Scalars in mechanics
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Transcript of Scalars in mechanics
Scalars in mechanics
Photo: Flickr
Quantities
Scalars in mechanics 2
Scalar quantities Vector quantities
mass m (kg)
energy E (J)
power P (W)
displacement 𝑠 (𝑚)
velocity 𝑣 (𝑚𝑠−1)
acceleration 𝑎 (𝑚𝑠−2)
force 𝐹 (𝑁)
momentum 𝑝 (𝑘𝑔𝑚𝑠−1) torque 𝑇 (𝑁𝑚)
• Magnitude • Magnitude• Direction• Point of application
History
Scalars in mechanics 3
17th centuryVector mechanics
GallileiHuygensNewtonLeibniz 19th century
Scalar mechanics
DavyRitter
OerstedFaradayKelvin
It was observed that ‘natural forces’ could transform.19th century natural philosophers (later physicists)Searched for a conserved ‘natural force’.
This resulted in the new concept ‘ENERGY’
Concept of energy
Scalars in mechanics 4
IT
• Is transformed in all physical changes
• Is required to make anything happen
• Is puts limits to any process or activity
• Is the most fundamental concept in modern science
pixabay
Concept of work
Scalars in mechanics 5
StudentHotel
HollandISC
you (today)
𝑠 = 4.3𝑘𝑚 𝐹𝑓𝑜𝑟𝑤𝑎𝑟𝑑
Assume constant 𝑣𝑏𝑖𝑘𝑒 = 3 𝑚𝑠−1 so Σ 𝐹 = 0 ⇒
𝐹𝑓𝑜𝑟𝑤𝑎𝑟𝑑 = − 𝐹𝑟𝑜𝑙𝑙 + 𝐹𝑎𝑖𝑟 = 12𝜌𝐴𝐶𝐷 𝑣𝑏𝑖𝑘𝑒+𝑣𝑤𝑖𝑛𝑑
2 + 𝐶𝑟𝑜𝑙𝑙 ∙ 𝐹𝑁 =
𝐹𝑟𝑜𝑙𝑙
𝐹𝑎𝑖𝑟+
1
2∙ 1.29 ∙ 0.4 ∙ 1 ∙ 3 + 5 2 + 0.02 ∙ 70 ∙ 9.81 = 16.5 + 13.7 = 30𝑁
Work done = 𝑊 = 𝐹 ∙ 𝑠 = 30 ∙ 4.3 ∙ 103 = 1.3 ∙ 105𝐽𝑜𝑢𝑙𝑒
Work vs. Impulse
Scalars in mechanics 6
Why not use FORCE TIME as a measure of work?
𝐹𝑓𝑜𝑟𝑤𝑎𝑟𝑑
𝐹𝑓𝑜𝑟𝑤𝑎𝑟𝑑
start
same force, same time,…same work?
Impulse is a vector: 𝐹 ∙ 𝑡 = 𝐼
Vector Scalar = Vector
Work is a scalar: 𝐹 ∙ 𝑠 = 𝑊
Vector Vector= Scalar
Vector ‘dot’ product
Force & displacement at an angle
Scalars in mechanics 7
𝐹
𝑠
𝜃
𝐹∕∕𝑠
𝐹⊥𝑠
𝑠
90°
𝐹∕∕𝑠 = 0
𝐹⊥𝑠 = 𝐹 𝐹
𝑠
𝜃
𝐹∕∕𝑠
𝐹⊥𝑠
𝑊 = 𝐹∕∕𝑠 ⋅ 𝑠 = 𝐹 ⋅ 𝐶𝑜𝑠𝜃 ∙ 𝑠 = 𝐹 ∙ 𝑠 ∙ 𝐶𝑜𝑠𝜃
Nett driving forceProvides energy
W > 0
Nett inhibiting forceWithdraws energy
W < 0
No driving forceNo energy transferred
W = 0
Non-constant forces
Scalars in mechanics 8
𝐹 pulling force
𝑠 𝑠
𝐹
𝑠𝑚𝑎𝑥
𝐹𝑚𝑎𝑥
𝐹𝑎𝑣𝑔
𝑊 = 𝐹𝑎𝑣𝑔 ∙ 𝑠
𝑊 = 0
𝑠
𝐹 ∙ 𝑑𝑠
𝑊 = 0
𝑠
𝐹 ∙ 𝐶𝑜𝑠𝜃 ∙ 𝑑𝑠
𝑊 = 0
𝑠
𝐹 ∙ 𝑑 𝑠
more general
more generalwith angle
complete formas vector dot product
W=Area
Work transforms energy
Scalars in mechanics 9
wikimedia
Work transforms energy
Scalars in mechanics 10
wikimedia
Work by the resistant force of the target transforms kinetic energy of the arrow to heat in the target.
