems

What is a “Force”?A force is ANYTHING that can make an object accelerate (speed up, slow down, or change direction)

The SI unit for force is the NEWTON (N). This unit is a derived unit, meaning it’s made up of base units.

m kg sec

sec sec kg

m N =

kgm

s2

The English unit for force is the POUND (lb).

1 lb = 4.44 N

Two types of forces

Contact forces (pushes & pulls)

Field forces (no contact … gravity

is an example) HOME

What did people long ago believe about motion?

Aristotle, the famous Greek philosopher (~350 BC), studied the motion of objects and concluded the following:

“It requires a continuous pushing or pulling” to keep an object such as a rolling stone moving. When the pushing or pulling is no longer applied, the stone comes to rest……A FORCE is required to produce a constant velocity” (Heath, pg. 128)

Sounds good right??? NO!!!!! Aristotle was wrong He thought that …

• With increased force objects move faster.• With decreased force objects move slower.• With No force objects will stop.

Common sense. Right?

HOME

What did Galileo think about motion?

Galileo Galilei, the great Italian physicist who lived during the Renaissance, used two different “thought experiments” about the motion of an object on an inclined plane to explain motion.

Experiment #1: Galileo imagined a ball rolling down a sloped plane. He figured that it would speed up. He then imagined a ball rolling up a slope. He figured that it would slow down. He reasoned that a ball rolled across a horizontal surface would neither speed up nor slow down, but rather continue to move with a constant velocity.

Speeds up

Slows Down

Wants to go at a constant speed

NEXT

Experiment #2: Galileo again imagined a ball rolling down a sloped plane. However, this time, he allowed the ball to roll up a plane afterward. He reasoned that no matter what the slope of the two planes, the ball would always attain the same height (the height it was rolled down from equals the height it rolls up to). Therefore, he concluded that if no“up” plane were present at the bottom of the “down” plane, the ball would roll on FOREVER, trying to but never reaching the height from which it were dropped.

Tries to get to this height again Should roll on

forever “trying” to get back to original height

NEXT

Both of Galileo’s thought experiments contradicted his observations of the same events in real-life. However, he attributed the differences

to “Resistance”, or what we today call FRICTION. Both thought experiments occurred on a frictionless surface. Galileo argued

that it was just as ‘natural’ for an object to move with a constant speed as it is to be at rest. This contradicted Aristotle’s view. Galileo Published these thoughts about motion in the early 17th century, and his contemporaries immediately dropped the Aristotelian view and embraced Galileo’s views of motion.

AT REST

MOVING AT A CONSTANT VELOCITY

The TWO (2) natural states that objects

want to be in (according to

Galileo) HOME

InertiaGalileo noticed that objects had a “natural tendency” to either stay at rest OR keep moving. This “tendency” to resist a CHANGE in motion is called INERTIA. To measure an objects “inertia”, we measure its MASS. Therefore, like mass, Inertia is a property of an object,

Examples:

• A large rock being difficult to “budge” from rest.

• A heavy grocery cart is difficult to slow down once it is moving down the grocery aisle.

• When you are driving and you come to a sudden stop, your body WANTS to go flying through the windshield. HOME

Mass vs Weight

Mass• The amount of matter in an

object• Mass does not change based

on location (moon vs earth)• Mass is measured in units of

Kg in the metric system• Mass is measured with an

instrument called a balance

Weight• The downward force of

gravity on that mass “Fg” (weight)

• Weight varies based on location (moon vs earth)

• Weight is measured in units of….Newtons in the metric system (Pounds in English)

• Weights is measured with an instrument called a scale

What is the difference between a balance and a scale?

Mass• A balance has a fulcrum,

and you need the same amount of “stuff” on each side to balance

Weight• A scale has something

inside that stretches or compresses under the force of gravity.

• Weight is calculated by factoring gravity into the mass of the object. Multiply mass by gravity.

Practice Questions• How much does a 6 kg object weigh on the

surface of the earth?– In the metric system?– In the English system?

