Upload
vokhanh
View
234
Download
19
Embed Size (px)
Citation preview
1
LINEAR KINETICS
Linear kinetics studies translation and its causes.
All bodies have:
if stopped, “reluctance” to be moved = property called “inertia” if moving, “reluctance” to get stopped
Definitions:
Mass is a direct measurement of inertia
Mass is a scalar
Units for mass: kilograms (Kg)
Force is the action of one body on another
Force is a vector
Units for force: Newtons (N)
If the force made on an object is zero, the object will not change its velocity.
2
Newton’s 1st Law
Galileo (16th century) discovered that bodies always stay moving at constant velocity
until you make a force on them to stop them.
Newton re-discovered this:
Newton’s 1st Law. Every body continues in its state of rest, or uniform motion in a
straight line, unless it is compelled to change that state by external forces applied upon
it.
Newton’s Law of Gravitation
Newton’s Law of Gravitation. Any two particles of matter attract one another with a
force that is directly proportional to the product of their masses and inversely
proportional to the square of the distance between them.
F =
€
k ⋅ m1 ⋅m2
d2
F is called the “gravitational force”.
k is an extremely small number.
3
For any body on the surface of the Earth:
m1 = mE = the mass of the Earth
m2 = mB = the mass of the body
d = R = the radius of the Earth
F = W = the “weight” of the body
So:
W =
€
k ⋅ mE ⋅mB
R2 =
€
(k ⋅ mE
R2 ) mB = k’ mB : weight is directly proportional to mass
k’ = 9.81 It’s also called “g”
So:
W = m ⋅ g or weight = mass ⋅ 9.81
The mass of an object is constant everywhere
but weight changes from one place to another
(NOTE: The mass of the Earth is distributed throughout a large volume. It is not concentrated at a single
point at the center of the Earth as assumed for our calculations. Because of this, the formula
W =
€
(k ⋅ mE
R2 ) mB is imperfect for the calculation of weight, but it still gives a good approximation. The
formula W = m ⋅ 9.81 is accurate.)
4
Newton’s 2nd Law
When force is exerted on a body, the body accelerates.
Newton’s 2nd Law. The acceleration of a body is directly proportional to the force
exerted on it, and inversely proportional to the mass of the body. The acceleration
produced is in the same direction as the force.
a =
€
Fm
5
F ⋅ t = Δ (m ⋅ v) or (This is another form of Newton’s 2nd Law) linear impulse = change in linear momentum
Units for linear impulse: Newton ⋅ second (N ⋅ s)
Units for linear momentum: kilogram ⋅ meter / second (Kg ⋅ m / s)
Hockey puck example problem
hockey puck mass:
2 kg (not realistic, it’s just an example!)
Forces on hockey puck:
10 N for 3 s
0 N for 2 s
6 N for 4 s
-8 N for 5 s
Initial linear velocity of the hockey puck is zero; calculate its final linear velocity.
6
METHOD #1 [Uses a =
€
Fm
]
m = 2 kg v0 = 0 m/s
F = 10 N for 3 s F = 0 N for 2 s F = 6 N for 4 s F = -8 N for 5 s
a =
€
Fm
=
€
10 N2 kg
= 5 m/s2 a =
€
Fm
=
€
0 N2 kg = 0 m/s2 a =
€
Fm
=
€
6 N2 kg = 3 m/s2 a =
€
Fm
=
€
−8 N2 kg = -4 m/s2
Δ v = a ⋅ t = 5 ⋅ 3 = 15 m/s Δ v = a ⋅ t = 0 ⋅ 2 = 0 m/s Δ v = a ⋅ t = 3 ⋅ 4 = 12 m/s Δ v = a ⋅ t = -4 ⋅ 5 = -20 m/s
v = v0 + Δv = 0 + 15 = 15 m/s v = v0 + Δv = 15 + 0 = 15 m/s v = v0 + Δv = 15 + 12 = 27 m/s v = v0 + Δv = 27 + (-20) = 7 m/s
METHOD #2 [Uses F ⋅ t = Δ (m ⋅ v) ]
m = 2 kg v0 = 0 m/s
m ⋅ v0 = 2 ⋅ 0 = 0 kg ⋅ m / s
Δ (m ⋅ v) = F ⋅ t = (10 ⋅ 3) + (0 ⋅ 2) + (6 ⋅ 4) – (8 ⋅ 5) = 14 N ⋅ s = 14 kg ⋅ m / s
m ⋅ v = m ⋅ v0 + Δ (m ⋅ v) = 0 + 14 = 14 kg ⋅ m / s
v =
€
m ⋅ vm
=
€
14 kg ⋅m/ s2 kg
7 m/s
7
Stopping a football player
m ⋅ v = 100 ⋅ 7 = 700 kg ⋅ m / s
To stop him, need Δ m ⋅ v = -700 kg ⋅ m / s, so that 700 + (-700) = zero
If Δ m ⋅ v = -700 kg ⋅ m / s then need: F ⋅ t = -700 N ⋅ s
Options for F ⋅ t = -700 N ⋅ s:
F t
-7000 N 0.1 s
-700 N 1 s
-350 N 2 s
-70 N 10 s
etc. etc.
8
Newton’s 3rd Law
Newton’s 3rd Law. When a body exerts a force on another body, the second body
exerts an equal and opposite force on the first.
9
Systems
System: an object or group of objects that we arbitrarily choose to designate as “the
system”.
The system is the bowling pin.
a =
€
F∑m
€
F∑ =
€
r F 1 +
r F 2 +
r F 3 +
r F 4
external forces and internal forces
€
F∑ =
€
r E 1 +
r E 2 +
r I 1 +
r I 2
but
€
r I 1 +
r I 2 = zero
so:
€
F∑ =
€
r E 1 +
r E 2
10
Free-body diagrams
Recipe:
Weight
Forces at contacts with external solid objects
Fluid forces
Magnetic forces
12
Airborne motion
Horizontal direction:
aH =
€
FH∑m
=
€
zerom
= zero m/s2
(uniform motion)
Vertical direction:
aV =
€
FV∑m
=
€
weightm
=
€
m ⋅ (−9.81)m
= -9.81 m/s2
(uniformly accelerated motion at –9.81 m/s2)
Some practical conclusions
The generation of a large velocity requires a large force and a long time.
So, to generate a large velocity you need a large force over a long range of motion.