Final Exam Review

December 9, 2009 1

December 9, 2009 2

What about vector subtraction?

Suppose you are given the vector relation

RULE:The resultant vector headAlways points toward the headof the vector that is beingsubtracted from (in this case ).

This can be turned into vectorAddition, by writing:

CBA

B

C

A

A

B

C

A

BCA

December 9, 2009 3

0rrr

ntDisplaceme

ott

r

velocityAverage

tt

r

0limVelocity ousInstantane

ott

ovv

onaccelerati Average

tt

v

0limonAccelerati ousInstantane

0v

v

Position vectors r, r0 at t, t0

There is an acceleration whenever there is a change of speed or direction

Speed, Velocity and AccelerationIn Two Dimensions

An airplane, initially flying at 150 m/s due east, makes a gradual turn at constant speed towards the south. After 15 seconds, the plane is pointing due south. What is its average acceleration?

December 9, 2009 4

December 9, 2009 5

tavv xxx 0

2

002

1tatvxx xx

tvvxx xx 002

1

2

0

22 xxox vvxxa

Equations of Kinematics in Two Dimensions

1)

2)

3)

4)

tavv yyy 0

2

002

1tatvyy yy

tvvyy yy 002

1

2

0

22 yyoy vvyya

Same as before, only with subscripts for each direction of motion

x y

Dec. 8, 2006 Final Exam #5

A cannonball is aimed 30 degrees above the horizontal and is fired with an initial speed of 125 m/s at ground level. How far away from the cannon will the cannonball hit the ground?

December 9, 2009 6

December 9, 2009 7

A car drives straight off the edge of a cliff that is 54 meters high. The police at the scene of the accident observe that the point of impact of the car is 130 meters from the base of the cliff.

How fast was the car traveling when it went over the cliff?

1. Velocity is constant if a zero net force acts

2. Acceleration is proportional to the net force and inversely

proportional to mass:

Force and acceleration are in the same direction

3. Action and reaction forces are equal in magnitude and

opposite in direction

Newton’s Laws of Motion

0if0 Fa

aFF

a

mm

so

00 Fa

if onAccelerati

constant Velocity v

Newton’s First Law (law of inertia)

Object of mass m

The velocity is constant if a zero net force acts on the mass.That is, if a number of forces act on the mass and their vector sum is

zero:

0....21 FFFnet

So the acceleration is zero and the mass remains at rest or hasconstant velocity.

Newton’s Law of Gravitation

Newton’s law of universal gravitation can be applied to extended (finite size) objects such as planets.

The prove requires the use of calculus (not done here).

The general equation:

works as long as the mass of the object is distributed around the center with spherical symmetry.

In this case r is the distance between the centers of the spheres (not the surfaces).

221

rmGm

Fg

Rw

EM

m

m106.38R

kgM6

E

2410985.

Earth

What is the gravitational force (weight, w) of a mass m on the earth’s surface?

26

2411

2 10386

10985106376

).(

).().(

mkg m

kg /NmR

GmMw 22E

mgm w 809.2R

GMg E At the earth’s surface

Above earth’s surface, weight decreases with distance r from thecentre of the earth as 1/r2.

Why is g=9.80 m/s2 ?

Problem 4.26 (6th edition): The weight of an object is the same on planets A and B. The mass of planet A is 60% that of planet B.

Find the ratio of the radii of the planets.

rw

m

M

Static friction occurs when there is no sliding or skidding:

• object at rest

• car moving without skidding or spinning wheels – where the tire meets the road, the tire is momentarily at rest, so

the friction is “static”

Static Friction

Kinetic Friction

There is sliding or skidding.

• usually less than the static friction force – it is hard to get an object sliding, and easier to keep it sliding.

The static friction force fs

opposes applied forces andincreases as the applied force Fis increased.

fs rises only to a maximum value

and then the block begins to slide.

Static Friction

yNsxs Ff ,,

force normal F

friction static of tcoefficien

N

s

Tension Force

Tension – the force within arope or cable that is used to pull an object.

