Physics 8A Midterm 2 Review Solutions
Problem 1
1.
θ
R
M
mg
T
f
s
x
y
N
Figure 1: Free Body Diagram
2. The cylinder is in static equilibrium, so the Σ
~
F
=
m~a
= 0 and Σ
τ
=
Iα
= 0.
Choose coordinates as indicated on figure 1, with counterclockwise as positive for torque.
Σ
F
x
=
f
+
Tcosθ

Mgsinθ
= 0
(1)
Σ
F
y
=
N

Tsinθ

Mgcosθ
= 0
(2)
Σ
τ
=
fR

TR
= 0
(3)
We can use (3) and (1) to find the magnitude of the friction. From (3), we get
f
=
T
.
Pluging into (1), We get
Σ
F
x
=
f
(1 +
cosθ
)

Mgsinθ
= 0
(4)
which simplifies to
f
=
Mgsinθ
(1 +
cosθ
)
(5)
3. To find the minimum
μ
s
that will prevent the block from slipping, use the fact that
f
≤
μ
s
N
. so the minumum will be when
f
=
μ
s
N
. The rest of is just a bunch of algebra,
which is probably more involved than you will see on the midterm. Don’t worry too much
if you couldn’t get it to simplify all the way.
Plug
N
=
f
μ
s
and
T
=
f
into (2)
N

Tsinθ

Mgcosθ
=
f
(
1
μ
s

sinθ
)

Mgcosθ
= 0
(6)
Using (5), we get
Mgsinθ
(1 +
cosθ
)
(
1
μ
s

sinθ
)

Mgcosθ
= 0
(7)
or
Mgsinθ
μ
s

Mgsin
2
θ

(1 +
cosθ
)
Mgcosθ
= 0
(8)
1
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multiplying by
μ
s
and dividing by Mg, we get
sinθ

μ
s
sin
2
θ

μ
s
cosθ

μ
s
cos
2
θ
= 0
(9)
using the fact that
sin
2
θ
+
cos
2
θ
= 1, we get
sinθ

μ
s

μ
s
cosθ
= 0
(10)
and solving for
μ
s
we get
μ
s
=
sinθ
1 +
cosθ
= 0
(11)
Problem 2
1.
ρ
ρ
l
a
c
F
mg
b
F
ext
Cube
Figure 2: the forces acting on the block are from gravity and the bouyancy force. A downward
external force must be applied to keep it submerged.
To find the minimum force required us Σ
~
F
= 0, which tells us
F
ext
=
F
b

Mg
=
ρ
l
V
disp
g

ρ
c
V
c
g
=
ρ
l
a
3
g

ρ
c
a
3
g
=
a
3
g
(
ρ
l

ρ
c
)
(12)
The external force is now coming from a volume of material being attached to it. From
the free body diagram of the material, we see that
T
=
m
m
g

F
bm
=
m
m
g

ρ
l
V
disp by m
g
(13)
so using (12) with
F
ext
=
T
and using the fact that
m
m
=
V
m
ρ
m
, we get
m
m
g

ρ
l
V
dispbym
g
=
V
m
ρ
m

ρ
l
V
disp by m
ga
3
g
(
ρ
l

ρ
c
)
(14)
and solving for
V
m
(assume that the material is fully submurged as well, so that
V
m
=
V
disp by m
)
V
m
=
a
3
(
ρ
l

ρ
c
)
ρ
m

ρ
l
(15)
3. Without doing any calculations, in order for the system of the block and material to have
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 Fall '07
 JACOBSEN
 Angular Momentum, Force, Static Equilibrium, Trigraph, µs

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