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Instructor(s): Field/Matcheva
PHYSICS DEPARTMENT
Exam 2
PHY 2048
October 31, 2014
Signature:
Name (print, last first):
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(1) Code your test number on your answer sheet (use lines 76–80 on the answer sheet for the 5-digit number).
Code your name on your answer sheet. DARKEN CIRCLES COMPLETELY. Code your UFID number on your
answer sheet.
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test, this exam printout is to be turned in. No credit will be given without both answer sheet and printout.
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Use g = 9.80 m/s2
Axis
Hoop about
central axis
R
Axis
Axis
Annular cylinder
(or ring) 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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PHY2048 Fall 2014
PHY2048 Exam 1 Formula Sheet
Vectors
r
r
r
r
a = axiˆ + a y ˆj + az kˆ b = bxiˆ + by ˆj + bz kˆ Magnitudes: a = a x2 + a y2 + a z2 b = bx2 + by2 + bz2
r
r r
r r r r
r
Scalar Product: a ⋅ b = a xbx + a y by + a z bz Magnitude: a ⋅ b = a b cos θ (θ = angle between a and b )
r r
Vector Product: a × b = (a y bz − a z by )iˆ + (a z bx − a x bz ) ˆj + (a x by − a y bx )kˆ
r
r r r r
r
Magnitude: a × b = a b sin θ (θ = smallest angle between a and b )
Motion
r r
r
Displacement: ∆x = x(t2 ) − x(t1 ) (1 dimension)
∆r = r (t2 ) − r (t1 ) (3 dimensions)
r r
r
r
∆x x(t2 ) − x(t1 )
∆r r (t2 ) − r (t1 )
Average Velocity: vave =
(1 dim)
=
=
vave =
∆t
t2 − t1
∆t
t2 − t1
(3 dim)
Average Speed: save = (total distance)/∆t
r
r
dx(t )
dr (t )
Instantaneous Velocity: v(t ) =
(1 dim)
(3 dim)
v (t ) =
dt
dt
r
r
r
Relative Velocity: v AC = v AB + vBC (3 dim)
r r
r
r
∆v v (t2 ) − v (t1 )
∆v v(t2 ) − v(t1 )
Average Acceleration: aave =
(1 dim)
(3 dim)
=
=
aave =
∆t
∆t
t2 − t1
t2 − t1
r
r
r
dv (t ) d 2 r (t )
dv(t ) d 2 x(t )
Instantaneous Acceleration: a (t ) =
(1 dim)
(3 dim)
=
a (t ) =
=
dt 2
dt
dt 2
dt
Equations of Motion (Constant Acceleration)
v y (t ) = v y 0 + a y t
v z (t ) = v z 0 + a z t
vx (t ) = vx 0 + axt
x(t ) = x0 + vx 0t + 12 axt 2
vx2 (t ) = vx20 + 2ax ( x(t ) − x0 )
y (t ) = y0 + v y 0t + 12 a y t 2
v y2 (t ) = v y20 + 2a y ( y (t ) − y0 )
z (t ) = z0 + v z 0t + 12 a z t 2
v z2 (t ) = v z20 + 2a z ( z (t ) − z0 )
Newton’s Law and Weight
Weight (near the surface of the Earth) = W = mg (use g = 9.8 m/s2)
r
r
Fnet = ma (m = mass)
Magnitude of the Frictional Force
(µs = static coefficient of friction, µk = kinetic coefficient of friction)
Kinetic: f k = µ k FN
(FN is the magnitude of the normal force)
Static: ( f s ) max = µ s FN
Uniform Circular Motion (Radius R, Tangential Speed v = Rω, Angular Velocity ω)
v2
2πR 2π
mv 2
2
Centripetal Acceleration & Force: a =
Period: T =
= Rω F =
=
= mRω 2
R
ω
v
R
Projectile Motion
(horizontal surface near Earth, v0 = initial speed, θ0 = initial angle with horizontal)
Range:
R=
v02 sin(2θ 0 )
g
Max Height:
H=
v02 sin 2 θ 0
2g
Time (of flight):
Quadratic Formula
If:
ax + bx + c = 0
2
− b ± b 2 − 4ac
Then: x =
2a
tf =
2v0 sin θ 0
g
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PHY2048 Fall 2014
PHY2048 Exam 2 Formula Sheet
Work (W), Mechanical Energy (E), Kinetic Energy (KE), Potential Energy (U)
r
r
r r
r r
r r
dW
2
r→F ⋅d
