Group Problem.
3kg
5kg
2kg
Two masses on a horizontal surface are joined by light strings over a light frictionless
pulley to a third hanging mass as shown in the diagram.
The coefficients of friction
between the surface and the masses are 0.3 (static) and 0.25 (kinetic).
If the system is
released from rest will the masses move?
If so determine the acceleration of the masses.
What are the tensions in the strings?
Draw force and acceleration diagram
Assuming the weights are moving and accelerating with a m/s
2
Resolving vertically for M
3
and applying second law
M
3
gT
3
=M
3
a
Eq 1
Pulley is frictionless therefore T
2
=T
3
Resolving horizontally for M
2
M
3
M
2
M
1
y
x
T
3
T
2
T
1
T
1
F
2
F
1
N
1
M
1
g
N
2
M
2
g
M
3
g
a
a
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T
2
T
1
F
2
=M
2
a
Eq2
The friction force is given by
F
2
=
µ
N
Resolving vertically N=M
2
g
Therefore
F
2
=
µ
M
2
g
Similarly resolving horizontally and vertically for M
1
T
1
F
1
=M
1
a
F
1
=
µ
M
1
g
Eq3
Substituting for T
1
,T
2
(=T
3
), F
1
and F
2
in equation 2
M
3
gM
3
a(M
1
a+
µ
M
1
g)
µ
M
2
g=M
2
a
g(M
3

µ
M
2

µ
M
1
)=(M
1
+M
2
+M
3
)a
a will be positive (i.e system will move) if
M
3
>
µ
(M
2
+M
1
)
where
µ
is the coefficient of
static
friction
i.e 3>0.3x7 >2.1
so it moves
When it is sliding
µ
is the coefficient of
sliding
friction =0.25
∴
a=(30.25x7)x9.8/10 =1.2 m/s
2
T
1
=M
1
(a+
µ
g)=2(1.225+0.25x9.8) =7.3N
T
2
=T
3
=M
3
(ga)=25N
Units, Newtons, OK.
Significant figures 2, OK
Is it reasonable?
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 Fall '08
 Marshak
 Force, Friction, Mass, Light, T1 N2 T2, equation T1sin1+T1cos1sin2/cos2=T3=mg T1

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