A force T applied to the end ofthe rope is transferred to otherend of the rope where it exertsthe same force on the block.

The block exerts an equal andopposite force on the rope(Newton’s 3rd law).

Problem 4.38 (edition 6):

A cup of coffee sits on a table in a plane ( s = 0.30 ). The plane accelerates.

What is the maximum acceleration before the cup starts to slide?

NF

g

m

pa

gFN

m ca

sf

Problem 4.26:

Can the person who is pulling the rope ever make the ropeperfectly horizontal?

To support the weight:

sin2Tmg

Dec. 17, 2005 Final Exam #22

A spring scale is loosely fastened to the ceiling of a railway car. When a 1.0-kg block is hung from the scale, it reads 12 Newton and is oriented as shown in the figure. What is the approximate acceleration of the car, as measured by an observer at rest on the ground, outside the car?

December 9, 2009 19

Dynamics of Uniform Circular Motion

Period of circular motion: T = 2r/v

• Centripetal acceleration: ac = v2/r

• Centripetal force: Fc = mac = mv2/r

• For motion in a horizontal circle,

– equilibrium in the vertical direction, vertical forces cancel– use Newton’s second law to relate net horizontal force to

the centripetal acceleration

Vertical Circular Motion

rv

mmgFN

21

1

rv

mmgFN

23

3

rv

mFN

22

2

rv

mFN

24

4

1) force toward centerweight larger than mg

2) force toward center

3) force toward centerrider falls off if

4) force toward center

03 NF

rv

mmg23

rgv 3The rider falls off if

Problem 5.40:

A motorcycle is traveling up one side of a hill and down theother side. The crest is a circular arc with a radius of 45 m.

Determine the maximum speed that the motorcycle can have while moving over the crest without losing contact with the road.

The net downward force on the bike atthe crest of the hill allows the motorbiketo remain in contact with the ground.Then FN > 0.

That is:net downward force = centripetal force, mv2/r.

Conservation of Mechanical EnergyIn the absence of applied forces and friction:The work done by a net zero applied force is zeroSo, PEKE 0

And KE + PE = E = total energy

is constant

Total Energy is Conservedin a Closed System

Other kinds of potential energy:

• elastic (stretched spring)

• electrostatic (charge moving in an electric field)

Work done in lifting an object

Alternative view: define a different form of energy-Gravitational potential energy:

Define:

Mechanical energy = kinetic energy + potential energyMechanical energy:

Then:

ymPE g

mgymvE 2

2

1

Work done by applied force, F, is (change in KE) + (change in PE)

So, PEKEFhW

Problem 6.40

A particle, starting from point A in the drawing, is projected down the curved runway. Upon leaving the runway at point B, the particle is traveling straight upward and reaches a height of 4.00 m above the floor before falling back down.

Ignoring friction and air resistance, find the speed of the particle at point A.

P 6.28

An extreme skier, starting from rest, coasts down a mountain that makes an angle of 21.3 ° with the horizontal. The coefficient of kinetic friction between her skis and the snow is 0.213. She coasts for a distance of 15.0 m before coming to the edge of a cliff. Without slowing down, she skis off the cliff and lands downhill at a point whose vertical distance is 2.46 m below the edge. How fast is she going just before she lands?

December 9, 2009 27

Oct. 25, 2006 Midterm #8

A 50 kg girl runs up a flight of stairs in a time of 12.5 sec. The stairs are 7.0 meters long and make an angle of 27 degrees above the horizontal. What is the average power that she produced?

December 9, 2009 28

Conservation of Momentum

Two isolated masses collide. Theinitial total momentum is:

With

While the masses are in contact, theyexert equal and opposite forces on eachother (Newton’s third law).

