Newtonian Forces
Tension is the pulling force away from an
object usually by a string, rope, or cable
Force is represented by mg
◦
M stands for the mass of the object
◦
G stands for the gravity constant on Earth (9.80
m/s
2
)
T

mg
If the mass is just hanging, then the
acceleration (ā) =
0
◦
Σ
𝐹
̅
=
0
◦
Σ
𝐹
=
mā
Where m is the inertial mass
mā
=
Σ
𝐹
= mg (down) + T (up)
T = mg (up)
M(
0
) = mg (down) + mg (up)
◦
Where the mass of “mg (up)” is related to gravity
This is called the Gravitational Mass: this tells how
much weight the object has
◦
In this class inertial mass ↔ gravitational mass
◦
𝐹
=
(GM
1
M
2
)
𝑥
r
2
◦
Where “r” is the distance between two objects with
different gravity
◦
Tension must be the same if the rope is the same
𝐹
= (down) m
1
g
(down) m
2
g
a
a
Atwood
Machine
T
T
Σ
𝐹
1
= m
1
ā
1
= m
1
g (down) + T (up)
Σ
𝐹
2
= m
2
ā
2
= m
2
g (down) + T (up)
T (up) = [m
2
ā
2
= m
2
g (down) = m
1
ā
1
= m
1
g (down)]
ā
2
= a (down)
m
2
(a(down)
–
g (down)
ā
1
= a (up)
m
1
(a(up)
–
g(down) = m
1
(

a(down)
–
g(down))
m
2
a

m
2
g + m
1
a + m
1
g
= (m
2
+ m
1
)a
–
(m
2
+ m
1
)g =
0
A
=
(m
2

m
1
)g
(
m
2
+ m
1
)
The difference in the weights makes the
acceleration go up
The sum of the weights makes the
acceleration go down
(
m
2
+ m
1
)a = (m
2
+ m
1
)g
Inertial mass =
(
m
sys
)a
Gravitational mass =
(
m
sys
)g
Inertial
mass of the system ≠ gravitational
mass of the system
A new rope means you have two separate
tensions
T
1
T
1
T
2
Always acts against the motion
θ

mg

F
fr
=
μ
s
N
or

F
fr
=
μ
k
N
◦
Where
μ
s
is the static friction constant and
μ
k
◦
is the kinetic friction constant
◦
μ
k
is like skating on ice
Pure slippage
◦
μ
s
N
is like rubber on concrete
Very little motion
Σ
𝐹
=
mg


F
fr
= ma
mg


μ
k
N
= ma
mg


μ
k
N
= ma
g


μ
k
g

=
a
g

= g
cos
θ
g

= g sin
θ
g
sin
θ

μ
k
g
cos
θ
= a
g(sin
θ

μ
k
cos
θ
) = a

F
fr
≤
μ
s
N
◦
The static friction has a maximum balancing force
θ
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