vectors in mechanics
DESCRIPTION
Newtonian mechanics in vector form: Elements of vectors, operations on vectors, kinematics, Laws of Newton, Momentum & ImpulseTRANSCRIPT
Vectors
in mechanics
Photo: Wikimedia.org
Quantities
Vectors in mechanics 2
Scalar quantities Vector quantities
mass m (kg)
energy E (J)
power P (W)
displacement π (π)
velocity π£ (ππ β1)
acceleration π (ππ β2)
force πΉ (π)
momentum π (ππππ β1) torque π (ππ)
β’ Magnitude β’ Magnitudeβ’ Directionβ’ Point of application
Vectors in mechanics 3
Vector notation
πΉ arrow above orF bold face
6.0 ππ
30 π
6.0ππ β 30π
1.0ππ β 5.0π Scale
Head
Tail
Point of application matters,β¦
Vectors in mechanics 4
πΉ
πΉ
butβ¦is often compromisedβ¦
Vectors in mechanics 5
πΉπ
πΉπππ
πΉππππ
πΉπ
πΉπ
πΉππππ
πΉπ
πΉπππ
πΉπ
πΉππππ
πΉπ
πΉπππ
β¦to make life easier
Basic operations with vectors
Vectors in mechanics 6
Adding vectors πΉπππ€ = πΉπππ1 + πΉπππ2
Subtracting vectors πΉπππ€ = πΉπππ1 β πΉπππ2
Multiplying a vector with a scalar πΉπππ€ = π β πΉπππ
Resolving a vector into components πΉπππ€1 + πΉπππ€2 = πΉπππ
a = 2
ππΉ β πΆππ π
πΉ β πππ π πΉ
Resolve a vector along 2 working lines
Vectors in mechanics 7
πΉπππ€1
πΉπππ€2
πΉπππ
copy
copy
πΉπππ = πΉπππ€1+ πΉπππ€2
πΉπππ€1 =?
πΉπππ€2 =?
πΉπππ
2 given working lines
Displacement vs. Distance
Vectors in mechanics 8
Type equation here.
π 1
π 2 = π 1 + β π
β π = π 2 β π 1
vector scalar
Do not interpret βsβas the distance, because it represents the magnitude of the displacement vector π
Velocity vs. Speed
Vectors in mechanics 9
π£ππππππ‘π¦ =πππ πππππππππ‘
π‘ππππ ππππ =
πππ π‘ππππ
π‘πππ
vector scalar
Average vs. Instantaneous
Vectors in mechanics 10
t(s)
s(m)
βπ
βπ‘
π£ππ£π 5π β 10π =π 10 β π 5
10 β 5=βπ
βπ‘
= gradient of the secant line= difference quotient
= gradient of the tangent= differential quotient
π£ 5π = limππ‘β0
π 5 + ππ‘ β π (5)
5 + ππ‘ β 5=ππ
ππ‘= π β²
Acceleration β notation issues
Vectors in mechanics 11
π =βπ£
βπ‘=π£ β π’
βπ‘
π =βπ£
βπ‘=π£2 β π£1βπ‘
Initial velocity = π’
Initial velocity = π£0
Movement along a straight line:β’ π β π ππ π₯β’ π£ β π£β’ π β πβ’ Choose an originβ’ Choose a + direction!
π£
π +
me
π =βπ£
βπ‘β βπ£ = π β βπ‘ β π£ β π£0 = π β π‘ β π‘0 β π£ = π£0 + π π‘ β π‘0
Uniformly accelerated motion - I
Vectors in mechanics 12
π‘
π£
π‘π
π£π
π > 0
π‘
π£
π‘π
π£π
π < 0
Uniformly accelerated motion - II
Vectors in mechanics 13
π‘
π£
π‘1
π£1
π‘2
π£2
π£ππ£π
π‘
π£
π‘1
π£1
π‘2
π£2
π£ππ£π
π£ππ£π =π£1 + π£22
π£ππ£π =βπ
βπ‘β βπ = π£ππ£π β βπ‘
Uniformly accelerated motion - III
Vectors in mechanics 14
π£ππ£π =π£ + π£02
βπ =π£ + π£02
β βπ‘
π£ = π£0 + π β βπ‘
βπ =π£0 + π β βπ‘ + π£0
2β βπ‘ = π£0 + 1 2π β βπ‘ β βπ‘ β π = π 0 + π£0 β π‘ β π‘0 + 1 2 π β π‘ β π‘0
2
Uniformly accelerated motion - IV
Vectors in mechanics 15
π = π 0 + π£0 β π‘ β π‘0 + 1 2π β π‘ β π‘02
π‘
π£
π‘π
π£π
π > 0
π£ = π£0 + π π‘ β π‘0
π‘
π
π‘π
π π
