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Transcript
Normal Force Example: Incline
m1
α
The angle of the frictionless incline is α = 30°. Mass m1
slides down the incline, starting from rest. What is the
speed of the mass after it slid 10 meters downhill?
[use g = 10 m/s2]
PHY2053, Fall 2013, Lecture 7 – Gravity, Contact Forces, Tension
Newton’s Third Law
LawIII:
Lex
III:Actioni
To every
contrariam
action there
semper
is always
et æqualem
an equalesse
and
Law
3: Inreaction:
ansive
interaction
objects,
each
reactionem:
opposite
corporum
or the between
forces
duorum
oftwo
two
actiones
bodiesinon
seeach
object
exerts
a esse
force
onand
the are
other.
Thesein
two
forces
mutuoare
other
semper
always
equal
æquales
et in
directed
partes
contrarias
opposite
are
equal
in
magnitude
and
opposite
in
direction.
dirigi.
directions.
m1
PHY2053, Fall 2013, Lecture 7 – Gravity, Contact Forces, Tension
m2
Example: Two Blocks
m1
m2
In the picture above, the two blocks are placed on a
frictionless surface, in contact with each other. The
masses of the two blocks are m1 = 10 kg and m2 = 15 kg.
A force F1 = 100 N is being applied to mass m1 from the
left, and force F2 = 200 N is being applied to mass m2
from the right. Compute the force F2,1 that mass m1 is
exerting on mass m2. Contact Forces, Tension
PHY2053, Fall 2013, Lecture 7 – Gravity,
Gravitational force
● Newton’s Law of Gravitation:
● where m1 and m2 are the masses of the two bodies
● r is the distance between them, measured from the
center of the (spherical) object
● G is the universal gravitational constant
G = 6.674 × 10-11 N m2 / kg2
● provides excellent description of planetary motion
PHY2053, Fall 2013, Lecture 7 – Gravity, Contact Forces, Tension
4
Example: Cranial Attraction
● The heads of neighboring students in the classroom
are separated by ~ 1 m. A typical human head weighs
about 5 kg. Compute the attractive force (due to
gravitation) between the heads of two students.
m1 m2
F =G 2
r
● m1 = m2 = 5 kg, r = 1 m, G = 6.674 × 10-11 N m2/kg2
● use in formula → obtain F = 1.67 × 10-9 N
PHY2053, Fall 2013, Lecture 7 – Gravity, Contact Forces, Tension
5
Example: Weight change while
airplane is in flight
● An airplane is cruising at the altitude of 10 km above
sea level. What factor less does a passenger weigh at
that altitude compared to their weight at sea level?
The radius of the Earth is 6370 km and
its mass is ME = 6 × 1024 kg.
m1 m2
F =G 2
r
● m1 = ME, m2 = m (unknown)
● Compare F(r) for r = RE to F(r) for r = (RE + 10 km)
PHY2053, Fall 2013, Lecture 7 – Gravity, Contact Forces, Tension
6
Example: “g” on surface of Mars
m1 m2
F =G 2
r
Declare this “g” when computed at the
surface of a specific planet.
Formula then simplifies to F = m2 g
● For all objects on the surface of a planet, one of the
masses (mass of the planet) and r are the same
● formula can then be simplified to F = m g, except g
is different on each planet (depends on M and R)
● for Mars, MMars = 1/9 ME , RMars = 0.533 RE
● we can then compute gMars in terms of gEarth
PHY2053, Fall 2013, Lecture 7 – Gravity, Contact Forces, Tension
7
Tension Force
● “Ideal String” concept:
● massless, infinitely thin, fixed length
● withstands any force without breaking
“String with tension T”
m1
PHY2053, Fall 2013, Lecture 7 – Gravity, Contact Forces, Tension
m2
Tension Example: Pulley + Weights
The masses in the depicted
Atwood machine are
m1 = 5 kg and m2 = 3 kg.
● What is the acceleration
of mass m1?
● What is the tension T2 of
the cable holding the
massless pulley?
PHY2053, Fall 2013, Lecture 7 – Gravity, Contact Forces, Tension
T2
m1
m2
Friction
● Force due to imperfections of surfaces in contact
● For a pair of surfaces, the frictional force depends
(only) on the normal force between the surfaces:
Frictional
force
Normal force
Coefficient of friction
● To a good approximation, does not depend on size or
shape of contact surface, only the materials in contact
● Always opposite to the direction of motion
● Static friction – objects not moving w.r.t. each other
● Kinetic friction – one object is being dragged across
Fall
2013,
Lecture
7 – Gravity,
Contact
Forces,
theTension
(surface of the) other
PHY2053,
Problem: Pushing a crate at an angle
A crate of mass 100 kg is
being pushed across a
horizontal surface. The
coefficient of friction between
the crate and the surface is
μ = 0.2. The force is being
applied downward on the
crate at a 30º angle with
respect to the vertical axis.
What is the minimum force at
which the crate will move
from its prone position?
[use g = 10 m/s2]
PHY2053, Fall 2013, Lecture 6 – Newton’s Laws
∘
30
F