Equation Sheet Physics 121 Midterm 2 Spring 2019
Kinematics in 1D Mathematics and Vectors
Position vector !⃗ = $%̂ + ()̂ + *+, Quadratic Equation -$ . + /$ + 0 = 0
;<
Δ$ = $4 − $5 = 6 78 9: −/ ± √/. − 4-0
$ −displacement Solution to quad eq’n. $=
;= 2-
Δ!⃗ = !⃗4 − !⃗5 Difference in Δ( = (4 − (5
Displacement vector
Δ!⃗ = Δ$%̂ + Δ()̂ + Δ*+, variable y: (B. = (. − (B
1
Distance between points DDDD⃗C
9 = CΔ! Integration of $ E 6 -$ E 9$ = -$ EIB
H+1
Δ!⃗ 9
Average velocity vector 7⃗JK = Differentiation of $ E (-$ E ) = H-$ ENB
Δ: 9$
9!⃗ 9$ P⃗ + QD⃗ =
Instantaneous velocity 7⃗ = ; 78 = Vector addition
9: 9: (P8 + Q8 )%̂ + RPS + QS T)̂
;<
Change in velocity P⃗ • Q
D⃗ = CP⃗CCQ
D⃗C cos Y
Δ78 = 6 -8 9: Scalar Product
$ − component ;= = P8 Q8 + PS QS + PZ QZ
Average acceleration Δ7⃗
-⃗JK = Vector Product CP⃗ × Q
D⃗C = CP⃗CCQ
D⃗C sin Y
vector Δ:
(
Instantaneous 97⃗ 978 9. $ Polar to rectangular ! = ^$ . + ( . ; tan Y =
-⃗ = ; -8 = = . $
acceleration 9: 9: 9: coordinates $ = ! cos Y ; ( = ! sin Y
Constant acceleration in 1D Circular Kinematics
1 Rotational to linear
position vs time $(:) = $5 + 75 : + -: . a = b!
2 position
Rotational to
Velocity vs time 78 (:) = 78,5 + -8 : de = 7; !
tangential accel.
. . Rotational to
Velocity vs position 78,4 = 78,5 + 2-8 Δ$ fe = -; !
tangential accel.
78,4 + 78,5 Centripetal 7;.
Displacement vs time Δ$ = Δ: -g = = de. !
2 acceleration !
Accel. on inclined plane -8 = h sin Y Rotational Inertia
.
Constant rotational acceleration Point masses i = ∑k5 !l,5
Rotational coordinate 1 Ring around ⊥ axis
b = b5 + de,5 : + fe : . i = no.
(angle) vs. time 2 through center
Rotational velocity vs Disk around ⊥ axis 1
de = de,5 + fe : i= no.
time through center 2
Rotational velocity vs . . Hollow sphere 2
de,4 = de,5 + 2fe Δb i = no.
angle around diameter 3
Solid sphere around 2
i = no.
diameter 5
Thin rod around ⊥ 1
axis through center i= nℓ.
12
Parallel axis theorem i = igs + k9.
,Equation Sheet Physics 121 Midterm 2 Spring 2019
Projectile Motion Energy
$ = $5 + 78,5 :
k.
Object position vs time 1 Energy Units tuvwx = +h
( = (5 + 7S,5 : − h: . a.
2
27S,5 27z sin Y Translational Kinetic 1 ~.
Time of flight :4y = = {;|JE} = k7 . =
h h Energy 2 2k
7z. sin 2Y Rotational Kinetic 1 J.
