PHYS 1110

Lecture 3

Professor Stephen Thornton

September 4, 2012

Reading QuizBoth boys jump off the cliff into the water at the same time. Boy 2 has an additional horizontal velocity. Which boy lands in the water first?

A) Boy 1.

B) Boy 2.

C) They land at the same time.

D) Cannot tell without further information.

A

B

Answer: C The horizontal and vertical motion is independent of each other. Both start with zero velocity in the y-direction and have the same acceleration. Their vertical motion is identical.

0

Position vector =

Displacement vector =

Average velocity vector =

Instantaneous velocity vector = = lim

Average acceleration vector =

Instantaneous accelera

f i

av

t

av

r

r r r

rv

tr

vt

va

t

0tion vector = lim

t

va

t

Two dimensional motion

Horizontal and vertical motions are independent! It is that simple.

We will find later that we only need to look at the force components along each direction. It is force that causes acceleration.

Constant-AccelerationEquations of Motion

x = x0 + v0xt

+ ½ axt2

vx = v0x

+ axt

vx2 = v0x

2

+ 2axx

y = y0 + v0yt

+ ½ ayt2

vy = v0y

+ ayt

vy2 = v0y

2

+ 2ayx

Position as a function of

time

Velocity as a function of

time

Velocity as a function of

position

Projectile motion

Assumptions: air resistance is ignored for now. acceleration of gravity is constant

and has value g = 9.81 m/s2. Earth’s rotation is ignored.

Set up solutions

Choose coordinate y

system.

ax = 0 and ay = gNow what will the equations in the table look like?

x

ay = g

Projectile motion equations0 0

20 0

0

0

2 20

2 20

1

2

2

x

y

x x

y y

x x

y y

x x v t

y y v t gt

v v

v v gt

v v

v v g y

Don’t memorize these equations!

Water fountains show projectile motion

Conceptual Quiz: Look at the demo (shoot and drop). Two balls are released at the same time by a spring loaded mechanism. Ball A falls straight down. Ball B is propelled out horizontally. Which ball lands first?

A) Ball A

B) Ball B

C) Impossible to determine.

D) Balls land at the same time.

Answer

D) Balls land at the same time.

Both balls feel the same vertical acceleration, -g. Therefore, they drop at the same rate.

Of course their horizontal motion is quite different.

Do demo.

Projectile Motion Applet/Physlets

http://physics.bu.edu/~duffy/semester1/semester1.html

Questions: Three ProjectilesMaximum heightTime of flightRangeMonkey and the Hunter

Conceptual QuizA battleship simultaneously fires two shells at enemy ships. If the shells follow the parabolic trajectories shown, which ship gets hit first?A) Ship A.B) Ship B.C) Both at the same time. D) Need more information.

Answer: B. Consider the time for the shell to reach its maximum height (or fall from its maximum height). Since shell A goes higher, it takes a longer time than shell B.

Projectiles with Air Resistance

Newton’s Laws of Motion

We now learn about FORCES

Operational definition: Forces are pushes and pulls.

Look at spring scale.

Newton’s First Law

A body in motion tends to stay in motion unless acted upon by an external (outside) net force.

Do demos

Include air track, air carts, scooter, tablecloth jerk, hanging masses (quizzes), bottle and pencil.

Conceptual Quiz.If we jerk on the bottom string, what happens?

A) Top string breaks.

B) Bottom string breaks.

C) Can’t possibly tell; it will happen randomly.

Answer: B. bottom string breaks.

The bottom string breaks, because the mass is large and has lots of inertia. The mass will not respond to a quick jerk.

Conceptual Quiz.Now we pull slowly on the bottom string. What happens?

A) Top string breaks.

B) Bottom string breaks.

C) Can’t possibly tell; it will happen randomly.

Answer: A) top string breaks.

The top string breaks now, because we are pulling slowly on it. The bottom string feels the force from our hand, but the top string feels the force from our hand plus the weight of the mass.

