FORCES AND CIRCULAR MOTION

A. Definition: a push or pull acting on a mass1. Force is a vector quantity with both magnitude (numeric value) and direction2. Force can be broken down into horizontal and vertical components3. Symbol:4. Units:

I. FORCE

Newtons ! (N)

F

B. Concurrent Forces: many forces acting on the same object at the same time.1. Treat all forces individually to find a resultant force (break into components)2. This resultant of all concurrent forces is called the Net Force

FnetSymbol:

C. Free Body Diagram: represents concurrent forces acting on an object

Two Physics students try pushing a car to see who is stronger. One student pushes west with a force of 500 Newtons. The other pushes East with a force of 700 Newtons.

Draw a free body diagram of the situation.

What is the Net Force?

What way does the car move?

EXAMPLE 1: NET FORCE

Two Physics students are again arguing and this time are in a tug of war. They are pulling on a box. One student pulls 30 degrees toward the northeast with a force of 400 Newtons and the other pulls at 20 degrees toward the Northwest with 500 Newtons.

Draw a free body diagram of the situation.

What is the Net Force?

EXAMPLE 2: NET FORCE

What is net force on a box of mice being pulled with a force of 20 Newtons due West toward a snake pit and another force of 30 Newtons pulling due East toward an alley fi lled with cats, a 50 Newton force pulling due North toward a cliff , and a 50 Newton force pulling due South toward a large pond? Draw a Free Body Diagram 1st!!!

JOURNAL #1110/1

D. Static Equilibrium: reached when the resultant of all forces acting on an object is ZERO (balanced)

1. At Equilibrium, objects remain at rest or constant velocity.

Fnet = ZeroFnet ≠ Zero

2. Net Force is equal to ZERO in static situations

Fnet =0

What forces MUST be added in order to produce static equilibrium in the free body diagram below?

EXAMPLE: STATIC EQUILIBRIUM

II. DYNAMICS

A. Newton’s First Law: An object maintains a state of equilibrium unless acted on by an unbalanced force. (at rest or constant velocity)

1. Any unbalanced force (Fnet ≠ 0) will produce a

change in an object’s velocity…either speed, direction, or both.

• the object will ACCELERATE

2. Newton’s First Law is also known as the Law of Inertia

• Inertia: the resistance of an object to a change in its motion

More Mass = More Inertia

Eff ects of forces acting on objects (Newtons Laws of Motion)

• Masses resist changes in motion…

What has more inertia? A 10 kg bag of feathers sitting still or a 5 kg gold bar moving along at 10 m/s?

What has more inertia? A 20 kg baseball sitting on a stand, or a 5 kg bowling ball moving along at 30 m/s?

EXAMPLES: INERTIA

B. Newton’s Second Law: the acceleration of an object is directly proportional to the net external force acting on an object and inversely proportional to the object’s mass.

• force is related to mass and acceleration using the famous expression:

• acceleration is produced by force(s)• increasing force will increase the

acceleration

maFnet

1. Units for Force…Yay!! Dimensional Analysis!

a. Newtons are the SI unit of force and are a derived unit (combination of fundamental units)

b. 1 Newton is equal to the force required to accelerate a 1 kilogram mass 1 meter per second squared

maFnet 2smkg N1

2. Increasing mass will increase the force needed to accelerate that mass

maFnet

larger m larger Fnet

*The equation must balance!

m

Fa net

m a

3. If the force is constant, then increasing the mass of an object will decrease the resulting acceleration

Fnet a

4. Graphing Fnet = ma :

Force (N)

Acce

lera

tio

n (

m/s

2) Direct

Relationship: Increasing Force produces more acceleration

A capybara with a mass of 100kg is tackled by a Jaguar with a steady force of 100 N along the ground. Assuming no friction, what is the acceleration of the rodent?

EXAMPLE: NEWTON’S 2ND LAW

5. On HORIZONTAL surfaces… only the HORIZONTAL component of a force will accelerate an object.

C. Newton’s Third Law: when one object exerts a force on a second object, the second object exerts a force on the fi rst that is equal in magnitude, but opposite in direction.

For every action there is an equal and opposite reaction!

• What happens to a boat when you step onto a dock?

Newton’s 3 rd Law!!!