• An object weighs 88N on the earth’s surface, what is it’s mass?

• An alien with a mass of 45 Kg weighs 700N on his home planet. What is the gravitational field strength on the surface of his planet?

• If a person weighs 150 lbs, what is their weight in the metric system?

Newton’s 1st LawIsaac Newton, an Englishman who lived later in the 17th century, began his theories of motion by looking at a concept that he called “Inertia”. It can be thought of as ‘object laziness’. Objects tend to keep doing what they are doing. It takes force to make an object start moving or change direction. The more massive an object is, the larger the force that is required for a given change.” (Holt,

teacher’s addition) Galileo was the first person to formalize this concept. However, Newton used it to develop his famous “Laws of Motion”, which he formally published in his book Principia Mathematica. This book is widely considered to be the greatest scientific work ever published. Newton’s “First Law” of motion, otherwise known as “Newton’s Law of Inertia”, informally states that …

NEXT

Newton’s 1st LawAn object in motion (or at rest) will remain in motion (or at rest) unless acted upon by

an unbalanced, external force.

“in motion” means “moving at a constant velocity”.

If all the forces acting on an object are BALANCED, then the object will NOT change its current motion state.Only external forces can cause a change in motion.

Sometimes called the “Law of Inertia”

Some Notes

NEXT

Newton’s 1st Law … continuedBut if this is in fact a “Law” of motion, why don’t we see this happen in the “real world”. When an object is pushed, it slows down, even when no other forces act upon it. Or so it would seem. FRICTION is the “invisible” force that acts on objects to slow them down. However, in a friction-free case (like in outer space), Newton’s 1st Law will be “observed” to hold true. And, it holds true in all cases, we simply have a hard time “observing” it because of friction.

Examples:

• No seatbelt … going through the windshield in a car accident.• WITH seatbelt … NOT going through the window.• A magician pulling the tablecloth off but leaving the place-setting.• Getting a car to stop rolling once it’s started. • Punching a light “speed bag” vs. punching a “heavy bag”.

NEXT

Newton’s 1st LawIn layman’s terms, please

Object’s “naturally” like to be either at rest or moving with a constant velocity. In order to make them do anything else (which obviously involves an acceleration), “something” needs to force them out of their natural motion state. This “something” is an unbalanced force that must be applied to the object

from the outside of the object. Forces that occur inside the object will NOT alter the object’s motion.

NEXT

Examples:Object at rest

Object at rest

Object at rest

50 N 50 N

The object stays at rest because the

forces acting on it are BALANCED.

50 N 20 N

The object will accelerate (to the right) because the NET force acting on it is not zero

(ie. The forces are UNBALANCED).

50 N 50 N

40 N

The object will accelerate (this way ) because the NET force acting on it is not zero

(ie. The forces are UNBALANCED).

NEXT

Bottom line … whichever direction the net force is

acting, that is the direction that the object will

accelerate.

NEXT

Examples:Object moving at a constant velocity to the right The object will keep doing this forever

unless its forced to change its motion.

50 N 20 N

The object will accelerate (to the right) and speed up because the NET force acting on it is not zero (ie. The forces are

UNBALANCED).

The object will accelerate (to the left) and slow down because the NET force acting on it is not

zero (ie. The forces are UNBALANCED).

Object moving at a constant velocity to the right

30 N 50 N

Object moving at a constant velocity to the right

NEXT

More Real-Life Examples of Newton’s 1st Law

A runaway truck on a highway vs. a runaway bike. Which one would you want to have to stop? I would much rather have to stop the runaway bike. It has

less mass, and therefore less inertia than the truck. Does it want to stop? No! It is in motion and wants to continue on in its motion. However, since its inertia is less than the truck’s, it has less of a tendency to avoid its motion change, and it will require less force to stop it.

