PHY 105
FORMULA SHEET
Useful information:
Acceleration due to gravity:
𝑔 = 10 m/s2
Values of trigonometric functions for common angles:
sin30!
sin37!
sin45!
sin53!
sin60!
= 0.5, cos30!
= 0.6, cos37!
= 0.7, cos45!
= 0.8, cos53!
= 0.9, cos60!
= 0.9
= 0.8
= 0.7
= 0.6
= 0.5
Formulas:
Kinematic quantities (one-dimensional):
Δ𝑥 = 𝑥" − 𝑥#
𝑣$,&'( =
)$
𝑎$,&'( =
)'!
)*
, 𝑣&'( =
)*
, 𝑎$ =
+
)*
, 𝑣$ =
+$
+*
+'!
+*
Kinematic equations for constant acceleration
(one-dimensional):
𝑣$" = 𝑣$# + 𝑎$ 𝑡
,
𝑥" = 𝑥# + -9𝑣$# + 𝑣$" :𝑡
,
𝑥" = 𝑥# + 𝑣$# 𝑡 + -𝑎$ 𝑡 𝑣$"
= 𝑣$#
+ 2𝑎$ 9𝑥" − 𝑥# :
Kinematic quantities (multi-dimensional):
Δ𝒓
=⃗ = =𝒓⃗" − =𝒓⃗#
/⃗
)𝒓
=⃗&'( = )* , 𝒗
𝒗
=⃗ =
=⃗ &'( =
𝒂
/⃗
)𝒗
)*
,𝒂
=⃗ =
=⃗" = 𝒗
𝒗
=⃗# + =𝒂⃗𝑡
/⃗
+𝒓
,
+*
=⃗" = =𝒓⃗# + =𝒗⃗# 𝑡 + -=𝒂⃗𝑡 𝒓
/⃗
+𝒗
+*
Centripetal (radial) and tangential acceleration:
𝑎3 = 𝑎4 =
'"
4
Kinematic equations for constant acceleration
(multi-dimensional):
+'
, 𝑎* = A +* A
Period and frequency:
𝑇=
-54
'
,𝑓=
,
6
Newton’s second law:
Forces:
D =𝑭⃗ = 𝑚𝒂
=⃗
𝐹( = 𝑚𝑔 (force of gravity)
𝑓7 ≤ 𝜇7 𝑛, 𝑓8 = 𝜇8 𝑛 (force of friction)
𝐹7 = −𝑘𝑥 (spring force; Hooke’s law)
Work:
Translational kinetic energy:
𝑊 = =𝑭⃗ ∙ Δ𝒓
=⃗ = 𝐹Δ𝑟 cos 𝜃 (constant force)
𝐾 = -𝑚𝑣 -
$
𝑊 = ∫$ # 𝐹$ 𝑑𝑥 (varying force, one-dimensional)
$
,
Potential energy:
Work-kinetic energy theorem:
𝑈( = 𝑚𝑔𝑦 (gravitational)
𝑊9:* = ∆𝐾
𝑈7 = %"𝑘𝑥 - (elastic)
𝑊#9* = −∆𝑈
𝐹$ = −&'
(one-dimensional)
&!
Conservation of mechanical energy:
Internal energy:
∆𝐸;:3< = ∆𝐾 + ∆𝑈 = 0
∆𝐸#9* = 𝑓8 𝑑
Conservation of energy:
Power:
∆𝐸7=7*:; = ∆𝐾 + ∆𝑈 + ∆𝐸#9* = D 𝑊:$*
𝑃&'( = ∆)
, 𝑃&'( = +
∆*
∆*
∆𝐾 + ∆𝑈 = −𝑓8 𝑑 + D 𝑊:$*
Linear momentum:
𝑃 = &)
, 𝑃 = &+
&*
&*
Impulse:
*#
=⃗ = 𝑚𝒗
𝒑
=⃗
𝑰⃗ = ∆𝒑
=⃗ = [ D =𝑭⃗ 𝑑𝑡
+𝒑
/⃗
+*
D =𝑭⃗ =
*$
Conservation of linear momentum:
Center of mass:
∆𝒑
=⃗*!* = 0
=⃗?@ = ,% ∑# 𝑚# =𝒓⃗# (system of particles)
𝒓
=⃗?@ = ,% ∫ =𝒓⃗ 𝑑𝑚 (extended object)
𝒓
Angular kinematic quantities:
Δ𝜃 = 𝜃" − 𝜃#
𝜔&'( =
)A
,𝜔=
)*
+A
𝛼&'( =
)B
+B
)*
,𝛼=
+*
+*
Rotational kinematic equations for constant angular
acceleration:
𝜔" = 𝜔# + 𝛼𝑡
,
𝜃" = 𝜃# + -9𝜔# + 𝜔" :𝑡
,
𝜃" = 𝜃# + 𝜔# 𝑡 + -𝛼𝑡 𝜔"- = 𝜔#- + 2𝛼9𝜃" − 𝜃# :
Relationship between translational and angular
quantities:
𝑠 = 𝑟𝜃
Torque:
𝜏 = 𝑟𝐹 sin 𝜙 = 𝐹𝑑
𝑣 = 𝑟𝜔
𝑎* = 𝑟𝛼
Moment of inertia:
Newton’s second law for rotation:
𝐼 = ∑# 𝑚# 𝑟#- (system of particles)
D 𝜏 = 𝐼𝛼