Kinetic Energy: KE = 12 mv
Work: W = F ⋅ dr   
Power: P =
=
F
⋅v

Cons tan t F
∫rr
dt
2
1
r
r2
r
∫
r
Potential Energy: ∆U = − F ⋅ dr
Work-Energy Theorem: KE f = KEi + W
r
r1
Work-Energy: W(external) = ∆KE + ∆U + ∆E(thermal) + ∆E(internal)
Fx ( x) = −
dU ( x)
dx
Work: W = -∆U
U ( y ) = mgy
Gravity Near the Surface of the Earth (y-axis up): Fy = −mg
Spring Force: Fx ( x) = −kx
U ( x) = 12 kx 2
Mechanical Energy: E = KE + U Isolated and Conservative System: ∆E = ∆KE + ∆U = 0
E f = Ei
Linear Momentum, Angular Momentum, Torque
r
t
r
r
r r dp
r f r
p2
Linear Momentum: p = mv F =
Kinetic Energy: KE =
Impulse: J = ∆p = ∫ F (t ) dt
dt
2m
ti
N
Center of Mass (COM):
M tot = ∑ mi
i =1
r
r
r
dPtot
Net Force: Fnet =
= M tot aCOM
dt
r
1
rCOM =
M tot
N
r
∑ mi ri
i =1
r
1
vCOM =
M tot
N
r
∑p
i
i =1
N
r
r
r
Ptot = M tot vCOM = ∑ pi
i =1
N
Moment of Inertia:
I = ∑ mi ri 2 (discrete) I = ∫ r 2 dm
(uniform)
Parallel Axis: I = I COM + Mh 2
i =1
r
θf
r r dL
Torque: τ = r × F =
Work: W = ∫ τ dθ
dt
θi
r
r
r
r
r
dp
Conservation of Linear Momentum: if Fnet =
= 0 then p = constant and p f = pi
dt
r
r
r
r
r
dL
Conservation of Angular Momentum: if τ net =
= 0 then L = constant and L f = Li
dt
Rotational Varables
r r r
Angular Momentum: L = r × p
Angular Position:
r
dω (t ) d 2θ (t )
θ (t ) Angular Velocity: ω (t ) = dθ (t ) Angular Acceleration: α (t ) =
=
2
dt
dt
Torque: τ net = Iα Angular Momentum: L = Iω
Arc Length: s = Rθ
Rolling Without Slipping: xCOM = Rθ
L
Power: P = τω
2I
Tangential Acceleration: a = Rα
Kinetic Energy: Erot = 12 Iω 2 =
Tangential Speed: v = Rω
dt
2
vCOM = Rω aCOM = Rα
2
KE = 12 MvCOM
+ 12 I COM ω 2
Rotational Equations of Motion (Constant Angular Acceleration α)
ω (t ) = ω0 + αt
θ (t ) = θ 0 + ω0t + 12 αt 2
ω 2 (t ) = ω02 + 2α (θ (t ) − θ 0 )
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1. A hill makes an angle of 15◦ with the horizontal. If a 50-kg jogger runs a
distance of d = 200 m down the hill as shown in the figure, how much work is
done by gravity on the jogger (in J)?
(1) 25,364
(2) 38,046
(3) 50,729
d
15o
(4) −25, 364
(5) −38, 046
2. A hill makes an angle of 15◦ with the horizontal. If a 50-kg jogger runs a
distance of d = 300 m down the hill as shown in the figure, how much work is
done by gravity on the jogger (in J)?
(1) 38,046
(2) 25,364
(3) 50,729
d
15o
(4) −25, 364
(5) −38, 046
3. A hill makes an angle of 15◦ with the horizontal. If a 50-kg jogger runs a
distance of d = 400 m down the hill as shown in the figure, how much work is
done by gravity on the jogger (in J)?
(1) 50,729
(2) 25,364
(3) 38,046
d
15o
(4) −25, 364
4. Near the surface of the Earth a block of mass M is released from rest at a
height h = 10 m on a frictionless incline as shown in the figure. The block
slides down the frictionless incline to reach a flat rough horizontal surface.
If the block slides a horizontal distance d = 50 m along the rough surface
before coming to rest, what is the kinteic coefficient of friction between
the block and the horizontal surface?
(5) −50, 729
M
h
d
(1) 0.2
(2) 0.4
(3) 0.5
(4) 0.3
5. Near the surface of the Earth a block of mass M is released from rest at a
height h = 10 m on a frictionless incline as shown in the figure. The block
slides down the frictionless incline to reach a flat rough horizontal surface.
