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Reza Nejat
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Physics

PHYSICS 1L03

Reza Nejat

Fall

Description

LUCY
Human evolution and Physics
Bipedalism, Quadrupedalism
Who is Lucy?
Concepts:
Center of Mass
Torque
Force
Equilibrium
Chapters: 4, 5
Pages: 95-101, 122-141
PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat
Lucy Figure 04.12c
• A partial hominid skeleton,
discovered at Hadar, Ethiopia,
Africa in 1974.
•Aliu.
•.atlt,d
•Crytr bipedal
locomotion
•irprl
humans (~3.2 million years)
PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Lucy Lucy
•W dwhik “Lucy” was bipedal
•4%omlltn (walking upright)?
of “Lucy” indicates that she
was bipedal.
•W rureieefr
effective and efficient bipedal posture
–WasShe?
and locomotion?
• Lucy’s entire skeleton was •L’mpewtnCiaze
reconstructed from the
(quadruped, knuckle walking), and
remains that were Human (biped, upright walking)
discovered.
PHYS 1L03, Fall 2013, McMaster University, R.NejaPHYS 1L03, Fall 2013, McMaster University, R.Nejat
Quadrupedalism vs. Bipedalism
Quadrupedalism vs. Bipedalism sInets
ela •uail are considered:
•Dsir ght need considerable
change to the shape of the pelvis bones?
–Doesthischangeinpelvisshapecreatesa
change in the position of the center of
mass.
•Dsihote
function of gluteal (abductor) muscles?
PHYS 1L03, Fall 2013, McMaster University, R.NejaPHYS 1L03, Fall 2013, McMaster University, R.Nejat Quadrupedalism vs. Bipedalism Center of Mass
•oseuteow sltetn equivalent
the physics of: single particle: Center of Mass
–Riojilecallandsofortharesystems
•Wits?a of particles
–Whatisweightline?
•Tremains constant in time.
•Wit?
–Themassofanobjectcanbedistributed
–Whatdoesforcedo? uniformly or non-uniformly
•Aladtitemiustr)pe
–Howcanwequantifythemassdistribution?
•Wit?
–Whatdoestorquedo? •e center of mass (CM) of a system is the point
at which all the mass of the system may be
•Witiillu considered to be concentrated.
PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat
Center of Mass Center of Mass
•Tstm may be either: se
–agroupofobjects(particles) extended object like our body.
–oranextendedobjectlikeourbody.
•olit
of the center of mass of
the system is the same
as if the mass of the
system is concentrated
at that point.
PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Center of Mass Center of Mass
•Fxodt, •Tcfsfa
different parts move with different speed.
a ball thrown in the air
•Smtadiid follows a parabolic
terms of the position and motion of the center
of mass of the body (point particle Model). path
•W sti
easily because the
ball is a simple, point-
like object
PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat
Center of Mass Center of Mass
•Tcflih
)alreafohTt•n also follows a a straight-line path, as though it were a
particle, but the wrench is rotating around its
parabolic path when the hammer is thrown in the air, but
this is harder to observe center of mass
hemolhrosphhtoT
PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Center of Mass Center of Mass
•Mytjtc
http://hippie.nu/~unicorn/tut/xhtml/ http://hippie.nu/~unicorn/tut/xhtml/
PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat
Center of Mass: Math Example 1: Center of mass
•hotf center of mass of a •wo.,s
system of two particles of unequal masses positions 6.0 m and – 4.0 m respectively from
is: the origin as shown below. Where is the center
of mass of these two masses?
-4 -2 0 2 4 6
x ▯m1x1▯ 2 2
m 1 ▯1m x 2 2 CM m1▯ m2
xCM ▯
m 1 m 2
•sotir
the system.
PHYS 1L03, Fall 2013, McMaster University, R.Nejat PPHHYYSS11LL0033,,FFaallll22001132,,MMccMMaasstteerrUUnniivveerrssiittyy,,RR..NNeejjaatt Center of Mass: Math Center of Mass
qtfiIs
masses, the x coordinate of the center of mass
y
is: m 2 1y2 m1
mx ▯ 1x 2 x1▯ x 2 xi
x CM ▯ ▯ 2 x1
m ▯ m 2
i
m1 1 ▯ m 2 2▯ m 3 3▯ ...▯ m n n m i
xCM ▯
•S, the position of the center of mass in this m 1 ▯ m 2 ▯ m3 ▯ ... m n
case is merely the average position of the
YCM ▯ m1y 1▯ m1y 2▯ m 3 3 ▯..▯ m n n
system. m 1 m ▯2m ▯ 3 ..▯ mn
PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat
Example 1: Center of mass
Center of Mass
• Three masses, 4.0 kg, 6.0 kg and 3.0 kg are
•Tefsify
placed at positions 6.0 m, – 2.0 m and 4.0 m
position in space described mathematically
respectively from the origin as shown below. as:
Where is the center of mass of these masses? x ▯ m 1 1m x2▯2 x 3 3 ... ▯ ▯ m ii
CM m ▯m ▯m ▯ ... m m1 m2
1 2 3 ▯ i (11, y x2y2
-4 -2 CM
0 2 4 6 m y m3
m 1 1 m 2 ▯2m x 3 3 yCM ▯ m1y1▯ m2y 2▯m 3 3▯... ▯ ▯ ii (3 3 y )
xCM ▯ m 1▯m 2▯m 3▯... ▯ mi
m1▯ m 2m 3
•baedijtie
many small particles, having specific mass
and specific coordinates.
PPHHYYSS11LL0033,,FFaallll22001132,,MMccMMaasstteerrUUnniivveerrssiittyy,,PHYS 1L03, Fall 2013, McMaster University, R.Nejat Center of Mass Center of Mass