Range o = 78,5 :4y = {|z; = id. =
h Energy 2 2i
Change in CM Kinetic
Momentum & Collisions Δ{ÄÅ = (∑ÇÉ8; ) • Δ!ÄÅ
Energy
Momentum vector ~⃗ = k7⃗ Mechanical Energy ÑsÉgÖ = { + Ü
Energy of closed Ñ4 = Ñ5
Impulse t⃗ = Δ~⃗
system ΔÑ = Δ({ + Ñ5E; ) = 0
Momentum of isolated ~⃗4 = ~⃗5 Energy change with
ΔÑ = á = ∑(ÇÉ8;,E8 Δ$àE )
system Δ~⃗ = 0 external forces
k. Δ7B8 Energy dissipated by
Inertia and velocity =− ΔÑ;Ö = Ç}â4 9äJ;Ö
kB Δ7.8 friction
k. -B8 åç
Inertia and acceleration =− Power ã= = Ç⃗É8; • 7⃗
kB -.8 å;
Gravitational
Forces |ΔÜ è | = |khΔ(|
potential energy
k t Spring potential 1
Force Units êxë:uH = +h = ΔÜ}ä|5Eí = +($ − $z ).
a . k energy 2
;SäÉ Relative velocity and 7⃗B. = 7⃗. − 7⃗B
Labelling Ç⃗âS zE speed 7B. = |7⃗B. | = |7⃗. − 7⃗B |
45EJy
9~⃗ 1-D Coefficient of 7B.
Second Law Σ Ç⃗ = = k-⃗ Restitution x=− 5E5;5Jy
9: 7B.
.
Interaction Pair Non-convertible 1 ~;z;
Ç⃗B. = −Ç⃗.B .
{ÄÅ = k;z; 7ÄÅ =
(Third Law) Kinetic Energy 2 2k;z;
{gzEK = { − {ÄÅ
Convertible Kinetic
Gravitational Force CÇ⃗çz
è
C = kz h 1 kB k. .
Energy {gzEK = 7
2 kB + k. B.
Spring Force
Ç⃗}y = −+($ − $z )%̂ Torque and Angular Momentum
(Hooke’s Law)
î⃗ = !⃗ × Ç⃗
Impulse t⃗ = 6 Ç⃗ 9: = Δ~⃗ Torque
î = !ÇaïHY = !Çl = !l Ç
Je = ide
Work DDDD⃗ = 6 Ç⃗ • DDDD⃗
á = Ç⃗ • Δ! 9! Angular Momentum Je = !~ sin Y = !l ~ = !~l
DJ⃗ = !⃗ × ~⃗
} E Rotational Dynamics 9Je
Static Friction ÇB. ≤ ó} ÇB. ∑îÉ8;,e =
Equation 9:
ò E te = R∑îÉ8;,e TΔ: = ΔJe
Kinetic Friction ÇB. = óò ÇB. Rotational Impulse
,CHAP 1 6 . PHYSICAL QUANTITIES & UNITS
HAP
2
acceleration
1) Displacement of 8)
Average
Vx
---
8
-V =
Of -
Vi Ax , av
= -
ot
2) Distance &
d = |X , - Xz) (14)
3) Position J
= xY(ID)
4) speed v
v =
10/
5) Unit rector motion I
B = bi (ID)
6) Average
velocity
o
Var = -
Xt
2) Instantaneous
velocity
= m
, chaps
. I
3 Acceleration
& If an
object's velocity is
changing ,
the object is
accelerating .
② A always points to the same direction to or
positive a doesn't mean
accelerating ,
consider direction of V:
Eq >
-
Vi
⑰
>
-Vf Wi
-
T))( 1 1)
③ When I & a direction >
speed up
are same -
- -
different- -
slows down
3 3
.
projectile
Projectile motion
*
a =
g = 9 0m/5 . trajectory :
#***)
& launch affects only the initial v
,
once the object is released .
affect
it
by gravity
.
↓
⑧ A
everywhere
In Pm . .
is the same
3. 5 Motion with constant acceleration
Formulas :
D Voy kit a ot
Veloci
ty t)
:
M
(by Vi ,
a ,
↓
ve--
② Xf-Xi El + -Vilot + !
