Final Exam Study Guide

What are the rules for dimensional analysis?

Rules for dimensional analysis 

1. Dimensional analysis when the dimensions of a quantity instead of the value  are used to solve a problem; this is often used to check a problem 2. Dimensions can be treated as algebraic symbols for example: a. V=l*w*h each is a unit of length so volume is equal to L3 

3. The quantities can only be added or subtracted if they have the same  dimensions

4. Each side of the equation must have the same dimensions

5. Trigonometric functions only apply to dimensionless quantities 6. Logarithms and exponential functions only apply to dimensionless quantities

This is a good way to check your answer or work to an answer if you don’t have a  clue what to do. Know what units compose a N [kgm/s2], J [Nm], W [J/s].  

What are the rules for significant figures?

Rules for significant figures 

1. When multiplying or dividing the answer will have the same amount of  significant figures as the number in the equation with the least amount of  significant figures

a. (12.5)*(2.0)=25

2. When adding or subtracting the answer has the same amount of decimal  places as the number with the least amount of decimal places in the problem 3. Numbers such as e and π don’t place restrictions on results

4. Scientific notation can be used to help avoid confusion with significant figures 5. Keep extra significant figures when solving the equation and only bring to the proper amount after the equation has been solved Don't forget about the age old question of What does factual knowledge mean?

6. When the answer begins with a 1 it is okay to keep an extra significant figure

What are the five kinematic equations?

We also discuss several other topics like What are the two important procedures that were initiated by the world trade organization?

Chances are this won’t be a huge factor in the final but it’s still a handy thing to  know.

Five Kinematic Equations

Five Rotational Equations 

These are very similar to the translational ones except d=θ, v=ω, and a=α. A very useful formula to use to work between the two formulas is: v=ωr

Newton’s Laws 

1. “Every body continues in its state of rest or of uniform motion in a straight  line, unless it is compelled to change that state by forces impressed on it.” 2. F=ma

3. F[Bon A]=−F[AonB] Equal and opposite reaction.

Two and Three Dimensional Motion 

∙ Position: r=xi+yj+zk We also discuss several other topics like What is an example of an intangible service?

∙ Displacement: ∆ r=∆ xi+∆ yj+∆ zk

∙ Average Velocity: v=∆r

∆ t

∙ Instantaneous Velocity: v=drdt

∆t Or a=dvdt

∙ Acceleration: a=∆ v

∙ Projectile motion:

v

(¿¿0 tcos(θ))i+(v0tsin (θ)−12g t2) j r=¿

∙ Range: Rmax=v02g

Forces 

Drawing a picture is especially useful when given a problem that involves multiple  forces. This will add a visual that will help with understanding the math you’ll be  doing. Don't forget about the age old question of What is the key issue of developing a county?

Frictional Force 

∙ Static friction  

∙Fsmax=μs FN 

oμs - The coefficient of friction (dependent on what the surface is) oF N - Normal Force

∙ Kinetic Friction  

∙Fk=μ k FN 

oμk -coefficient of kinetic friction (unitless)

∙ Acceleration

∙ax=FT−Fk Don't forget about the age old question of What is the 1913 alien land act?

m

o FT- The force of tension Don't forget about the age old question of What argument does russell use to defend his claim that the “sense-data” we are immediately aware of in sensory experience are “signs” of external objects rather than the external objects themselves?

∙ Rolling friction  

∙Fr=μrF N 

oμr -coefficient of rolling friction

∙ Drag force

∙F D=−bv

o b- the experimental constant. It is negative to show it’s moving  opposite velocity (v).