If the block slides a horizontal distance d = 25 m along the rough surface
before coming to rest, what is the kinteic coefficient of friction between
the block and the horizontal surface?
(5) 0.6
M
h
d
(1) 0.4
(2) 0.2
(3) 0.5
(4) 0.3
6. Near the surface of the Earth a block of mass M is released from rest at a
height h = 10 m on a frictionless incline as shown in the figure. The block
slides down the frictionless incline to reach a flat rough horizontal surface.
If the block slides a horizontal distance d = 20 m along the rough surface
before coming to rest, what is the kinteic coefficient of friction between
the block and the horizontal surface?
(2) 0.2
(3) 0.4
M
h
(4) 0.3
7. Near the surface of the Earth, an ideal spring with spring constant k is on
a frictionless horizontal surface at the base of a frictionless inclined plane as
shown in the figure. A block with mass M = 0.5 kg is pressed against the
spring, compressing it 6 cm from its equilibrium position. The block is then
released and is not attached to the spring. If the block slides a distance d = 2 m
up the inclined plane with θ = 30◦ before coming to rest and then sliding back
down, what is the spring constant k (in N/m)?
(1) 2,722
(2) 3,267
(3) 3,811
x-axis
(5) 0.6
d
(1) 0.5
x-axis
(4) 1,958
(5) 0.6
k
d
M
θ
(5) 4,611
x-axis
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8. Near the surface of the Earth, an ideal spring with spring constant k is on
a frictionless horizontal surface at the base of a frictionless inclined plane as
shown in the figure. A block with mass M = 0.6 kg is pressed against the
spring, compressing it 6 cm from its equilibrium position. The block is then
released and is not attached to the spring. If the block slides a distance d = 2 m
up the inclined plane with θ = 30◦ before coming to rest and then sliding back
down, what is the spring constant k (in N/m)?
(1) 3,267
(2) 2,722
(3) 3,811
(2) 2,722
(3) 3,267
d
θ
M
(4) 1,958
9. Near the surface of the Earth, an ideal spring with spring constant k is on
a frictionless horizontal surface at the base of a frictionless inclined plane as
shown in the figure. A block with mass M = 0.7 kg is pressed against the
spring, compressing it 6 cm from its equilibrium position. The block is then
released and is not attached to the spring. If the block slides a distance d = 2 m
up the inclined plane with θ = 30◦ before coming to rest and then sliding back
down, what is the spring constant k (in N/m)?
(1) 3,811
k
(5) 4,611
k
d
θ
M
(4) 1,958
(5) 4,611
10. The potential energy function of a point mass is given by the expression: U (x) = 2x2 − 8x, where x is the coordinate of the
body and U is measured in Joules. Only conservative forces are acting. Find the kinetic energy of the body (in J) at the
point of stable equilibrium if the mechanical energy of the body is 20 J.
(1) 28
(2) 30
(3) 32
(4) 12
(5) 14
11. The potential energy function of a point mass is given by the expression: U (x) = 2x2 − 8x, where x is the coordinate of the
body and U is measured in Joules. Only conservative forces are acting. Find the kinetic energy of the body (in J) at the
point of stable equilibrium if the mechanical energy of the body is 22 J.
(1) 30
(2) 28
(3) 32
(4) 12
(5) 14
12. The potential energy function of a point mass is given by the expression: U (x) = 2x2 − 8x, where x is the coordinate of the
body and U is measured in Joules. Only conservative forces are acting. Find the kinetic energy of the body (in J) at the
point of stable equilibrium if the mechanical energy of the body is 24 J.
(1) 32
(2) 28
(3) 30
(4) 12
13. Three uniform square slabs are arranged in the xy-plane as shown in the figure.
Each of the three squares have mass M and sides of length L. If L = 6 m, what
are the x and y components of the center-of-mass of the three slab system?
(1)
(2)
(3)
(4)
(5)
xcom
xcom
xcom
xcom
xcom
=5
=7
=5
=7
=4
m,
m,
m,
m,
m,
ycom
ycom
ycom
ycom
ycom
=7
=5
=5
=7
=7
xcom
xcom
xcom
xcom
xcom
=7
=5
=5
=7
=4
m,
m,
m,
m,
m,
ycom
ycom
ycom
ycom
ycom
=5
=7
=5
=7
=7
y-axis
L
L
m
m
m
m
m
14. Three uniform square slabs are arranged in the xy-plane as shown in the figure.
Each of the three squares have mass M and sides of length L. If L = 6 m, what
are the x and y components of the center-of-mass of the three slab system?
