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.

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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

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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.

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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.

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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

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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.