Newton’s 3rd Law also applies in space when making objects move

A. Weight: gravitational force exerted on a small mass by a planet/large body

1. Weight CHANGES based on what planet/object you are on… MASS does NOT CHANGE

2. Symbol:

3. Units:

4. Equation:

III. NATURAL FORCES

mgFg

Newtons ! (N)

How much do you weigh?

EXAMPLE: WEIGHT

The fattest, ugliest Capybara has a mass of 66 kg. What is the weight of the rodent on Earth?

Convert the mass to pounds if 1 kilogram = 2.2 pounds

B. Newton’s Universal Law of Gravitation: Describes the force of attraction between diff erent masses. Any two bodies attract each other with a force that is directly proportional to the product of their masses, and inversely proportional to the square of the distance between them

2

21

r

mGmFg

2

21

r

mGmFg

Fg = Gravitational Force

G = Universal Gravitational Constant = 6.67 x 10 -11

N•m2/kg2

m1 = mass of object 1

m2 = mass of object 2

r = distance between the two masses

On Your Reference Tables!!

(Front Cover)

• Graphical Representation:

What is the force of gravitational attraction between the Earth and the Moon?

m1 = Earth = 5.98 x 1024 kg

m2 = Moon = 7.35 x 1022 kg

r = 3.84 x 108 m

G = 6.77 x 10 -11 N•m2/kg2

2

21

r

mGmFg

EXAMPLE: NEWTON’S UNIVERSAL LAW OF GRAVITATION

2

21

r

mGmFg 28

222411

)1084.3(

)1035.7)(1098.5)(1067.6( 2

2

m

kgkgkg

mN

2217

2237

1047.1

)1093.2(

kgm

kgmN

NFg

201099.1

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A 200 kg box of rodents is sitting on the road. A truck pulls the crate with a force of 300 Newtons to the East while another truck pulls the crate with a force of 150 Newtons to the West.

Draw a free body diagram and label all forces…What is the net force on the crate?Disregarding friction, what will the acceleration of the

crate be?Calculate the weight of the crate of the rodents?What is the force of gravitational attraction between

the crate of rodents and the Moon?

2. Gravitational Fields: vectors are used to show gravitational force

A “unit test mass” will accelerate along gravitational field lines, toward the center of the source of gravity

C. The Normal Force: force exerted on an object perpendicular to the surface of contact

a. Prevents objects from accelerating due to gravity

b. Usually denoted FN Fn or N

c. For horizontal surfaces, the normal force

perfectly balances/cancels Fg (weight)

90°

d. If the object is on an incline, the normal force is angled from vertical the same amount as the incline

90°

θ

θ

e. The components of the Normal force

coscos mgFF gg y

f. Component of Fg that is perpendicular to

the surface always balances FN

g. If the object “slides” down the surface, the force responsible is the component of gravity acting along the ramp (if it doesn’t slide, it must have another force balancing gravity!)

D. Applied Force: the actual direction of the push or pull• an object may accelerate in the direction of the applied force if it is strong enough

appliedF

E. Force of Friction: opposes applied force or relative motion of two objects in contact with each other

a. If applied force exceeds frictional force, the object will accelerate in the direction of the applied force

Symbol:

Ff

b. Factors Aff ecting Force of Friction

1. Compression: Increasing the force pressing objects together increases force of friction

on Earth, weight and normal force usually compress objects

friction is directly proportional to the normal force and weight

BEWARE: normal force and weight are only equal on horizontal, flat

surfaces!

2. Surface Area of contact between surfaces of materials has NO eff ect on force of friction because an object’s weight is distributed over the surface of contact regardless of size

3. Type of Surface

Roughness = Friction

4. Speed vs. Stationary: Moving objects experience less resistance while continuing movement than to start moving

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A 20 kg crate is sitting on a 35 degree incline. There is friction between the ramp and the crate.

Draw a free body diagram and label all forces…Calculate the weight of the crateWhat is is the normal force of the incline on the

crate? (Break apart Gravity!)

c. Static Friction: opposes applied force onobjects at rest and resists motion

d. Kinetic Friction: opposes motion of objects in contact that are already moving

e. Kinetic friction is LESS than Static friction due to action of molecular forces

f. Coeffi cient of Friction: ratio of frictional

force to normal force – denoted μ

Nf FF

N

f

F

F

so the force of friction is equal to:

essentially, the force of friction is a fraction of the normal force

A large wooden crate of (you fi ll in the blank) is sitting on a level wooden plank. The mass of the crate is 100 kg. A super strong physics teacher pushes on the crate with a force of 500 Newtons East. FRICTION opposes that force.

