Week 5
Applications of Newton's Laws
Dr Sohail Amjad
Outline
Free-Body diagrams
Applying Newton's First Law: Equilibrium
Applying Newton's Second Law:Dynamics
Frictional Forces
Dynamics of Circular Motion
Fundamental forces of Nature
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Free-Body diagrams
Newton's rst and second laws apply to a specic body.
ΣF~ = 0 for an equilibrium
~
ΣF = m~a , for a non-equilibrium
Whenever you use Newton's rst law,
situation or Newton's second law,
situation, you must decide at the beginning to which body you are
referring.
Only forces acting on the body matter.
Free-body diagrams are essential to help identify the relevant forces.
A free-body diagram is a diagram showing the chosen body by
itself, free of its surroundings, with vectors drawn to show the
magnitudes and directions of all the forces applied to the body by
the various other bodies that interact with it.
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Free-Body diagrams
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Free-Body diagrams
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Free-Body diagrams
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Free-Body diagrams
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Free-Body diagrams
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Free-Body diagrams
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Applying Newton's First Law - Equilibrium
A body is in equilibrium when it is at rest or moving with constant
velocity in an inertial frame of reference.
For now, we will consider only equilibrium of a body that can be
modeled as a particle.
When a particle is in equilibrium, the net force acting on it (the
vector sum of all the forces acting on it) be zero:
ΣF~ = 0
We can also use the component form:
ΣFx = 0,
ΣFy = 0
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Example
A gymnast with mass suspends herself from the lower end of a hanging
rope of negligible mass
mG = 50.0 kg .
The upper end of the rope is
attached to the gymnasium ceiling.
(a) What is the gymnast's weight?
(b) What force (magnitude and direction) does the rope exert on her?
(c) What is the tension at the top of the rope?
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Example
A gymnast with mass suspends herself from the lower end of a hanging
rope of negligible mass
mG = 50.0 kg .
The upper end of the rope is
attached to the gymnasium ceiling.
(a) What is the gymnast's weight?
(b) What force (magnitude and direction) does the rope exert on her?
(c) What is the tension at the top of the rope?
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Example
a)
wG = mG g = 50kg × 9.8m/s2 = 490 N
b)
ΣFy = 0 =⇒ TR
on G
+ (−wG ) = 0
=⇒ TRo nG = wG = 490N
c) Your answers?
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Example
A gymnast with mass suspends herself from the lower end of a hanging
rope of negligible mass
mG = 50.0 kg .
The upper end of the rope is
attached to the gymnasium ceiling.
(a) What is the gymnast's weight?
(b) What force (magnitude and direction) does the rope exert on her?
(c) What is the tension at the top of the rope?
Solve the problem if weight of Rope is
120 N .
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Example
A car engine with weight
w
hangs from a chain that is linked at ring
O
to
two other chains, one fastened to the ceiling and the other to the wall.
Find expressions for the tension in each of the three chains in terms of
w.
The weights of the ring and chains are negligible compared with the
weight of the engine.
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Example
A car engine with weight
w
hangs from a chain that is linked at ring
O
to
two other chains, one fastened to the ceiling and the other to the wall.
Find expressions for the tension in each of the three chains in terms of
w.
The weights of the ring and chains are negligible compared with the
weight of the engine.
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Example
A car engine with weight
w
hangs from a chain that is linked at ring
O
to
two other chains, one fastened to the ceiling and the other to the wall.
Find expressions for the tension in each of the three chains in terms of
w.
The weights of the ring and chains are negligible compared with the
weight of the engine.
T1 = w
T3 = 1.2w
T2 = 0.58w
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Example
A car of weight
w
rests on a slanted ramp attached to a trailer. Only a
cable running from the trailer to the car prevents the car from rolling o
the ramp. (The car's brakes are o and its transmission is in neutral.)
Find the tension in the cable and the force that the ramp exerts on the
car's tires.