1 2 3
𝐸𝑐ℎ 𝐸𝑝 𝐸𝑘 𝑄
3
Work by the elastic force transforms elastic potential energy in the bow to kinetic energy of the arrow.
2
Work by the pulling force transforms chemical potential energy in master Kim’s arm to elastic potential energy in the string.
1
Gravitational (potential) energy
Scalars in mechanics 11
+ℎ1
ℎ2
𝑚 ∙ 𝑔
𝐹
𝑊 = 𝐹 ∙ 𝑠 = 𝑚 ∙ 𝑔 ∙ ℎ2 − ℎ1 = ∆𝐸𝑝
To lift the object, the average lifting force needs to equal 𝑚 ∙ 𝑔The lifting force performs positive workGravitational potential energy is built up
Strictly 𝐸𝑝 = 0 at the Earth’s centre.
Since this is not practical, we set 𝐸𝑝 = 0 and ℎ = 0
at the lowest point in your context.
Gravitational (potential) energy
Scalars in mechanics 12
ℎ1
ℎ2
𝑚 ∙ 𝑔
𝐹
𝑚 = 10𝑘𝑔𝑔 = 9.81𝑚𝑠−2
ℎ1 = 5.0𝑚ℎ2 = 7.0𝑚
To lift the object from the floor to ℎ1, the lifting force performs work:𝑊 = 𝑚 ∙ 𝑔 ∙ ℎ1 = 10 ∙ 9.81 ∙ 5.0 = 4.9 ∙ 10
2𝐽
Arrived at ℎ1the object has a potential energy 𝐸𝑝 = 4.9 ∙ 102𝐽
To lift the object from ℎ1 to ℎ2, the lifting force performs work:𝑊 = 𝑚 ∙ 𝑔 ∙ ℎ2 − ℎ1 = 10 ∙ 9.81 ∙ 2.0 = 2.0 ∙ 102𝐽
Arrived at ℎ2the object has a new potential energy: 𝐸𝑝 = 4.9 ∙ 10
2𝐽 + 2.0 ∙ 102𝐽 = 6.9 ∙ 102𝐽
Example
𝐸𝑝 = 4.9 ∙ 102𝐽
𝐸𝑝 = 6.9 ∙ 102𝐽
𝐸𝑝 = 0
4.9 ∙ 102𝐽Work done
2.0 ∙ 102𝐽Work done
Elastic (potential) energy
Scalars in mechanics 13
𝑠
𝐹
𝐹𝑚𝑎𝑥
𝐹 = 𝐶 ∙ 𝑠
𝑠
𝐹𝑚𝑎𝑥2
𝑊 = 𝐹𝑎𝑣𝑔 ∙ 𝑠 =1
2𝐶 ∙ 𝑠 ∙ 𝑠 =
1
2𝐶𝑠2 = ∆𝐸𝑝
To pull the object, the average pulling force equals 1
2𝐶 ∙ 𝑠
The pulling force performs positive workElastic potential energy is built up
𝑠
Elastic (potential) energy
Scalars in mechanics 14
𝐹 = 𝐶 ∙ 𝑠
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 1 2 3 4 5 6 7 8 9 10
F (N
)
s (cm)
Example with spring 𝐶 = 0.20𝑁/𝑐𝑚
The spring starts without potential energy: 𝐸𝑝 = 0
To extend the spring 4.0 𝑐𝑚, the pulling force does work:
𝑊 = 𝐹𝑎𝑣𝑔 ∙ 𝑠 =0 + 0.8
2∙ 0.040 = 0.016𝐽
The spring then has a potential energy: 𝐸𝑝 = 0.016𝐽
To extend the spring 3.0 𝑐𝑚 more, the pulling force does work:
𝑊 = 𝐹𝑎𝑣𝑔 ∙ 𝑠 =0.8 + 1.4
2∙ 0.030 = 0.033𝐽
The spring then has a new potential energy: 𝐸𝑝 = 0.016 + 0.033 = 0.049𝐽
Area =1st W
Area =2nd W
Kinetic energy
Scalars in mechanics 15
𝑡
𝑣1
𝑡1 𝑡2
𝑣2
∆𝑣
∆𝑡
Total work done on an object ≡Work done by the overall force !