(1)
(2)
(3)
(4)
(5)
(5) 16
m
m
m
m
m
x-axis
y-axis
L
L
x-axis
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15. Three uniform square slabs are arranged in the xy-plane as shown in the figure.
Each of the three squares have mass M and sides of length L. If L = 6 m, what
are the x and y components of the center-of-mass of the three slab system?
(1)
(2)
(3)
(4)
(5)
xcom
xcom
xcom
xcom
xcom
=5
=5
=7
=7
=4
m,
m,
m,
m,
m,
ycom
ycom
ycom
ycom
ycom
=5
=7
=5
=7
=7
y-axis
L
m
m
m
m
m
L
x-axis
16. A small piece of cheese is placed at the center of a thin horizontal 2 kg rod
of length L. The rod rotates horizontally around its center of mass with an
angular velocity 10 rad/sec as shown in the figure. A 0.5 kg mouse originally
standing at the edge of the rod runs towards the cheese. What is the angular
velocity (in rad/s) of the rod when the mouse reaches the cheese?
Axis of
Rotation
Mouse
L /2
L
(1) 17.5
(2) 21.0
(3) 24.5
(4) 5.7
17. A small piece of cheese is placed at the center of a thin horizontal 2 kg rod
of length L. The rod rotates horizontally around its center of mass with an
angular velocity 12 rad/sec as shown in the figure. A 0.5 kg mouse originally
standing at the edge of the rod runs towards the cheese. What is the angular
velocity (in rad/s) of the rod when the mouse reaches the cheese?
(5) 6.9
Axis of
Rotation
Mouse
L /2
L
(1) 21.0
(2) 17.5
(3) 24.5
(4) 5.7
18. A small piece of cheese is placed at the center of a thin horizontal 2 kg rod
of length L. The rod rotates horizontally around its center of mass with an
angular velocity 14 rad/sec as shown in the figure. A 0.5 kg mouse originally
standing at the edge of the rod runs towards the cheese. What is the angular
velocity (in rad/s) of the rod when the mouse reaches the cheese?
(5) 6.9
Axis of
Rotation
Mouse
L /2
L
(1) 24.5
(2) 17.5
(3) 21.0
19. A block of mass M1 starts from rest at a height h = 9 m above the level
surface and slides down a smooth ramp as shown in the figure. The
block slides down the ramp, across the level surface, and collides with
a block of mass M2 which is at rest. The two blocks stick together and
travel up a smooth ramp. If the maximum height that the combined 2block reaches H = 4 m and all the surfaces are smooth and frictionless,
what is the mass M2 ?
(1)
1
M1
2
(2)
1
M1
3
(3)
2
M1
3
20. A block of mass M1 starts from rest at a height h = 16 m above
the level surface and slides down a smooth ramp as shown in the
figure. The block slides down the ramp, across the level surface, and
collides with a block of mass M2 which is at rest. The two blocks stick
together and travel up a smooth ramp. If the maximum height that
the combined 2-block reaches H = 9 m and all the surfaces are smooth
and frictionless, what is the mass M2 ?
1
(1) M1
3
1
(2) M1
2
2
(3) M1
3
(4) 5.7
(5) 8.0
M1
h
H
M2
x-axis
(4) M1
(5) 2M1
M1
h
H
M2
x-axis
(4) M1
(5) 2M1
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21. A block of mass M1 starts from rest at a height h = 25 m above
the level surface and slides down a smooth ramp as shown in the
figure. The block slides down the ramp, across the level surface, and
collides with a block of mass M2 which is at rest. The two blocks stick
together and travel up a smooth ramp. If the maximum height that
the combined 2-block reaches H = 9 m and all the surfaces are smooth
and frictionless, what is the mass M2 ?
(1)
2
M1
3
(2)
1
M1
2
(3)
1
M1
3
M1
h
H
M2
x-axis
(4) M1
(5) 2M1
22. Near the surface of the Earth, a wooden block with mass m = 2 kg is attached
to a string. The string is wrapped around a frictionless pulley with a radius
R = 0.5 m, and rotational inertia I = 2.5 kg·m2 as shown in the figure. The
pulley and the block are initially at rest. When the system is released and the
string begins to unwind, what is the tension in the string (in N)?
R I
m
(1) 16.33
(2) 22.62
(3) 28.00
(4) 19.60
(5) zero
23. Near the surface of the Earth, a wooden block with mass m = 3 kg is attached
to a string. The string is wrapped around a frictionless pulley with a radius
R = 0.5 m, and rotational inertia I = 2.5 kg·m2 as shown in the figure. The
pulley and the block are initially at rest. When the system is released and the
string begins to unwind, what is the tension in the string (in N)?