- - -- +
=
Vist
act +
-displacement
Vi
X + Xi-
= Vio+
Displacement &
(by Vi ,
a , +) *+ = Xi + aot" + Vict
Velocity ③ Vit =
Vi + 2aox
Ity Vi ,
a ,
ox
Kinematics in 1D Mathematics and Vectors
Position vector !⃗ = $%̂ + ()̂ + *+, Quadratic Equation -$ . + /$ + 0 = 0
;<
Δ$ = $4 − $5 = 6 78 9: −/ ± √/. − 4-0
$ −displacement Solution to quad eq’n. $=
;= 2-
Δ!⃗ = !⃗4 − !⃗5 Difference in Δ( = (4 − (5
Displacement vector
Δ!⃗ = Δ$%̂ + Δ()̂ + Δ*+, variable y: (B. = (. − (B
1
Distance between points DDDD⃗C
9 = CΔ! Integration of $ E 6 -$ E 9$ = -$ EIB
H+1
Δ!⃗ 9
Average velocity vector 7⃗JK = Differentiation of $ E (-$ E ) = H-$ ENB
Δ: 9$
9!⃗ 9$ P⃗ + QD⃗ =
Instantaneous velocity 7⃗ = ; 78 = Vector addition
9: 9: (P8 + Q8 )%̂ + RPS + QS T)̂
;<
Change in velocity P⃗ • Q
D⃗ = CP⃗CCQ
D⃗C cos Y
Δ78 = 6 -8 9: Scalar Product
$ − component ;= = P8 Q8 + PS QS + PZ QZ
Average acceleration Δ7⃗
-⃗JK = Vector Product CP⃗ × Q
D⃗C = CP⃗CCQ
D⃗C sin Y
vector Δ:
(
Instantaneous 97⃗ 978 9. $ Polar to rectangular ! = ^$ . + ( . ; tan Y =
-⃗ = ; -8 = = . $
acceleration 9: 9: 9: coordinates $ = ! cos Y ; ( = ! sin Y
Constant acceleration in 1D Circular Kinematics
1 Rotational to linear
position vs time $(:) = $5 + 75 : + -: . a = b!
2 position
Rotational to
Velocity vs time 78 (:) = 78,5 + -8 : de = 7; !
tangential accel.
. . Rotational to
Velocity vs position 78,4 = 78,5 + 2-8 Δ$ fe = -; !
tangential accel.
78,4 + 78,5 Centripetal 7;.
Displacement vs time Δ$ = Δ: -g = = de. !
2 acceleration !
Accel. on inclined plane -8 = h sin Y Rotational Inertia
.
Constant rotational acceleration Point masses i = ∑k5 !l,5
Rotational coordinate 1 Ring around ⊥ axis
b = b5 + de,5 : + fe : . i = no.
(angle) vs. time 2 through center
Rotational velocity vs Disk around ⊥ axis 1
de = de,5 + fe : i= no.
time through center 2
Rotational velocity vs . . Hollow sphere 2
de,4 = de,5 + 2fe Δb i = no.
angle around diameter 3
Solid sphere around 2
i = no.
diameter 5
Thin rod around ⊥ 1
axis through center i= nℓ.
12
Parallel axis theorem i = igs + k9.
,Equation Sheet Physics 121 Midterm 2 Spring 2019
Projectile Motion Energy
$ = $5 + 78,5 :
k.
Object position vs time 1 Energy Units tuvwx = +h
( = (5 + 7S,5 : − h: . a.
2
27S,5 27z sin Y Translational Kinetic 1 ~.
Time of flight :4y = = {;|JE} = k7 . =
h h Energy 2 2k
7z. sin 2Y Rotational Kinetic 1 J.