Σ𝑊 = Σ𝐹 ∙ 𝑠 = 𝑚 ∙ 𝑎 ∙ 𝑠
= 𝑚 ∙𝑣2 − 𝑣1∆𝑡
∙𝑣2 + 𝑣12
∙ ∆𝑡
= 𝑚 ∙∆𝑣
∆𝑡∙ 𝑣𝑎𝑣𝑔 ∙ ∆𝑡
=𝑚
2∙ 𝑣2 − 𝑣1 ∙ 𝑣2 + 𝑣1
=1
2𝑚𝑣2
2 −1
2𝑚𝑣1
2 = ∆𝐸𝑘Σ𝑊 = Δ𝐸𝑘
2nd law of Newton (rephrased)
Scalars in mechanics 16
Vector mechanics: the acceleration of an object equals the overall force divided by its mass
Scalar mechanics: the increase of object’s kinetic energy equals the overall work done on the object
Σ𝑊 = Δ𝐸𝑘
Σ 𝐹 = 𝑚 𝑎
Energy transfer & efficiency
Scalars in mechanics 17
Photo: wikimedia
Aggregate
GeneratorEfficiency 𝜂2
𝐸𝑐ℎ
𝑊
𝑄1
𝐸𝑒
𝑄2
Chemical 100J Work 30J
Heat 70J
Heat 8J
Electric 22J
Flows out of the system
The engine converts chemical energy to work with an efficiency
𝜂1 =𝑊
𝐸𝑐ℎ=30
100= 0.30 (30%)
The generator converts work to electrical energy with an efficiency
𝜂2 =𝐸𝑒𝑊=22
30= 0.73 (73%)
Overall efficiency =𝜂 = 𝜂1 ∙ 𝜂2 =
0.30 ∙ 0.73 = 0.22 (22%)
EngineEfficiency 𝜂1
Energy conservation
Scalars in mechanics 18
EngineEfficiency 𝜂1
GeneratorEfficiency 𝜂2
𝐸𝑐ℎ
𝑊
𝑄1
𝐸𝑒
𝑄2
Chemical 100J Work 30J
Heat 70J
Heat 8J
Electric 22J
Surrounding air
Aggregate
Closed system
Loss for the aggregate
First law of thermodynamics (energy conservation law): In a closed system the total energy is conserved
Conservation of energy – Example I
Scalars in mechanics 19
A ball is thrown vertically upwards with a velocity of 8.0 m/s. Which height will the ball reach (above the point of release)?
𝐸𝐴 = 𝐸𝐵𝐸𝑝𝐴 + 𝐸𝑘𝐴 = 𝐸𝑝𝐵 + 𝐸𝑘𝐵
0 +1
2𝑚𝑣𝐴
2 = 𝑚𝑔ℎ𝐵 + 0
1
2𝑣𝐴2 = 𝑔ℎ𝐵
1
2∙ 8.02 = 9.81 ∙ ℎ𝐵
ℎ𝐵 = 3.3𝑚
Strategy:
Indicate 2 points:A = point of releaseB = highest point
Set ℎ = 0 in A (lowest point)So, in A there is NO potential energy
In B there is no kinetic energy
Solve the energy conservation law
B
A
𝑣𝐴
ℎ𝐴 = 0
𝑣𝐵 = 0
Solution:
Conservation of energy – Example II
Scalars in mechanics 20
𝐸𝐴 = 𝐸𝐵
𝐸𝑝𝐴 + 𝐸𝑘𝐴 = 𝐸𝑝𝐵 + 𝐸𝑘𝐵
0 +1
2𝑚𝑣𝐴
2 = 𝑚𝑔ℎ𝐵 +1
2𝑚𝑣𝐵
2
1
2𝑣𝐴2 = 𝑔ℎ𝐵 +
1
2𝑣𝐵2
1
2∙ 8.02 = 9.81 ∙ 2.0 +
1
2∙ 𝑣𝐵2
𝑣𝐵 = 5.0𝑚𝑠−1
Strategy:
Indicate again 2 points:A = point of releaseB = highest point
Set ℎ = 0 in A (lowest point)So, in A there is NO potential energy
In B there is both potential and kinetic energy
Solve the energy conservation law
B
A
𝑣𝐴
ℎ𝐴 = 0
Solution:
A ball is thrown upwards with a velocity of 8.0 m/s at a certain angle.The ball climbs 2.0 m. Calculate the magnitude of the velocity at the highest point
𝑣𝐵
ℎ𝐵
Making work easier (but not less) - I
Scalars in mechanics 21
ℎ1
ℎ2
100𝑘𝑔 ∙ 9.81𝑚𝑠−2
𝐹1
∆ℎ = 2.0𝑚𝐹2 𝑠
There is NO other force than weight.Only the vertical displacement requires work.