R I
m
(1) 22.62
(2) 16.33
(3) 28.00
(4) 29.40
(5) zero
24. Near the surface of the Earth, a wooden block with mass m = 4 kg is attached
to a string. The string is wrapped around a frictionless pulley with a radius
R = 0.5 m, and rotational inertia I = 2.5 kg·m2 as shown in the figure. The
pulley and the block are initially at rest. When the system is released and the
string begins to unwind, what is the tension in the string (in N)?
R I
m
(1) 28.00
(2) 16.33
(3) 22.62
(4) 39.20
25. Starting from rest at time t = 0, a circular wheel with radius R = 3 m is pulled
to the right along a horizontal surface at a constant acceleration of a = 2 m/s2
as shown in the figure. If the wheel rolls without slipping, how long (in s) does
it take the wheel to make 6 revolutions?
(1) 10.6
(2) 12.3
(3) 13.7
(2) 10.6
(3) 13.7
a
R
(4) 5.3
26. Starting from rest at time t = 0, a circular wheel with radius R = 3 m is pulled
to the right along a horizontal surface at a constant acceleration of a = 2 m/s2
as shown in the figure. If the wheel rolls without slipping, how long (in s) does
it take the wheel to make 8 revolutions?
(1) 12.3
(5) zero
(4) 5.3
(5) 15.1
a
R
(5) 15.1
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27. Starting from rest at time t = 0, a circular wheel with radius R = 3 m is pulled
to the right along a horizontal surface at a constant acceleration of a = 2 m/s2
as shown in the figure. If the wheel rolls without slipping, how long (in s) does
it take the wheel to make 10 revolutions?
(1) 13.7
(2) 10.6
(3) 12.3
(4) 5.3
28. Near the surface of the Earth a solid sphere of mass M , radius R = 0.5 m,
and moment of inertia I = 2M R2 /5 rolls along a horizontal surface and up a
ramp of height H = 0.7 m that makes an angle of θ = 20◦ with the horizontal
surface as shown in the figure. If the ball rolls without slipping on both the
horizontal surface and the ramp, what is the minimum number of revolutions
per second the ball must have while rolling on the horizontal surface in order
to reach the top of the ramp?
(1) 1.0
(2) 1.2
(3) 1.5
(2) 1.0
(3) 1.5
(2) 1.0
(3) 1.2
R
θ
(4) 0.5
H
(5) 2.0
R
θ
(4) 0.5
30. Near the surface of the Earth a solid sphere of mass M , radius R = 0.5 m,
and moment of inertia I = 2M R2 /5 rolls along a horizontal surface and up a
ramp of height H = 1.5 m that makes an angle of θ = 20◦ with the horizontal
surface as shown in the figure. If the ball rolls without slipping on both the
horizontal surface and the ramp, what is the minimum number of revolutions
per second the ball must have while rolling on the horizontal surface in order
to reach the top of the ramp?
(1) 1.5
(5) 15.1
(4) 0.5
29. Near the surface of the Earth a solid sphere of mass M , radius R = 0.5 m,
and moment of inertia I = 2M R2 /5 rolls along a horizontal surface and up a
ramp of height H = 1.0 m that makes an angle of θ = 20◦ with the horizontal
surface as shown in the figure. If the ball rolls without slipping on both the
horizontal surface and the ramp, what is the minimum number of revolutions
per second the ball must have while rolling on the horizontal surface in order
to reach the top of the ramp?
(1) 1.2
a
R
H
(5) 2.0
R
θ
(5) 2.0
FOLLOWING GROUPS OF QUESTIONS WILL BE SELECTED AS ONE GROUP FROM EACH TYPE
TYPE 1
Q# S 1
Q# S 2
Q# S 3
TYPE 2
Q# S 4
Q# S 5
Q# S 6
TYPE 3
Q# S 7
Q# S 8
Q# S 9
TYPE 4
Q# S 10
Q# S 11
Q# S 12
TYPE 5
Q# S 13
Q# S 14
Q# S 15
TYPE 6
Q# S 16
Q# S 17
Q# S 18
H
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TYPE 7
Q# S 19
Q# S 20
Q# S 21
TYPE 8
Q# S 22
Q# S 23
Q# S 24
TYPE 9
Q# S 25
Q# S 26
Q# S 27
TYPE 10
Q# S 28
Q# S 29
Q# S 30
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