Force due to spring is given by:

Use FBD for spring problems, the sign of the force should be clear from the diagram

Equation sheet Exam 1 PHY2048

Newton’s Laws of Motion

1. A body at rest will remain at rest a body in motion will remain in motion

unless acted upon by an external force

2. The net sum of forces accelerates an object by an amount proportional to

its mass and in the direction of the net forces.

3. For every action there is an opposite and equal reaction. Action reaction

pairs never act on the same object

mF = a

0F =

A on B B on A F F

Force of friction comes in two flavors. Static frictional forces apply when the object is at rest

with respect to the surface. Kinetic frictional forces apply when the object is moving with

respect to the surface. Both frictional forces always act parallel to the surface and are

proportional to the normal force.

2 2

1tan

x x x

y y y

x y

y

x

=

A B C

A B C

C C

C

C

A+ B C

C

2 2

1tan

x x x

y y y

x y

y

x

=

A B C

A B C

C C

C

C

A B C

C

B

Chapter 1: Measurements, Estimation, Vectors

Chapter 4&5: Newton’s Law and Applications

Definition of velocity and acceleration. note: velocity and acceleration are vector

quantities

0

2

0 0

2 2

0 0

1

2

2

fx x x

f x x

fx x x f

v v a t

x x v t a t

v v a x x

2

rad

va

R

For motion in a circle the acceleration vector points towards

the center of the circle with a magnitude given by:

Equations of motion in 2 or 1 dimension are given below. These apply only when the

acceleration is constant, gravitational acceleration is an example of a constant a. The

time t is the quantity that is common to both dimensions for 2-D problems

Chapter 2&3: Motion in a Straight Line and in a Plane

0

0

, lim

, lim

f i f ii

average instt

f i f i

average instt

t t dt

t t dt

x x x x dxv v

v v v v dva a

friction k

friction s

F N

F N

AC

B

Distance: A “scalar” quantity that describes the length of the path taken between

two locations

Displacement: A “vector” that points between two locations. The vector begin at theinitial location and ends at the final.

i

i

dist s

f i x = x xA

B

C

|A|

|A|cos

|A|sin

0

2

0 0

2 2

0 0

1

2

2

fy y y

f y y

fy y y f

v v a t

y y v t a t

v v a y y

( |0) |0

( |0) |0

0

cos

sin

y

x

f x f

f y f

a g

a

v v

v v

For projectile motion in the plane

v

x

y

Spring k F x

0 0 0

0 0 0

0

0

( ) ( )

( ) ( )

v t t

v t t

x t t

x t t

d t dt t dt

d t dt t dt

v a v = v a

x v x = x v

𝑣

equations of motion that apply generally

Equation sheet Exam II

Chapter 8: Momentum, Impulse, Conservation of Momentum

2

1

, ,

: ,

Momentum is conserved if there are no external forces

0

Impulse is defined as:

Impulse-momentum theorem:

x x y y

ext T initia

t

t

l T final

m p mv p mv

t

dt

p v

pF = p p

J = F

J = 12 p p

Collisions (momentum is cnrv’d)

Elastic: KE is conserved implies

the following velocity relation

Inelastic: KE is not conserved

Completely inelastic: objects stick

together after they collide and

1 1 2 2 3 3

1 2 3

1 1 2 2 3 3

1 2 3

cm

cm

x m x m x mx

m m m

y m y m y my

m m m

Center of Mass and Momentum

, , , ,B f A f B i A i v v v v

, ,B f A fv v

2 2

1 1

2 2

2 1 2 1

Work definition

cos

Work by varing force or curved path

Work-energy relation, definition of

1 1

2

cos

2

s s

s s

W Fs

W

KE

W K KE KE m

d

v m

d s

v

F

F s

F s

||

||

Definition of Power

,

,

av av av

WP P Fv

t

dWP P Fv

dt

Chapter 6: Work, Energy and Power

Chapter 10: Rotational Dynamics

The torque is given by the magnitude of where is a vector

pointing from the pivot point to the where the acts. The angle is the smallest angle between

, s

the vect

in ,

r F r F r

F

ors when located tail to tail.

, where is the moment of inertia and is the angular velocity

, for a single particle where is perpendicular distance from

axis a

Angular Mom

nd is

entu

the linear mo

m

mentu

L I

L mv

I

l

mv

l

2 2

2 1

,1 ,2

Work Energy theorem for angular motion

1

m

Conservation of Angular Momenta follows from Newton's 2

(2

thus if and onf if 0, ., the are

nd

no ext

Law

e

)

Total Total

ext

CM

ext

d dI

dt dt

W I

LI

L L ie

rnal torques

, for rigid bodies the relation is useful a rI

Chapter 9: Rotational Motion

2 1

2

0

2 2

1 12 2

2 1

Angular velocity and acceleration

;

Angular kinematic relati

1

on

2

( )

s

2

inst inst

d d

dt dt

t

t t

2 2 2 2

1 1 2 2 3 3

Potential Energy

Moment of Inertia, general form, see table 9.2

...i

CM

i ir r r m

U Mgy

I m rm m

2

2

Angular to linear relations

arc length

tangential

tangential

tange

radial

ntial

r

r

s r

v

a

va r

r

Rolling without slip

,

ping

0, 2bottom CM topv v vR R

2 2

,1 1 2

,2 ,1

,1

,1 ,2 1 2

,2

2

Potential Energy ( ) (conservative forces)

( )

1( )

2

Conservation of Energy

ela

gravity grav grav grav

elasstic elas

Total Total

Total elas grav other

elas

U

W U U mg y

W U U k x x

E E

E KE U U W

U

U

KE

y

,2 ,2 1 ,1 ,1grav elas grav elas otherU U KE U U W

Chapter 7: Potential Energy and Conservation of Energy

Force from Energy is

( ) ( )

In 3-D

ˆ ˆ ˆ

x

dU xF x

dt

U U U

x y x

F i + j + k

2 21 12 2

Kinetic Energy of object with rotational and linear motion

total cm cmKE Mv I

Equillibrium of rigid body

0 : 0, 0 and 0x yF F F

FBD for man w/ladder on friction less wall, The torque is computed

about axis at B but you can use anywhere else as axis of rotation

To solve equilibrium problems

begin by using FBD approach.