Newton’s first law is also known as the law of inertia.

Inertia means the body wants to keep its present motion, whether at rest or not.

If a body is moving at constant velocity, it wants to remain moving at constant velocity.

If at rest, it wants to remain at rest.

It keeps its inertia unless a net force acts on it!

Newton’s second law of motion

Let’s do some experiments on the air track with a constant force (fan car). We will use the fan to push various masses and observe the acceleration of the masses.

Do experiments.

We learn that the acceleration is proportional to F/m.

Because there may be several forces on the object, we have to take the net force.

Fam

Newton’s Second Law

Determine the net force.

Now Newton’s Second Law appears as

Unit is newton. 1 N = 1 kg·m/s2

net ii

F F

netF ma

Forces

Lots of things to learn about forces.

Find net force – free-body diagrams are very helpful.

Different kinds of forces:

W weight

N normal force, also FN

T tension, for example, a rope

f friction

several others

Forces are vectors

In some cases we will need to use the vector notation:

, , ,W N T f

Newton’s Third Law

When an object 1 exerts a force on object 2, then object 2 will exert an equal, but opposite, force on object 1.

Forces always come in pairs and are equal and opposite.

Math form of Newton’s 3rd Law

Force on body 1 due to 2, , is equal and opposite to the force on 2 due to 1, .

We often say “for every action there is an equal and opposite reaction”.

Law of action and reaction.

12 21F F

12F

21F

Examples of Action-Reaction Force Pairs

Do demos

Air track reaction cars

Two carts with students

Important Points

Action-reaction force pairs always act on different objects!

When dealing with forces, we want the on a particular object. In a 3rd law force pair, one force acts on our object and the other force acts on another object. This is a big source of confusion!

netF

Example – which object is the force acting upon?

Quiz: Which of Newton’s laws refers to an action and a reaction force?

A) First law.

B) Second law.

C) Third law.

D) This is a trick question. There is only one Newton’s law.

Answer: C

Newton’s third law is more commonly known by its abbreviated form, “for every action there is an equal and opposite reaction”.

Weight

Weight is a force

W = mg called gravitational force

Note that weight is not a mass!

W mg

Weight and Mass

The Normal Force May Equal the Weight

Quiz: Is the normal force on a body always equal to its weight? A) Yes.B) No: it depends on the net external force due to other forces acting on the body.C) No: it depends on the shape of the body.D) No: it depends on the area of contact between the body and the surface.

Answer: B

There can be other external forces that reduce the normal force on a body.

An Object on an Inclined Surface

W

Force pushing downhill

Rubbing surfaces cause friction

Facts about friction

• Leonardo da Vinci performed experiments on friction (~1500). He found it was the normal force that was important, not the surface area.

• Show block on table.

• Friction is bad. $1B of research/yr

• Friction is good. Many examples; we walk, drive cars, etc.

Kinetic Friction and the Normal Force

Constant velocity.

2 2

k

k

F f

F f

More experimental results

Friction is proportional to the magnitude of the normal force.

Friction is independent of the relative speed of the surfaces.

Friction is (to first order) independent of the area of contact between surfaces.

We are examining kinetic friction.

Static friction

In kinetic or sliding friction the two objects are moving with respect to each other.

In static friction the object is at rest, but we want it to move. We have to overcome friction to have it move. We find there is a certain amount of force needed to do this.

Example of static friction

Experimental result for static frictionThere is an upper limit of static friction that we have to overcome in order to move an object.

Static friction can have any value up to this maximum amount.

s, max s

s s,max0

f N

f f

These are concepts!

Facts about static frictionStatic friction can have any value from zero to its maximum amount.

Value:

Static friction is independent of area of contact between surfaces.

