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Instructor(s): Profs. Acosta, Hamlin, Muttalib
PHYSICS DEPARTMENT
Exam 2
PHY 2048
Name (print, last first):
March 24, 2017
Signature:
On my honor, I have neither given nor received unauthorized aid on this examination.
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Axis
R
Axis
Axis
Annular cylinder
(or ring) about
central axis
Hoop about
central axis
R1
Solid cylinder
(or disk) about
central axis
R2
L
R
1
1
I = 2 M(R 12 + R 22 )
I = MR 2
Axis
Solid cylinder
(or disk) about
central diameter
Axis
L
L
I = 2 MR2
Thin rod about
axis through center
perpendicular to
length
Axis
Solid sphere
about any
diameter
2R
R
1
1
Axis
2
1
I = 4 MR2 + 12 ML2
I = 12 ML2
Thin spherical
shell about
any diameter
Axis
R
I = 5 MR2
Hoop about
any diameter
Axis
Slab about
perpendicular
axis through
center
2R
b
a
2
I = 3 MR2
1
I = 2 MR2
1
I = 12 M(a2 + b2)
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Formula-sheet: Exam 1
• For constant acceleration ~a:
~v = ~v0 + ~at
2
vx2 = vx0
+ 2ax (x − x0 )
1
~r = ~r0 + ~v0 t + ~at2
2
2
vy2 = vy0
+ 2ay (y − y0 )
Acceleration due to gravity: g = 9.8 m/s2 = 32 ft/s2 vertically down.
• For force acting on a body of mass m: F~ = m~a
• Frictional forces: fs,max = µs N ; fk = µk N. N : normal force.
• For uniform circular motion: centripetal acceleration is ac =
• Kinetic energy: K =
v2
r
1
mv 2
2
• Work done by a constant force: W = F~ · d~ = F d cos (angle between F~ and d~ )
• Work-kinetic energy theorem: W = Kf − Ki
~ = îAx + ĵAy ;
• Vectors (2d): A
A=
q
A2x + A2y ;
~ and î);
Ax = A cos (angle between A
~ and î)
Ay = A sin (angle between A
Formula-sheet: Exam 2
• Potential Energy: ∆U = −W = −
Z
~
r2
F~ · d~r
Fx = −
~
r1
Gravitational (y=up): Fy = −mg
dU
dx
U (y) = mgy
Elastic (x from spring equilibrium): Fx = −kx
U (x) =
1 2
kx
2
• Mechanical Energy: Emec = K + U
Emec = constant for an isolated system with conservative forces
• Work-Energy: W (external) = ∆K + ∆U + ∆E(thermal)
• Center of Mass: ~rcom =
N
1 X
mi~ri
Mtot i=1
• Linear Momentum: p~ = m~v
d~
p
F~ =
dt
P~tot
Mtot =
N
X
mi
i=1
Impulse: J~ = ∆~
p=
Z
tf
F~ (t)dt = F~av ∆t
ti
d~
p
If F~ =
= 0 then p~ = constant and p~f = p~i
dt
N
X
dP~tot
= Mtot ~acom
F~net =
pi
~
= Mtot ~vcom =
dt
i=1
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• Elastic Collisions of Two Bodies, 1D
v1f =
m1 − m2
2m2
v1i +
v2i
m1 + m2
m1 + m2
• Rockets: Thrust = M a = vrel
2m1
m2 − m1
v1i +
v2i
m1 + m2
m1 + m2
Mi
∆v = vrel ln
Mf
v2f =
dM
dt
• Rotational Variables
angular position: θ(t)
angular velocity: ω(t) =
dθ(t)
dt
d2 θ(t)
dω(t)
=
dt
dt2
• For constant angular acceleration α:
angular acceleration: α(t) =
ω 2 = ω02 + 2α(θ − θ0 )
1
θ = θ0 + (ω + ω0 )t
2
ω = ω0 + αt
1
θ = θ0 + ω0 t + αt2
2
• Angular to linear relationships for circular motion
arc length: s = rθ
velocity: v = rω
tangential acceleration: aT = rα
centripetal acceleration: ac = rω 2
• Rotational Inertia: I =
N
X
mi ri2 (discrete)
i=1
R
I = r2 dm (uniform)
Parallel Axis: I = Icom + Mtot d2 (d is displacement from com)
1 2
1
1
2
Iω Kroll = M vcom
+ Icom ω 2
2
2
2
= rω acom = rα
• Rotational, Rolling Kinetic Energy: Krot =
Rolling without slipping: xcom = rθ
vcom
• Torque: ~τ = ~r × F~
(where k̂ = î × ĵ gives directions for cross product)
τ = rF sin (angle between ~r and F~ ) = rF⊥
~
dL
dt
L = rp sin (angle between ~r and ~
p) = rp⊥
~
dL
~ = constant and L
~f = L
~i
= 0 then L
If ~τnet =
dt
~ = ~r × ~
• Angular Momentum: L
p
~τ =
• Work done by a constant torque: W = τ ∆θ = ∆Krot =
1 2 1 2
Iω − Iω
2 2 2 1
• Power done by a constant torque: P = τ ω
• For torque acting on a body with rotational inertia I: ~τ = I~
α
• Stress and Strain (Y = Young’s modulus, B = bulk modulus)
F
∆L
Linear:
=Y
A
L
F
∆V
Volume: P =
= −B
A
V
• For torque acting on a body with rotational inertia I: ~τ = I~
α