NEXT

Real-Life Examples of Newton’s 1st Law

A car in a collision with a brick wall (and the person inside not wearing a seatbelt)

The car and he person have inertia. When the car hits the wall, the car has an unbalanced force applied to it, and it slows down and stops. The person inside, who is in motion, wants to stay in motion. Without the seatbelt supplying an unbalanced force, they will continue in their motion and fly through the windshield. NEXT

Real-Life Examples of Newton’s 1st Law

Trying to get a loaded grocery cart to move from rest, and then trying to get it to stop once its moving.

When the cart is at rest, it has inertia. This tends to keep it at rest, and it requires a pretty large force to get the cart to move. Once the cart is moving, it wants to keep moving (since it still has inertia) and it will take a pretty large force to slow it down.

HOME

Newton’s 2nd LawLet’s Review. Newton’s 1st Law says “An object in motion (or at rest) tends to stay in motion (or at rest) unless acted upon by an unbalanced, external force”. Included in this statement is the fact that objects naturally like to be either at rest or moving at a constant velocity. Their inertia keeps them in one of these two natural motion states, and it requires an unbalanced, external force to “knock them out” of their preferred motion state. Many forces can act on an object at rest, but unless the forces are unbalanced, the object will not move. The same can be said for objects moving at a constant velocity. Now for the one million dollar question. What happens to an object IF an unbalanced, external force DOES act upon it? The answer is simple: its motion will change. In other words, it will accelerate. Newton’s 2nd Law explains what will happen to this object. Stated in the simplest terms, it says …

NEXT

Newton’s 2nd Law

F = ma

FNet = ma

Do NOT memorize Newton’s 2nd Law as F = ma!!!!

NEXT

FNet = maNewton’s 2nd Law has a vector nature.

Object initially at rest

y

x

FNET,x = max

FNET,y = may10 kg

50 – 30 = 10ax ax = 2 m/s2

60 – 60 = 10aY aY = 0 m/s2

Since an unbalanced force acts on the object, it will accelerate in the +x-direction at 2 m/s2.

60 N

60 N

30 N 50 N

Applied forces

NEXT

In order to apply Newton’s 2nd Law to a situation, we

often start by making a “free-body” diagram for the object in question. We will refer to

them as FBD’s, for short.

HOME

Example Free Body Diagrams (FBD)s:Object sitting at rest on a surface.

Object falling through the air.

Force due to gravity (Fg)

Force that the surface pushes back on the object with (called the NORMAL FORCE, FN)

Fg

Air Resistance Force, FA

REAL LIFE SCENARIO

FBD

NEXT

Example FBDs:Object being pulled across a surface

Object being pushed across a surface

Fg

FN

FPUSHFFRICTION

Fg

FN

FPULLFFRICTION

Notice that friction always opposes the

motion of the object! NEXT

Example FBDs:Object being pulled at an angle across a surface

Notice that friction always opposes the

motion of the object!

Object being pushed at an angle across a surface

Fg

FN

FPULL

FFRICTION FP,x

FP,y

Fg

FN

FFRICTION

FPUSHFP,y

FP,x

HOME

There are two “VIEWPOINTS” from which we can look at a problem.

Example: An object is pulled across a surface

VIEW FROM ABOVE VIEW FROM THE SIDE

Fg

FN

FPULLFF

Fg points into the pageFN points into the page

FPULLFF

HOME

How do you know when to use the “VIEW FROM ABOVE” view?

Example: A 3 kg box is pulled by three force. The 1st is 70 N [E], the 2nd is 45 N [NW], and the 3rd force is 15 N [E 20o S]. Find the acceleration of the block.

Notice how this problem is using words like “North, South, East, and West”, and not “up and down”. When a problem uses words like “North and East” (which are perpendicular to each other), then you must use the VIEW FROM ABOVE.

If you see the words “up and down” and “East and West”, OR “North and South” (but not both together), then use the VIEW FROM THE SIDE. HOME

Before we can practice using Newton’s 2nd Law (along with Free-Body Diagrams, FBDs) in order to tackle problems, we

need to learn how to calculate the FORCE DUE TO

GRAVITY (Fg) acting on an object.

