Exam 3 Solutions now posted here

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PHYSICS 160
Spring 2015
NAME________________________
ID_______________
Exam 3: Version A
When you are done, PLACE YOUR SCANTRON SHEET
INSIDE YOUR TEST and put these on the correct color pile at
the front of the room
Scalar Product:
! ! ! !
A ⋅ B = A B cos θ AB = Ax Bx + Ay By + Az Bz
Vector Product:
! !
! !
A × B = A B sin θ AB ,
! !
A × B = (Ay Bz − Az B y )iˆ + ( Az Bx − Ax Bz ) ˆj + (Ax B y − Ay Bx )kˆ
Kinematic equations of motion:
v fx = vix + a x t
1
x f = xi + vix t + a x t 2
2
v fx 2 = vix 2 + 2a x ( x f − xi )
ω = ω 0 + αt
1
θ = θ 0 + ω 0t + αt 2
2
2
2
ω = ω 0 + 2α (θ − θ 0 )
Radial Acceleration:
a rad =
v 2 4π 2 r
=
r
T2
Newton’s second law
!
!
∑ F = ma
Magnitude of kinetic friction
fk = µ k N
Definition of work
! !
W = ∫ F ⋅ ds
Definition of kinetic energy:
KE =
1 2
mv
2
Change in gravitational potential energy:
ΔU g = mg Δy
Magnitude of static friction
f s ≤ µs N
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Force of a compressed spring:
Fs = −kx
Elastic potential energy:
U el =
1 2
kx
2
Work-Energy Theorem:
Wnet ,ext = ΔU + ΔKE
Center-of-mass position
X COM =
1
M
n
∑x m
i
i
i =1
Definition of momentum
!
!
p = mv
Conservation of momentum
!
!
pi = p f
Definition of torque
! ! !
τ = r×F
Newton’s second law for rotation
!
∑τ
!
= Iα
Rolling:
aCOM = αR
vCOM = ωR
! ! !
!
!
Angular momentum: L = r × p or L = I ω
with
I = ∑ mi ri 2
i
Idisk = MR2/2
Ihoop = MR2,
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14 questions total = 12 regular + 2 extra credit
All questions are 2 pts each
_____________________
1. A ball hits a wall and rebounds with the same speed, as illustrated in the diagram.
The changes in the components of the momentum of the ball are:
a. Δpx > 0, Δpy > 0
b. Δpx < 0, Δpy > 0
c. Δpx = 0, Δpy < 0
d. Δpx = 0, Δpy > 0
e. Δpx > 0, Δpy < 0
2. If the radii of the rear and front and rear
sprockets of a bike are rrear and rfront, as
shown, then what is the relation between
the angular accelerations ωrear and ωfront?
a. ωrear = ωfront
b. ωrear = ωfront rrear / rfront
c. ωrear = ωfront rfront / rrear
2
d. ωrear = ωfront (rrear / rfront )
2
e. ωrear = ωfront (rfront / rrear )
3. Three identical objects, each of mass M, are fastened to a massless rod of length L
as shown. The rotational inertia about one end of the rod of this array is:
a. ML2/2
b. ML2
c. 3ML2/2
d. 5ML2/4
e. 3ML2
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4. A man is marooned at rest on frictionless ice. In desperation, he hurls his shoe to the
right at 15 m/s. If the man weighs 720 N and the shoe weighs 4.0 N, the man moves to
the left at approximately:
a. 0
b. 8.3x10-2 m/s
c. 1.7x10-1 m/s
d. 15 m/s
e. 2.7x103 m/s
5. Two boys, with masses 40 kg and 60 kg, respectively, stand on a frictionless frozen
lake holding the ends of a massless 10 m long rod. The boys pull themselves together
along the rod. When they meet, how far will the 40 kg boy have moved? (Hint: making
a drawing may help)
a. 4 m
b. 5 m
c. 6 m
d. 10 m
e. need more information
In the figure below, the disk, hoop, and solid sphere are made to spin about a fixed axis
through their centers by means of a force as shown. Note that there’s no translation!
Each object has the same mass and radius and starts from rest.
6. After the same force is applied for 10 seconds on each object, rank the objects
according to their angular momentum:
a)
b)
c)
d)
e)
Disk > Sphere > Hoop
Disk < Hoop, Hoop = Sphere
Disk < Hoop < Sphere
Disk > Hoop, Hoop = Sphere
Disk = Hoop = Sphere
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7. After the same force is applied for 10 seconds on each object, rank the objects
according to their angular speed:
a)
b)
c)
d)
e)
Disk > Sphere > Hoop
Disk < Hoop, Hoop = Sphere
Disk = Hoop = Sphere
Disk < Hoop, Hoop < Sphere
Disk > Hoop, Hoop = Sphere
8. A rocket is fired straight upward on a windless day. At the peak of its trajectory, it
explodes into two parts, one with three times the mass as the other. Both pieces strike
the ground at the same time. You find the heavy piece 10m to the East of the launch
site. Where should you look for the other piece?
a) 3 m to the West of the launch site.
b) 10 m to the West of the launch site.
c) 20 m to the West of the launch site.
d) 30 m to the West of the launch site.
e) not enough information to tell.
A 16 kg block is attached to a cord that is wrapped around the rim of a flywheel of
diameter 0.40 m and hangs vertically, as shown in the figure. The rotational inertia of
the flywheel is 0.50 kg.m2. Take g=10 m/s2
9. What is the torque on the pulley in Nm terms of the tension T (in newtons) in the
string?
a. τ = 4T
b. τ = 0.4T
c. τ = 5T
d. τ = 0.4(T – 160)
e. τ = 0.5( T – 160)
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10. What is the relation between the linear acceleration of the block, a, and α, the
angular acceleration of the pulley?
a. a = α
2
b. a = (0.4) α
c. a = 0.4α
d. a = α/(0.4)
2
e. a = α/(0.4)
11. When the block is released and the cord unwinds, what is the magnitude of the
block’s acceleration, a (in m/s2), and the tension in the cord, T (in newtons)?
Take g=10 m/s2
a. a = 0.2, T = 0.6
b. a = 8.4, T = 26
c. a = 10, T = 160
d. a = 5, T = 13
e. a = 10, T = 260
12. A diver of weight 750 N stands at the end of a 4.5 m
diving board of negligible mass. The board is attached to
two pedestals 1.5 m apart.
What are the magnitude and direction of the force on the
board from the left pedestal?
A) 2250 N downward
B) 1125 N downward
C) 3000 N upward
D) 3000 N downward
E) 1125 N upward
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13. Extra Credit. A rod of mass M and length L is attached to a support about which it
can rotate. The rod is temporarily held in a
horizontal position by a vertical stick as shown. The
stick is suddenly removed.
What is the instantaneous angular acceleration, α,
of the rod in terms of g and L?
a.
b.
c.
d.
e.
g/L
3g/(2L)
2g/(3L)
gL
gL2
14. Extra Credit A merry-go-round with a radius of 2 m and a moment of inertia 200
kgm2 is rotating at 0.2 revolutions per second. A child of mass 50 kg runs at the merrygo-round in a line tangent to its edge, and grabs onto it. After the child is on board, it
rotates at 0.3 revolutions per second. How fast was the child running?
a.
b.
c.
d.
e.
10 m/s
12 m/s
4 m/s
25 m/s
5 m/s
ω
2m
50kg