∙ For the drag force on blunt objects (What we mostly will work with) drag force is shown by:

∙F D=12C Aρ v2

o C- The drag coefficient

o ρ- The density of the medium

A- The object’s perpendicular cross-sectional area

o v- The speed of the objects

Centripetal Force 

The magnitude of centripetal force is found by:

F C=mv2r

If the origin is in a polar coordinate system

F C=−mv2 

rr^

Keplers Laws

Planetary Motion

a=r p+r A 

2

1. rp- The perihelion distance

2. rA- The aphelion distance

3. This answer comes in Astronomical Units or AU

4. 1 AU=1.5E11 meters

o Were only going to consider the first and the third since they’re the ones  formulas apply to.

o In Kepler’s third law it says that the square of a planet’s period T is  proportional to the cube of its semi-major axis

2 =a[AU ] 

3

oT[ yr ] 

o Newton’s law of universal gravity shows:

oF G=Gm1m2 

r2 

o G- The gravitational constant= 6.673E-11 o The gravitational field of any source can be found from: Unit vector pointing away

g (r)=−GM

o

r2(¿ particle)

o G-Gravitational constant o M- Mass

o r- Radius

Energy

o Kinetic Energy

oK=12m v2 

o This answer is in Joules

o Potential Energy  

o PE=mgh

o To find gravitational potential energy:

o UG (r)=−Gm1m2 

r

o Elastic potential energy:

oUe=12k x2 

o Or

oUe=12k y2 

o K= the spring constant

o The sum of a systems Potential and Kinetic energy is its mechanical energy.  This is shown by:

o E=K+U

o The conservation of mechanical energy is shown by:

o∆ K+∆U =0

Momentum 

o Momentum moves in the same direction as velocity and is given  by the equation:

p=mv

o The initial momentum is always the same as the final momentum.   In the problems that ask for recoil motion use the following  formula often times in momentum problems they are paired with  translational kinematic equations:

m1v1=m2v2

Rocket 

An open systems is a system that gains or loses mass.  This is the  formula to us for the rocket or a similar system:

Fthrust=v(∆ M

∆t)

Torque and Inertia 

Torque can also be found by taking the cross product of two vectors.   The magnitude of Torque is given by:

R=ABsinφ

Inertia is the torque divided by the angular acceleration.

In translational motion the more massive particle has the most rotational  inertia

Mass distribution with respect to the rotational axis also impacts  rotational inertia; the farther away the mass is from the rotational axis  the more rotational inertia will exist.

Rotational inertia can be found by:

n

miri2

M is the mass of the particle

I=∑ i=1

r is the distance from the rotational axis

On a continuous object the rotational inertia is:

I=∫r2dm

The parallel axis theorem where M is mass, h is the perpendicular  distance between the new axis and the axis through the center of mass  and ICM is the rotational inertia around the center of mass

I=I CM +M h2 

The center of mass is found by:

xCM =(m2 

m1+m2) x2 

For center of mass multiply the associated mass with the associated x  coordinate.

Kinetic Energy for rotating object:

Kr=12I ω2 

Conservation of energy holds true for rotational motion similar to  translational motion

Ki+Kri+Ui+W=K f +Krf +U f +∆ Eth 

Ki­ is the translational kinetic

Kri­ is rotational kinetic

Newtons second law is shown through torque and is the sum of all  torque.

If the system is conserved torque initial is equal to torque final

Equilibrium 

Two conditions need to be met for an object to be at equilibrium: 1. Ftot=dpdt Ftot=0

2. τtot=dLdtτtot=0

These are all that is needed to solve and equilibrium problem.  To  determine the radius for the torque always set a reference point for the  initial radius.  I think this is easier if an endpoint is chosen.

Cross Product 

You may need to know this for torque.

Cross Product: To do cross product the simplest way is to use  matrices. If r=1i +1j and F=2i+3j the r x F=

i j k 1 1 0 2 3 0

=i1 0

3 0− j1 0

2 0+k1 1

2 3

The discriminate of each matrix is found. (ac-bd) So the answer to this problem is k.

Fluid Mechanics 

Density ρ=(mass/volume)

Force

Pressue=

Area

The density of water is 1000 kg/m3 

Most problems can be solved using the following two equations in some fashion: A1v1=A2v2 

And Bernoulli’s Equation:

P1+12ρ v12+ρg y1=P2+12ρ v22+ρg y2 

ρ= density of fluid

y= height

Think back to conservation of energy to use this one. It’s the same thing. This  formula can also be used for flowing air.