NEXT

Weight and the Force due to Gravity

Force due to gravity: A field force (a vector quantity) that always is directed towards the center of the earth.

Weight: The magnitude of the Force due to gravity.

mgamFW gg

mgmaF gg

Mass is an inherent property of an object. It stays the same no matter where you move. Weight depends on the acceleration due to gravity (g). It changes depending on your location. NEXT

Examples: On the earth your mass is 80 kg. What is your weight?

If you take a trip to the moon, what will your new mass and weight be?

NkgmgWsm 7848.980 2

Your mass doesn’t change when you change locations. Therefore, it is still 80 kg. Your weight depends on the acceleration due to gravity. On the moon, it is 1.6 m/s2 instead of 9.8 m/s2. Therefore, your weight on the moon is …

NkgmgWsm 1286.180 2

HOME

A 20 kg box is pushed across a frictionless surface by a force of 500N. Find the box’s acceleration as well as the

normal force acting on the block (from the ground).

FNET,x = max

FNET,y = may

500 – 0 = 20ax ax = 25 m/s2

FN - 196 = 20(0)FN = 196 N(20)(9.8) = 196N

FN

500N20 kg

y

x

ax

NEXT

A 40 kg box is pushed across a rough surface at a constant velocity by a force of 76 N. Find the force of friction acting on the block (from the ground).

FNET,x = max

76 – Ff = 40(0) Ff = 76N

(20)(9.8) = 196N

FN

76N40 kg

y

x

Ff

From Newton’s 1st Law, constant velocity means that FNET = 0, or, in other words, that a = 0.

HOME

A man pushes a 30 kg block with a constant force of 200N at an angle of 30o above the horizontal. If the surface provides

a constant frictional force of 30N, find the acceleration of the block and the normal force acting on the block.

(30)(9.8) = 294N

FN

Ff = 30N

200N100N

173.130 kg

y

x

FNET,x = max

FNET,y = may

173.1 – 30 = 30ax ax = 4.77 m/s2

FN - 100 - 294 = 30(0)FN = 394 Nax

NEXT

A man pulls a 20 kg block with an unknown force at an angle of 40o above the horizontal. A constant frictional force of 60 N acts on the block. If the block moves at a

constant velocity, find the unknown pulling force.

FNET,x = max

Fcos - 60 = 20(0)Fcos = 60

F = 60 / cos (40) = 78.3 N

Fg

FN

Ff = 60N

F

Fcos20 kg

y

x

Fsin

HOME

Newton’s 3rd LawFor every action force there is

an EQUAL, but OPPOSITE reaction force.

NEXT

Examples:Action Force Reaction Force

Pushing against a wall.

Pushing against the ground.

Feet pushing backward against the ground.

Swimmer pulling the water backward with his hand.

Rocket engines pushing exhaust gas out into the air at the back end of the rocket.

The wall pushes back (normal force).

The ground pushes back (normal force).

Ground pushes you forward (allowing you to move forward when walking).

Water pushes the swimmer forward through the water.

Air pushes the rocket forward through the sky.

NEXT

Example: Two astronauts are sitting next to each other in deep space. One has a mass of 80 kg and the other

has a mass of 120 kg. The more massive astronaut pushes the les massive one with a force of 200 Newtons.

Find the acceleration of each astronaut.

120 kg200 N

a less massivea more massive

267.1

120200

sma

akgN

25.2

80200

sma

akgN

80 kg200 N

NEXT

The NORMAL ForceThe NORMAL FORCE is a “reactionary” force exerted on one object by another that is always PERPENDICULAR to the surface of contact.

It is a direct result of NEWTON’S 3RD LAW

Example:

Object sitting at rest on a surface.

Fg

FN

Notice how the normal force is

the ground’s “reaction” to

the gravity force.

NEXT

Example: A 50 kg block sits at rest on the ground. Find the normal force acting on the block.