So the impulse acting on m1 is equal in magnitude and opposite indirection to the impulse acting on m2

21 ppp

0222

0111

vp

vp

m

m

2112 FF

Therefore, (change in momentum = impulse)

After the collision:

So, the total momentum after the collision is:

That is, total momentum is conserved:

21 pp

12222

111

ppppvp

ppvp

f2

f1

m

m

p

pp

ppppppp

21

121121

2121 pppp

2211011 ff vmvmvm :Momentum Conserved

So,

Then use conservation of energy

Combine the twoequations and aftersome algebra:

2

110112 m

vmvmv f

f

222

211

2011 2

1

2

1

2

1ff vmvmvm

21

1012

21

21011

2

mmm

vv

mmmm

vv

f

f

0121

21

0 vvv then

mmif

ff ,

,

P 7.34

The drawing shows a collision between two pucks on an air-hockey table. Puck A has a mass of 0.0240 kg and is moving along the x axis with a velocity of +5.61 m/s. It makes a collision with puck B, which has a mass of 0.0480 kg and is initially at rest. The collision is not head-on. After the collision, the two pucks fly apart with the angles shown in the drawing. Find the speed of (a) puck A and (b) puck B.

December 9, 2009 32

Correspondence between Linear and Angular Motion

to

2

2

1ttoo

too 2

1

222 oo

atvv o

2

2

1attvxx oo

tvvxx oo 2

1

222 oo vvxxa

xv

a

Angular Linear

In addition we now also have:

andrl rv

Example:

The wheels of a bicycle have an angular velocity of 20 rad/s. The brakes are applied, bringing the bicycle to a uniform stop.During braking, the angular displacement of the wheels is 15.92revolutions.

a) How much time does it take to stop?b) What is the angular acceleration of the wheels?

P 9.68

A platform is rotating at an angular speed of 2.2 rad/s. A block is resting on this platform at a distance of 0.30 m from the axis. The coefficient of static friction between the block and the platform is 0.75. Without any external torque acting on the system, the block is moved toward the axis. Ignore the moment of inertia of the platform and determine the smallest distance from the axis at which the block can be relocated and still remain in place as the platform rotates.

December 9, 2009 35

A “How-to” Approach To Torque Problems

1. Identify your point of rotation (P) (usually either given or obvious) Not so obvious for free falling bodies – center of gravity

2. Identify the point at which a force is applied and establish the distance (d) to the point of rotation

3. Identify a line (the lever arm) from the point of rotation to the line of action of the force, that is perpendicular to the line of action

4. Identify the angle (θ) between the line (d) and the force direction

5. Calculate the length of the lever arm from (d) and (θ): l = d sin (θ)

6. Calculate the torque: T = F l = F d sin (θ)

P 10.22

An object attached to a horizontal spring is oscillating back and forth along a frictionless surface. The maximum speed of the object is 1.48 m/s, and its maximum acceleration is 7.13 m/s2. How much time elapses betwen an instant when the object's speed is at a maximum and the next instant when its acceleration is at a maximum?

December 9, 2009 41

(a) The mass and the radius of the sun are 1.99 × 1030 kg and 6.96 × 108

m. What is its density?

(b) If a solid object is made from a material that has the same density as the sun, would it sink or float in water? Why?

(c) Would a solid object sink or float in water if it were made from a material whose density was the same as that of the planet Saturn (mass = 5.7 × 1026 kg, radius = 6.0 × 107 m)? Provide a reason for your answer.

Problem 11.44

P 11.75

A uniform rectangular plate is hanging vertically downward from a hinge that passes along its left edge. By blowing air at 11.0 m/s over the top of the plate only, it is possible to keep the plate in a horizontal position, as illustrated in part a of the drawing. To what value should the air speed be reduced so that the plate is kept at a 30.0° angle with respect to the vertical, as in part b of the drawing?

December 9, 2009 50

A large tank is filled with water to a depth of 15 m. A spout located 10.0 m above the bottom of the tank is then opened as shown in the drawing. With what speed will water emerge from the spout?

December 9, 2009 51

Thermal Expansion

Linear expansion – the increase in length, width or thickness whenan object is heated.

= coefficient oflinear expansionTypical values formetals 1510

-6per C

o.

TLL 0

That’s all !

Make sure you:

1) Understand the concepts presented in this review and know how to apply them.

2) Work some problems that are similar to those worked in this review

3) Understand that it is still your responsibility to review and understand all of the material presented in this course !

Good Luck …