Range o = 78,5 :4y = {|z; = id. =
h Energy 2 2i
Change in CM Kinetic
Momentum & Collisions Δ{ÄÅ = (∑ÇÉ8; ) • Δ!ÄÅ
Energy
Momentum vector ~⃗ = k7⃗ Mechanical Energy ÑsÉgÖ = { + Ü
Energy of closed Ñ4 = Ñ5
Impulse t⃗ = Δ~⃗
system ΔÑ = Δ({ + Ñ5E; ) = 0
Momentum of isolated ~⃗4 = ~⃗5 Energy change with
ΔÑ = á = ∑(ÇÉ8;,E8 Δ$àE )
system Δ~⃗ = 0 external forces
k. Δ7B8 Energy dissipated by
Inertia and velocity =− ΔÑ;Ö = Ç}â4 9äJ;Ö
kB Δ7.8 friction
k. -B8 åç
Inertia and acceleration =− Power ã= = Ç⃗É8; • 7⃗
kB -.8 å;
Gravitational
Forces |ΔÜ è | = |khΔ(|
potential energy
k t Spring potential 1
Force Units êxë:uH = +h = ΔÜ}ä|5Eí = +($ − $z ).
a . k energy 2
;SäÉ Relative velocity and 7⃗B. = 7⃗. − 7⃗B
Labelling Ç⃗âS zE speed 7B. = |7⃗B. | = |7⃗. − 7⃗B |
45EJy
9~⃗ 1-D Coefficient of 7B.
Second Law Σ Ç⃗ = = k-⃗ Restitution x=− 5E5;5Jy
9: 7B.
.
Interaction Pair Non-convertible 1 ~;z;
Ç⃗B. = −Ç⃗.B .
{ÄÅ = k;z; 7ÄÅ =
(Third Law) Kinetic Energy 2 2k;z;
{gzEK = { − {ÄÅ
Convertible Kinetic
Gravitational Force CÇ⃗çz
è
C = kz h 1 kB k. .
Energy {gzEK = 7
2 kB + k. B.
Spring Force
Ç⃗}y = −+($ − $z )%̂ Torque and Angular Momentum
(Hooke’s Law)
î⃗ = !⃗ × Ç⃗
Impulse t⃗ = 6 Ç⃗ 9: = Δ~⃗ Torque
î = !ÇaïHY = !Çl = !l Ç
Je = ide
Work DDDD⃗ = 6 Ç⃗ • DDDD⃗
á = Ç⃗ • Δ! 9! Angular Momentum Je = !~ sin Y = !l ~ = !~l
DJ⃗ = !⃗ × ~⃗
} E Rotational Dynamics 9Je
Static Friction ÇB. ≤ ó} ÇB. ∑îÉ8;,e =
Equation 9:
ò E te = R∑îÉ8;,e TΔ: = ΔJe
Kinetic Friction ÇB. = óò ÇB. Rotational Impulse
,CHAP 1 6 . PHYSICAL QUANTITIES & UNITS
HAP
2
acceleration
1) Displacement of 8)
Average
Vx
---
8
-V =
Of -
Vi Ax , av
= -
ot
2) Distance &
d = |X , - Xz) (14)
3) Position J
= xY(ID)
4) speed v
v =
10/
5) Unit rector motion I
B = bi (ID)
6) Average
velocity
o
Var = -
Xt
2) Instantaneous
velocity
= m
, chaps
. I
3 Acceleration
& If an
object's velocity is
changing ,
the object is
accelerating .
② A always points to the same direction to or
positive a doesn't mean
accelerating ,
consider direction of V:
Eq >
-
Vi
⑰
>
-Vf Wi
-
T))( 1 1)
③ When I & a direction >
speed up
are same -
- -
different- -
slows down
3 3
.
projectile
Projectile motion
*
a =
g = 9 0m/5 . trajectory :
#***)
& launch affects only the initial v
,
once the object is released .
affect
it
by gravity
.
↓
⑧ A
everywhere
In Pm . .
is the same
3. 5 Motion with constant acceleration
Formulas :
D Voy kit a ot
Veloci
ty t)
:
M
(by Vi ,
a ,
↓
ve--
② Xf-Xi El + -Vilot + !
- - -- +
=
Vist
act +
-displacement
Vi
X + Xi-
= Vio+
Displacement &
(by Vi ,
a , +) *+ = Xi + aot" + Vict
Velocity ③ Vit =
Vi + 2aox
Ity Vi ,
a ,
ox