50 kg

Fg = mg = (50kg)(9.8 m/s2) = 490N

FN

Since the block does not move UP or DOWN, the normal force’s job is to provide balance. Therefore, UPS = DOWNS. And, as a result, in the problem the

normal force is equal to the weight (490N). NEXT

Finding the Normal Force

Example: A 50 kg block sits at rest on the ground. A man pulls upward on the block with a force of 200N using a rope at an angle of 30o above the horizontal. Find the normal force acting on the block.

50 kg

Again, since the block does not move UP or DOWN, UPS must equal DOWNS. Therefore, FN + 100 = 490.

The Normal Force equals 390 N.NEXT


FN 200 N

FFRICTION 173N

100N

490 N

Example: A 50 kg block sits at rest on the ground. A man pushes downward on the block with a force of 200N (like it’s a lawnmower) at an angle of 30o below the horizontal. Find the normal force.

50 kg

Again, since the block does not move UP or DOWN, UPS must equal DOWNS. Therefore, FN = 490 + 100.

The Normal Force equals 590 N.NEXT


FN 200 N

FFRICTION 173N

100N

490 N

So, Let’ Review ….

When an object is simply sitting on flat ground, the normal force is equal to the weight. The ground feels the full weight, and it reacts by pushing back.

When an object on flat ground is pulled upward, at an angle, the normal force is reduced. WHY? Because the process of pulling up relieves some of the weight that the ground feels … and the thus the block “feels” lighter to the ground, which has to react less to hold it up.

Vice Versa, when an object on flat ground is pushed downward at an angle, the normal force is increased, since the pushing force is “forcing” the block into the ground more. The ground reacts accordingly, increasing the normal force.

NEXT

Example: A 50 kg block is being pushed up against a wall by a force of 1000 N, at an angle of 30o above the horizontal. Find the normal force exerted in the block.

Since the block is not moving INTO the wall or OUT OF it, INTOs =

OUTOFs

Therefore, FN = 865 N

Finding the Normal Force … one more time

50kg

FN

1000 N

FFRICTION

490 N865 N

HOME

Frictional ForcesLet’s review what we probably already know about friction:

1)Friction USUALLY slows things down, instead of speeding them up. It USUALLY resists motion.

2)Friction happens when two surfaces rub together. The types of surfaces determines whether there is a lot or a little friction.

3)The harder you press the surfaces together, the larger the force of friction. For example, if you are using sandpaper to sand wood, the harder you push the paper into the wood, the more “sanding” takes place.

NEXT

Frictional ForcesNow, for what you might not know yet:

1)If something is moving at a CONSTANT VELOCITY (a = 0), the frictional forces must be in balance with any forces pushing or pulling the block forward.

2)There are two types of friction: The first type acts on an object when it is at rest (trying to keep it at rest), and the second type acts on objects when they are moving (trying to slow them down).

3)When an object sits (at rest) on a flat surface, there is no friction acting on it. When an object sits at rest on an incline, friction DOES act on it (to keep it from sliding down).

NEXT

The Coefficient of Friction …… tells you how much friction exists between different surfaces.

NEXT

alot of friction

very little friction

Calculating FrictionThe amount of friction between two surfaces can be calculated using ….

NEXT

The coefficient of friction (which is given the greek

letter “mu”, )

The larger the normal force between the two surfaces, the larger the friction between the

surfaces.

Ff FN

mu is pronounced “myoo”

Two different kinds of friction

NEXT

STATIC friction occurs when an object is trying to be moved from

rest. It is larger than KINETIC friction. Once the maximum

static frictional force is exceeded, the object moves.

KINETIC friction occurs when a there is movement

between the two surfaces which are in contact.

“OLD” Friction Problems

(friction is given simply use)

1) A 50 kg box is pulled across a surface by a force of 100 N. A constant force of friction of 25 N acts against the object. Find the objects acceleration.

(constant velocity find the frictional force)

2) A wagon is pulled at a constant velocity by a force of 100 N at an angle of 30o above the horizontal. Find the frictional force acting on the block.