Each force is now applied on

extended object. Next write down

S F = 0, in x and y, and then write

down S = 0, take the axis of

rotation for the torques to be

anywhere on the object that is the

most convenient. Finally solve for

the unkwnown(s) from the

resulting three equation.

Chapter 11: Equilibrium

The speed of a mechanical wave is:

, where is frequency and is the wavelength.

The magnitude of depends only on the physical properties

of the media through which the wave is propagating.

wavev f f

v

Thus for a string of

length the wave speed

, where is the linear mass density or mass per unit length of the string.wave

l

Fv

Equation sheet Final Exam, PHY2048

Chapter 15: Mechanical Waves

Equation that describes a mecha

,

,

nical wave

( , ) cos 2

( , ) cos

y x t A

y x t A

x t

T

kx t

These eq. describe waves propagating to the right

with a phase (), amplitude A, angular frequency

𝜔 and wave number k

2

2

2

Velocity and acceleration of any particle on a transverse

( , ) sin

( , )

wa

o

e

s

v

c

y

y

dxv x t A kx

dt

d xa x t A kx

dt

t

t

2 2

2 2 2 2

max a e

2

v

Power in a sinousoidal wave,

( , ) sin

1,

2

P x t A kx t

P A

F

F P F A

1 2

2N

m m

rF = G r

2

22

2

G rad G Sat

Sat Earth EarthN Sat N

vF = ma => F = m

r

m m mvG m G v

r r r

Newton’s Law of Universal Gravity. F is the force exerted by objects of mass

m1 and m2 on one another at a distance r apart. The direction of force is along

the line joining the two objects. These force obey Newton’s third law and can

be considered an action reaction pair.

Equations for satellite motion in a circular orbit. Here r is the distance between the

satellite and the object being orbited. GN is Newton’s constant and T is the period

or time it takes the satellite to go one time around.

Chapter 13: Gravitation

2 2

Potential Energy ( ) of a mass a distance from center of the Earth

The weigth and gravitational constant g at surface of the earth is

, i

N EE

N E N Eg

E E

U r

G m mU

r

G m m G mw F g

R R

Kepler’s Laws

1. Each planet moves in an elliptical orbit with the sun at one of the foci of the ellipse

2. A line from a planet to sun sweeps out equal area in equal times

3. The periods of the planets are proportional to the 3/2 power of the major axis

length of their orbit

2 2

2 2

22

2

Simple harmonic motion, restorative force proportional to

, ,

Equations of motion for a mass attached to a spring

( ) cos( ); sin ; cos ;

x x

xx

x

d x d x kF kx ma kx m kx x

dt dt m

dx d xx t A t v A t a A t

dt dt

k

m

Amplitude ( max displacement from equillibrium)

Period, time it takes to complete one cycle

Frequency, how many cycles per sec Hz or 1

Angular frequency which is equal to 2

1, freque

A m

T s

f s

f rad s

fT

ncy and period are inverse of each other

Chapter 14: Periodic Motion

The physical pendulum

, where is gravitational constant and is the distance btw center-of-mass and axis, is the mass and is the moment of inertiamgd

d m II

g

Angular simple harmonic motion is

, where is the torsion constant and is the moment of inertiaII

2 2 2 2

Total Energy of a mass attached to a spring on a frictionless surface

1 1,

2 2T x

kE mv kx v A x

m

2

3/2

11 2 2

2 2

6.67 10

EarthN

N Earth

N

m rG T r

r T G m

G N m kg

The simple pendulum

, where g is the gravitational acceleration, L is the length of the pendulumg

L

2

1 2

2 2

2 1

Power and Intensity and the Inverse square law

; 4

r

r

PI

I r

I

2 2with and k

T

1 2

Wave superposition

( , ) ( , ) ( , )totaly x t y x t y x t

1

1

( , ) sin sin

( 1,2,3...)2

Standing waves on a string

2 1( 1,2,3...)

SW

n

n

y x t A kx t

vf n nf n

L

Ln

n n

Chapter 12: Fluid Mechanics

Density and press

;

ure

p

m

V

dF

dA

2 2

Pressure in a fluid at rest (Pascal's Law)

, p p gh Where the p2 and p1 are the

pressures at pts 2 and 1, 𝜌 is the density of the fluid, g is

the gravitational acceleration and h is the distance in the

vertical btw pts 2&1

Archimede's principle

F B f gV The Buoyant is theupward forceexcerted by fluid on abody immersedequal to amount offluid displaced.

0

F 0 B mg

F

2 21 11 1 1 2 2 22 2

Bernoulli's equation

= p gy v p gy v

gauge ab atmp p p

Absolute pressure: The total pressure including

atmospheric. Gauge pressure: The excess pressure

above atmospheric pressure