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Example
A car of weight
w
rests on a slanted ramp attached to a trailer. Only a
cable running from the trailer to the car prevents the car from rolling o
the ramp. (The car's brakes are o and its transmission is in neutral.)
Find the tension in the cable and the force that the ramp exerts on the
car's tires.
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Example
ΣFx = T + (−wsinα) = 0 =⇒ T = w sinα
ΣFy = n + (−wcosα) = 0 =⇒ n = w cosα
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Applying Newton's Second Law Non-Equilibrium
Let's discuss dynamics (opposed to statics - not kinematics)
problems. In these problems, we apply Newton's second law to
bodies on which the net force is not zero.
These bodies are not in equilibrium and hence are accelerating. The
net force on the body is equal to the mass of the body times its
acceleration:
ΣF~ = m~a
We can also use the component form:
ΣFx = max ,
ΣFy = may
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Example
An iceboat is at rest on a frictionless horizontal surface. A wind is
blowing along the direction of the runners so that 4.0 s after the iceboat
is released, it is moving at
6 m/s.
What constant horizontal force
FW
does the wind exert on the iceboat? The combined mass of iceboat and
rider is 200 kg.
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Example
An iceboat is at rest on a frictionless horizontal surface. A wind is
blowing along the direction of the runners so that 4.0 s after the iceboat
is released, it is moving at
6 m/s.
What constant horizontal force
FW
does the wind exert on the iceboat? The combined mass of iceboat and
rider is 200 kg.
FW = 300 N
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Example
Let's complicate things a bit
An iceboat is at rest on a frictionless horizontal surface. A wind is
blowing along the direction of the runners so that 4.0 s after the iceboat
is released, it is moving at
6 m/s.
What constant horizontal force
FW
does the wind exert on the iceboat? The combined mass of iceboat and
rider is 200 kg.
Suppose a constant horizontal friction force
with magnitude 100 N opposes the motion of
the iceboat. In this case, what constant force
FW
must the wind exert on the iceboat to
cause the same constant x-acceleration
ax = 1.5 m/s2 ?
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Example
An iceboat is at rest on a frictionless horizontal surface. A wind is
blowing along the direction of the runners so that 4.0 s after the iceboat
is released, it is moving at
6 m/s.
What constant horizontal force
FW
does the wind exert on the iceboat? The combined mass of iceboat and
rider is 200 kg.
Suppose a constant horizontal friction
force with magnitude 100 N opposes the
motion of the iceboat. In this case, what
constant force
FW
must the wind exert
on the iceboat to cause the same
constant x-acceleration
ax = 1.5 m/s2 ?
FW = 400 N
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Example
An elevator and its load have a combined mass
of
800 kg .
The elevator is initially moving
downward at
10 m/s
it slows to a stop with
constant acceleration in a distance of
What is the tension
T
25.0 m.
in the supporting cable
while the elevator is being brought to rest?
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Example
800 kg . The elevator
10 m/s it slows to a stop with constant
25.0 m. What is the tension T in the
An elevator and its load have a combined mass of
is initially moving downward at
acceleration in a distance of
supporting cable while the elevator is being brought to rest?
T = 9440 N .
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Example
Let's complicate things a bit ....
800 kg . The elevator
10 m/s it slows to a stop with constant
25.0 m. What is the tension T in the
An elevator and its load have a combined mass of
is initially moving downward at
acceleration in a distance of
supporting cable while the elevator is being brought to rest?
A 50.0-kg woman stands on a bathroom
scale while riding in the elevator. What
is the reading on the scale?
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Example
Let's complicate things a bit ....
800 kg . The elevator
10 m/s it slows to a stop with constant
25.0 m. What is the tension T in the
An elevator and its load have a combined mass of
is initially moving downward at
acceleration in a distance of
supporting cable while the elevator is being brought to rest?
A 50.0-kg woman stands
on a bathroom scale while
riding in the elevator.
What is the reading on the
scale?
390 N .
Note: Real weight of
woman = 490 N
Apparent weight of
woman when elevator is
slowing down = 590 N
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