FNET ma 100N 25N (50kg)a

a 1.5m

s2

Fg

FN

100 N25 N

a

FNET ma m(0) 0

Ff 73N50 N

Fg

FN 100 N

Ff 73 N

a

NEXT

Example: A 10 kg box is motionless on the floor. If the coefficient of static friction is 0.4 and the coefficient of kinetic friction is 0.3 (between the box and the floor), find the force required to start the block in motion.

Fg mg 490N

UPsDOWNs FN 490N

Ff FN (.4)(490) 196N

FNET ma F 196 m(.00000001)

F 196NFg

FN

FFf

a

In order to move, it only needs to accelerate a little. NEXT

Example: Continuation of the last problem. Once the block (10 kg) starts moving, find the force needed to keep it moving. (remember, the coefficient of static friction is 0.4 and the coefficient of kinetic friction is 0.3 (between the box and the floor).

Fg mg 490N

UPsDOWNs FN 490N

Ff FN (.3)(490) 147N

FNET ma F 147 m(.00000001)

F 147NFg

FN

FFf

a

Again, in order to keep moving, it only needs to accelerate a little.

NEXT

Notice that the force needed to make something start moving is greater then the force required to keep it moving!

Example: It takes a 50 N horizontal force to pull a 20 kg object along the ground at a constant velocity. What is the coefficient of kinetic friction?

Fg mg (25kg)(9.8m

s2 ) 245N

UPs DOWNs FN 245N

Ff FN (245)

FNET ma 50 245 (25)(0)

50 245 .204Fg

FN

50 N Ff

a

Notice that mu doesn’t have any units.

NEXT

Example: A cart with a mass of 2.0 kg is pulled across a level desk by a horizontal force of 4.0 N. If the coefficient of kinetic friction is 0.12, what is the acceleration of the cart?

Fg mg (2kg)(9.8m

s2 ) 19.6N

UPsDOWNs FN 19.6N

Ff FN (.12)(19.6) 2.352N

FNET ma 4 2.352 (2)a

a .824m

s2

Fg

FN

4 N Ff

a

NEXT

Example: A small 10 kg cardboard box is thrown across a level floor. It slides a distance of 6.0 m, stopping in 2.2 s. Determine the coefficient of friction between the box and the floor.

x 6m, t 2.2s, v2 0 v1 5.455m

s

t 2.2s, v2 0, v1 5.455m

s a 2.48

m

s2

UPs DOWNs FN Fg 98N

Ff FN 98FNET ma 98 10(2.48)

.253

Fg

FN

Ff

a

NEXT

Notice that the acceleration arrow points backward because the

block is slowing down.

Example: A farmer is pushing down a 4 kg shovel with a force of 40 N at an angle of 60 o with the ground. Determine the acceleration of the shovel if the coefficient of friction between the shovel and the icy ground is 0.15.

UPsDOWNs FN Fg 34.6

FN 39.2 34.6 73.8N

Ff FN (.15)(73.8) 11.07N

FNET ma 20 11.07 4a

a .223m

s2

Fg

FN

Ff

a6034.6 N

20 N

40 N

HOME

Block-pushing-block problems

NEXT

50 kg 150 kg200 N

EXAMPLE #1: Assuming a frictionless surface, find the force applied by the 50 kg box onto the 150 kg box.

200 200a

a 1m

s2

Step #1: Look at the entire set of masses at the same time (MEGA MASS!!!) 200 N

1,960 N

1,960 N

a200 kg


NEXT

50 kg 150 kg200 N

EXAMPLE #1 … CONTINUED Assuming a frictionless surface, find the force applied by the 50 kg box onto the 150 kg box.

Step #2: Look at the masses individually (MINIs!!!)

200 N

490 N

490 N

Ra

200 R 50a

50R

1,470 N

1,470 N

a

R 150a 150(1) 150N

150


NEXT

40 kg10 kg

FEXAMPLE #2: If = 0.4 and if the force between the blocks is 50N, find the force pushing the blocks.