,umttro
•ymbjemis
having irregular shape, the center of mass may be
distributed uniformly (i.e. constant density), evaluated by suspending as shown below:
the center of mass is at the geometric center
of the object
msth body of any shape can be
located by suspending it from at least two points.
PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat
Example 1: Center of mass of the Arm Center of mass of the Arm
elnFin etiri
The masses of the upper arm, the lower arm and ▯ mii ▯ miyi
the hand are 2.5 kg, 1.6 kg and 0.64 kg respectively. CM ▯ ▯ mi , yCM▯ ▯ mi
Their center of mass are at 0.18 m above the elbow,
and 0.12 m and 0.40 m to the right of elbow
respectively, as illustrated below.
▯ ▯ m x
▯xCM ▯ i i
▯ ▯ mi 0.18 m
▯ elbow
▯y ▯ ▯ miyi 0.12 m
▯ CM ▯ mi 0.40 m
PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Center of Mass (review) Physics: Torque
•heCs)a of a system is the
point at which all the mass of the system may •Qeis:
be considered to be concentrated.
•hltf the center of mass
of the system is the same as if the mass of What is Torque?
the system is concentrated at that point.
What does Torque do?
•nljte
center, nor will it divide an object into equal
mass points.
PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat
Pop-Quiz : Torque
Physics: Torque
Q: Where on a door do
you push to open it?
A
C E
B
D
A: Far from the hinge.
Ywtte this door most easily
(i.e. with the smallest possible force),
along which line should you push?
PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Physics: Torque Physics: Torque
•Truittdnyfct •Tqe, ▯ (Greek tau), due to a force F
applied at a distance r from the axis of
produce rotation about an axis. rotation (pivot, hinge, fulcrum), at an
angle ▯ to the radial line is:
•Trum yeo nsidered as the F
rotational equivalent of force. ▯ ▯ r Fsin▯ F sin ▯
Axis ▯
F sin▯ ▯ r
•Tqeruetileitn , r
like Force that produces translational F ▯ Fsin▯
▯
acceleration •:i•t•)
PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat
Physics: Torque Physics: Torque
•qeitifi
and point of application of force) x •Teu ▯ ,sign convection for direction:
(component of force perpendicular
to this line)
= (r) x (Fsin▯)
F
F sin ▯
• ▯ > 0, if rotation is counter clockwise.
Axis ▯
• ▯ < 0, if rotation is clockwise.
r
PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Physics: Torque (Review) Physics: Torque
• The ability of a force to cause a rotation or a
▯ We quantify the effectiveness of a force twisting motion depends on:
in creating rotation using the quantity – The magnitude, F, of the force.
called torque. – The distance, r, from the pivot to the point at
which the force is applied.
– The angle, ▯,atwhc ihthefoiapild
▯ Torque plays the same role in
rotational motion that force plays in ▯ = r F sin▯ F F
translational motion. ▯
Axis Axis ▯
▯ = r F sin▯
r r
Larger Torque Smaller Torque
PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat
Physics: Calculating Torque ▯ Physics: Calculating Torque ▯
F
F
▯ F ▯ = F r sin▯ r
▯ = r F ▯ ▯
Axis
Axis ▯ r = r sin▯ Le
F ▯ F sin▯ F▯ ▯ ▯ ver e
= r ar l
r sn m , ce ti
F ▯ F cos▯ ▯ = F r ▯ ▯ For
of ac
F is the perpendicular component of the force F
▯ The distance r ▯ is called Lever arm
which contributes to the torque.
(moment arm).
F ▯s the parallel component of the force F which
passes through the axis and does not contribute to The Lever arm is the perpendicular
the torque. distance to the line of action of the force.
PPHHYYSS11LL0033,,FFaallll22001132,,MMccMMaasstteerrUUnniivveerrssiittyy,PHYS 1L03, Fall 2013, McMaster University, R.Nejat Example 4: Torque Torque: Conceptual Question
stdgir•sis •emhitf
leg at an angle of 30.0 from the horizontal with an string, which makes angle ▯ with the vertical.
89 N weight attached to the ankle. The distance Which of the following are true?
between the ankle weight and the hip joint is 84 cm.
What is the torque due to the ankle weight on the The gravitational torque on the pendulum
about the pivot point:
leg. pivot
a. depends on the length of R
b. Is zero at the max displacement ▯ R
of the swing
c. Is constant since g is constant m
d. Is smaller when the angle is smaller
Try to work through it at home. A way of tesging!
PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat
Physics: Mechanical equilibrium Physics: Mechanical equilibrium
•Aojdiib mechanical •arn extended object to be in static or
equilibrium if it has neither translational dynamic mechanical equilibrium, the
acceleration nor rotational acceleration
following two conditions must be satisfied:
–itcanstillbetranslatingand/orrotating
–The net external force must be zero:
at constant speeds (dynamic ▯
mechanical equilibrium) ▯ F ▯ 0
–Thenetexternaltorquemustbezero:
•Aoniit static mechanical
equilibrium if it is neither rotating

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