Sketch a free body diagram of the situation showing ALL forces acting on the crate.

What is the Weight of the crate?What is the Normal Force on the crate from the roadway?What is the force of Static Friction on the crate?Why does the crate move?What is the new Force of Friction on the moving crate?What is the Net Force on the moving crate?What is the acceleration of the crate?

EXAMPLE: FRICTION!

1. Coeffi cients of friction are listed in the Reference Tables!!

Reference Tables – Front Cover!!

2. Since the force of static friction is greater than the force of kinetic friction, the coeffi cients change as well

3. Equations become:

N

f

s F

Fs

N

f

k F

Fk

Nssf FF Nkf FFk

• treat static and kinetic situations separately

A 25 kg crate of dead rodents some material, initially at rest on a horizontal floor requires a 75 N horizontal force to the West to set it in motion Draw a free body diagram Find the coeffi cient of static friction

between the crate and the floor.

N

f

s F

Fs

N

N

245

75 30.0

sfF

NF

gF

appF

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g. Friction and Motion1. Any force problems require us to

think of Net Force!2. Force of friction only opposes motion

along a surface!!!3. Break down gravity and simplify

diagrams

Ff

θ

Re-orient

4. Then “tally” forces in x and y directions

5. Remember: if there is no net force in a direction, then acceleration in that direction is zero

6.RULES:

If object is moving at CONSTANT speed the friction force is equal to the force making the object move

If object speeds up (Accelerates), the friction force is LESS than the force making the object move

If the object slows down (decelerates), the friction force is GREATER than the force making the object move

EXAMPLE: FRICTION ON AN INCLINE

A 150 kg GOLD plated guinea pig sits on a ramp at 40 degrees to the horizontal.

Draw a free body diagram and label all forces…What is the weight of the trophy?What is the normal force?What is magnitude of the force of friction preventing

the trophy from moving? (It is equal to the part of gravity pulling it down the ramp!)

What is the coeffi cient of Friction between the trophy and the ramp?

F. Tension: The force in ropes, strings, or wires

1. Symbol: FT or T

2. Tension is equal to weight Fg if the object

hangs vertically3. Objects moving the opposite direction as

tension can reduce the tension in a rope (elevators moving down)

4. Objects being forced UPWARD increase the tension in a rope based on the force applied

5. Tension can be split by angled rope

F. UNIFORM CIRCULAR MOTION

1. For an object moving at a constant velocity, a force that acts perpendicular to the direction of the velocity will change the direction of the velocity, causing uniform circular motion

2. Remember: velocity is a vector…and can be changed in both magnitude and direction

a. A change in velocity is an acceleration!

b. Therefore, an object in UCM is always accelerating toward the center of its circular path

at

vvv if

***Note that speed can remain the same but object is still accelerating***

c. The constant acceleration of an object toward the center of its circular path is called centripetal acceleration.

d. Magnitude of centripetal acceleration can be found using the following equation:

r

vac

2

ac = centripetal acceleration

v = magnitude of velocity

r = radius of circular path

3. The force directed toward the center of a circular path that causes a centripetal acceleration is called the centripetal force.

a. Centripetal force is denoted Fc

b. Newton’s 2nd Law can be used to determine centripetal force:

becomes maF cc maF

c. Using the formula for centripetal acceleration:

can be re-written

cc maF

r

vac

2

r

mvFc

2

4. What about “centrifugal force”?

a. This is a FICTICIOUS FORCE (does not exist)

b. Misconception comes from the apparent motion of an object tangent to the circular path…

c. Tangential motion only occurs if the centripetal force goes to zero!

then the object will obey Newton’s Law of Inertia

With Centripetal Force

object continues in circular path

Centripetal Force Removed

object continues in a path tangent to its initial velocity

5. DIAGRAM!

6. Example: A rope is tied around a 0.5 kg rat which is then twirled around overhead. The rope is 0.75 m and it makes a full circle in 0.55 seconds. Calculate the speed, centripetal acceleration, and centripetal force.

JOURNAL #X

A fuzzy bunny rabbit hanging from a tree with a rope 2.1 m in length. Assuming the mass is 10 kg and swinging with a tangential speed of 2.5 m/s…

What is the magnitude of the centripetal acceleration?

What is the magnitude of the centripetal force?What supplies the centripetal force and what is it’s

direction?