F 196 50a

a ???

Step #1: Look at the

MEGA MASS!!!

F

490 N

490 N

a 50 kg

Ff FN (.4)(490)

196N


NEXT

40 kg10 kg


Step #2: Look at the MINIs!!!

F

392 N

392 N50 N

a

F 50 156.8 40a

a ???

40

Ff (.4)(392)

156.8N

50 N

98 N

98 N

a

50 39.2 10a

a 1.08ms2

10

Ff (.4)(98)

39.2N


NEXT

40 kg10 kg


Step #3: Plug “a” back into the other equations to find the missing force.

F 50 156.8 40a

F 50 156.8 40(1.08)

F 250N

a 1.08m

s2

RECAP of Block-pushing-block problems#1) Look at the entire set of masses at once (the MEGA MASS). Draw an accurate FBD, making sure to account for all forces. Use F = ma. If you can solve for an unknown, great! If not, we can always come back to this equation later.

#2) Look at each individual part separately (the MINIs). Draw an accurate FBD for each, accounting for ALL the forces. Use F = ma for each one. #3) Whenever one block pushes another, the pushing force and the “push back” force are the same (Newton’s 3rd Law).

#4) If you got information using the MEGA MASS, plug it into the MINIs (since both the MEGA and the MINIs accelerate at the same rate). Otherwise, solve one of the MINIs equations for an unknown and plug it into the MEGA equation.

HOME

Train-Style problems

NEXT

250 kg 350 kg400 N EXAMPLE #1: Assuming a frictionless

surface, find the acceleration of the train as well as the tension in the rope between the blocks.

400 600a

a 0.67ms2

Step #1: Look at the entire set of masses at the same time (MEGA MASS!!!) 400 N

5,880 N

5,880 Na

600 kg

NEXT

Step #2: Look at the masses individually (MINIs!!!)


250 kg 350 kg400 N EXAMPLE #1: Assuming a frictionless

surface, find the acceleration of the train as well as the tension in the rope between the blocks.

400 N

2,450 N

2,450 N

Ta

400 T 250a

250T

3,430 N

3,430 N

a

T 350a 350(0.67) 234.5N

350


NEXT

30 kg 40 kgF

EXAMPLE #2: The train shown starts from rest and covers 50 meters in 8 seconds, all the while accelerating at a constant rate on a frictionless surface. Find the force pulling the train as well as the tension between the blocks.

x 1

2at 2 v1t

50 1

2a(8)2 (0)(8)

a 1.5625m

s2

Step #1: Use an equation of motion to get “a”.


NEXT

F 70a

F 70(1.5625) 109.375N

Step #2: Look at the entire set of masses at the same time (MEGA MASS!!!)

F

686 N

686 Na

70 kg

30 kg 40 kgF



NEXT

NT

aT

5.62)5625.1(40

40

Step #3: Look at a MINI

T

392 N

392 Na

40 kg

30 kg 40 kgF



NEXT

30 kg 40 kg800 N

EXAMPLE #3: If = 0.3 for the train at the right, how far will it travel, starting from rest, in 4 seconds. Then, find the tension in the rope.

800 205.8 70a

a 8.489m

s2

Step #1: Use the MEGA mass.

800 N

686 N

686 Na

70 kg

Ff (.3)(686)

205.8N


30 kg 40 kg800 N


Step #2: Use an equation of motion.

x 1

2at 2 v1t

1

2(8.489)(4)2 (0)(4) 67.9m

NEXT


30 kg 40 kg800 N


Step #3: Use a MINI.

HOME

T

392 N

392 Na

40 kg

Ff (.3)(392)

117.6N

T 117.6 40a

T 117.6 40(8.489)

T 457.2N

Documents

Newton’s Laws of Motion Forces Some WRONG views about MotionSome WRONG views about Motion Galileo’s “Thought Experiments” Inertia Newton’s 1 st LawNewton’s