∆𝐸𝑝 = 𝐹1 ∙ ∆ℎ = 𝐹2 ∙ 𝑠
981𝑁 ∙ 2.0𝑚 = 𝐹2 ∙ 6.0𝑚
𝐹2 = 3.3 ∙ 102𝑁
Feels like 33kg
Making work easier (but not less) - II
Scalars in mechanics 22
= 40𝑁 120𝑁
The pulley block carries the weight on 3 ropes.The load must be lifted 10 m up.
The weight is split in 3. One only needs to pull with 40𝑁 !
But!!
∆𝐸𝑝 = 120𝑁 ∙ 10𝑚 = 40𝑁 ∙ 30𝑚
You need to haul in 30 m of rope
Collisions
Scalars in mechanics 23
Elastic
Partlyinelastic
Before After Σ 𝑝 Σ𝐸𝑘Σ𝐸
Conservation of
Totallyinelastic
+ heat
++ heat
Power
Scalars in mechanics 24
𝑃 =Δ𝐸
Δ𝑡
Power is the rate of energy conversion
1𝑊𝑎𝑡𝑡 = 1𝐽/𝑠
𝑃 =Δ𝑊
Δ𝑡=𝐹 ∙ ∆𝑠
∆𝑡= 𝐹 ∙
∆𝑠
∆𝑡= 𝐹 ∙ 𝑣
Power
Scalars in mechanics 25
𝐹𝑓𝑜𝑟𝑤𝑎𝑟𝑑
𝐹𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑣
A car travels at a constant velocity 𝑣. Newton: 𝐹𝑓𝑜𝑟𝑤𝑎𝑟𝑑 = 𝐹𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒
The required power to keep the car driving at this velocity equals:
𝑃 =Δ𝑊
Δ𝑡=𝐹𝑓𝑜𝑟𝑤𝑎𝑟𝑑 ∙ ∆𝑠
∆𝑡= 𝐹𝑓𝑜𝑟𝑤𝑎𝑟𝑑 ∙
∆𝑠
∆𝑡= 𝐹𝑓𝑜𝑟𝑤𝑎𝑟𝑑 ∙ 𝑣 = 𝐹𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ∙ 𝑣
Power - Example
Scalars in mechanics 26
𝐹𝑓𝑜𝑟𝑤𝑎𝑟𝑑
𝐹𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑣
A BMW 7-series car has:Drag coefficient 𝐶𝐷 = 0.30Frontal area 𝐴 = 2.2𝑚2
Calculate the mechanical power required to drive 160 km/h.
𝑣 = 160𝑘𝑚ℎ−1 =160
3.6𝑚𝑠−1 = 44.4𝑚𝑠−1
Density of air = 1.29 𝑘𝑔𝑚−3
𝐹𝑎𝑖𝑟 =1
2𝜌𝐴𝐶𝐷𝑣
2 =1
2∙ 1.29 ∙ 2.2 ∙ 0.30 ∙ 44.42
𝐹𝑎𝑖𝑟 = 841𝑁
𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ⇒ 𝐹𝑓𝑜𝑟𝑤𝑎𝑟𝑑 = 𝐹𝑎𝑖𝑟
𝑃 = 𝐹𝑎𝑖𝑟 ∙ 𝑣 = 841 ∙ 44.4 = 37 ∙ 103𝑊 = 37𝑘𝑊
Solution
Power & efficiency
Scalars in mechanics 27
𝜂 =𝐸𝑢𝑠𝑒𝑓𝑢𝑙
𝐸𝑡𝑜𝑡𝑎𝑙=
𝐸𝑢𝑠𝑒𝑓𝑢𝑙 𝑡
𝐸𝑡𝑜𝑡𝑎𝑙 𝑡=𝑃𝑢𝑠𝑒𝑓𝑢𝑙
𝑃𝑡𝑜𝑡𝑎𝑙
Efficiency can be related to a ratio of energies or a ratio of powers. Look at the context and choose!
END
Scalars in mechanics 28
DisclaimerThis document is meant to be apprehended through professional teacher mediation (‘live in class’) together with a physics text book, preferably on IB level.