Static friction is parallel to surface of contact, and in the direction that opposes relative motion.

s s,max0 f f

Typical Coefficients of Friction

Rubber on concrete (dry) 0.80 0.90

Steel on steel 0.57 0.74

Glass on glass 0.40 0.94

Wood on leather 0.40 0.50

Copper on steel 0.36 0.53

Rubber on concrete (wet) 0.25 0.30

Steel on ice 0.06 0.10

Waxed ski on snow 0.05 0.10

Teflon on Teflon 0.04 0.04

Materials Kinetic, k Static, s

s kNote that

When we walk, is it static friction or kinetic friction when our feet are in contact with the ground?

Answer: static

Same with car tires.

Conceptual QuizAn object is held in place by friction on an inclined surface. The angle of inclination is slowly increased until the object starts to move. If the surface is kept at this angle, the object A) speeds up.B) moves at uniform speed.C) slows down.

Answer: A

Remember that the coefficient of static friction is greater than that of kinetic friction. So when static friction is overcome and the object starts sliding, the frictional force becomes smaller, and the net force (due to gravity) is down the plane.

Tension in a Heavy Rope

Heavy rope:

Light rope:

3 2 1T T T

3 2 1T T T

We usually consider light ropes.

g

= mg

mass m

A Pulley Changes the Direction of a Tension

Tension in a String

Spring Forces

Spring Forces

F kx

Notice signs of the force in both cases.

Spring Forces

Equation F = -kx is known as Hooke’s Law.

The force is always in the direction to restore the spring to equilibrium.

The minus sign simply indicates that the force is a restoring force.

Concepts!

Quiz: A person hoists a bucket from a well using a rope. Let the bucket be at rest. She then ties the other end of the rope to the handle. In which case is the tension in the rope the greatest? 1 2

A) Case 1 B) Case 2C) They are the same

 

Answer: A) The tension is greatest in the first case. See the next slide.

Circular motion Do demo with

string and ball.

Note that the direction of the velocity is changing. The ball is accelerating!

f iv v v

Notice that tends to point towards the center of the circle. As becomes smaller and smaller, points directly to center. Therefore the acceleration points towards the center of the circle.

v

v

Centripetal accelerationCentripetal means “center seeking”.

The derivation is straightforward, but we will not do it. The result is that the magnitude of the centripetal acceleration acp is

where r is the radius and v is the speed.

2 1av

v v va

t t

2

cp

va

r

v

a

Circular motion

Results for circular motion: Consider an object moving in a

circle of radius r with a constant speed v.

A centripetal acceleration of magnitude v2/r must cause it.

There must be a centripetal force Fcp of value 2

cp cp

mvF ma

r

Centripetal forceWhere in the world did this centripetal force come from?

There has to be a force to keep the object moving in a circle. In the case of the ball and string, it was the tension in the string. The tension always pointed towards the center!

The direction of the centripetal force must also be towards the center!

The moon rotates around the Earth in a circle. What is the centripetal force that causes this?

If you drive around in a circle with a bicycle or even with a car, what is the centripetal force?

In a simple atomic model of the hydrogen atom, the electron rotates around the proton in a circle. What is the centripetal force?

Conceptual QuizA ball is attached to a string and swung in a horizontal circle of constant radius. Immediately after the string is released the ball will move in what direction?

A

EB

C

D

·

Answer: B

Remember that the velocity is always tangent when we have circular motion. This is the instantaneous velocity. So right when the string is released, it has to go in the direction of the velocity at that instant. Therefore it must go in its tangential direction.

DO DEMO!

What other forces are exerted on the ball besides mg?A) FrictionB) TensionC) A normal force perpendicular to mg.D) A normal force perpendicular to the surface of the cone at the ball.

Answer: D

The only other possible force is the normal force, and it must be perpendicular to the surface that the ball is rolling upon.

Quiz: What is the direction of the net force?

A) towards the center of the dashed circle at the ball (radially).B) away from the center of the circle at the ball.C) up at the ball.D) down at the ball.E) cannot tell with information given.

Answer: A

Because the ball is moving at constant speed in a circle, the net force must be along the radial direction, towards the center of the circle. This is the centripetal force.