Gradient of tangent = π£0
Uniformly accelerated motion - V
Vectors in mechanics 16
π = π 0 + π£0 β π‘ β π‘0 + 1 2π β π‘ β π‘02
π‘
π£
π‘π
π£π
π < 0
π£ = π£0 + π π‘ β π‘0
π‘
π
π‘π
π π
Uniformly accelerated motion - VI
Vectors in mechanics 17
π = π 0 + π£0 β π‘ β π‘0 + 1 2π β π‘ β π‘02
π£ = π£0 + π π‘ β π‘0General equations
π‘0 = 0 Generally
π 0 = 0 Accelerationfrom restπ£0 = 0
π = π 0 + π£0 β π‘ + 1 2π β π‘2
π£ = π£0 + ππ‘
π = 1 2π β π‘2
π£ = ππ‘
The math behind kinematics
Vectors in mechanics 18
π π‘ =ππ£
ππ‘= π£β² π‘ = π β²β² π‘
π£ π‘ =ππ
ππ‘= π β² π‘
π 0
π
ππ = π‘0
π‘
π£ β ππ‘
π£0
π£
ππ£ = π‘0
π‘
π β ππ‘
π π‘ Displacement
Velocity
Acceleration π π‘
Differentiation
Differentiation
Integration
Integration
Displacement diagrams
Vectors in mechanics 19
π‘
π
π‘
π
βπ‘
βπ‘
βπ
βπ
Determine π‘ read π‘-axis
Determine π read π -axis
Determine π£ gradient of the tangent
Determine π
Velocity diagrams
Vectors in mechanics 20
Determine π‘ read π‘-axis
Determine π area under the graph
Determine π£ read π£-axis
Determine π gradient of the graph
π‘
π£1
π‘
π£
π‘1 π‘2
π‘1 π‘2
π£2
βπ£
βπ‘
Acceleration diagrams
Vectors in mechanics 21
π
π
Determine π‘ read π‘-axis
Determine π
Determine π£ area under the graph
Determine π read π-axis
π‘1 π‘2
π‘1 π‘2
Relative velocity - I
Vectors in mechanics 22
How much time does it take A to catch up on B?
π£π΄π΅ = π£π΄ β π£π΅ = 15 β 10 = 5ππ β1 β βπ‘ = βπ π΄π΅
π£π΄π΅ = 1005 = 20 π
π£π΄ = 15 ππ β1 π£π΅ = 10 ππ
β1
100 π
How much time does it take A to βmeetβ B?
π£π΄π΅ = π£π΄ β π£π΅ = 15 β β10 = 25ππ β1 β βπ‘ = βπ π΄π΅π£π΄π΅ = 150
25 = 6.0 π
π£π΄ = 15 ππ β1 π£π΅ = 10 ππ
β1
150 π
A
A
B
B
Relative velocity - II
Vectors in mechanics 23
How fast does A approach B? π£π΄ = 15 ππ β1
π£π΅ = 10 ππ β1
B
30Β°
30Β°
π£π΅
π£π΄
π£π΄π΅
π£π΄π΅π₯
π£π΄π΅π¦
π£π΄π΅π₯ = 15πΆππ 30Β° β 10 = 3.0 mπ β1
π£π΄π΅π¦ = 15πππ 30Β° = 7.5 mπ β1
π£π΄π΅ = 3.02 + 7.52 = 8.1 ππ β1
Forces
Vectors in mechanics 24
Name Symbol Value Direction Point of application
Weight πΉπ πΉπ = m β π To the centre of the Earth Center of mass
Friction πΉπΉ πΉπΉ = π β πΉπ β₯ to surface Contact surface
Air resistance πΉπ΄ππ πΉπ΄ = 1 2ππΆπ·π΄π£2 β π£ Front of moving object
Rolling resistance πΉπ πΉπ = πΆπ β πΉπ β π£ Contact surface
Buoyancy, Upthrust πΉπ πΉπ = π β π β π Along pressure gradient π»π Lowest point of object
Tension πΉπ Reaction force β₯ the cord End of the cord
Contact force πΉππ’π β πΉπ Reaction force β₯ surface Contact surface
Tension, Compression, Shear
Vectors in mechanics 25
Compression
Tension
Shear
Resultant, Net, Overall force
Vectors in mechanics 26
πΉπ
πΉπ΄ππ
Ξ£ πΉ
πΉππΈππ
πΉπ πΈπ
Ξ£ πΉ1st choice
Linear momentum
Vectors in mechanics 27
π = π π£ Momentum is conserved in an isolated system Ξ£ π = constant
ππ΄ = 5.0 π
π£π΄ = 300 ππ β1
ππ΅ = 1.0 ππ
π£π΅ = 0
π£πππ ?
Ξ£πππππππ = Ξ£ππππ‘ππ β ππ΄ β π£π΄ +ππ΅ β π£π΅ = ππ΄ +ππ΅ β π£πππ
5.0 β 10β3 β 300 + 1.0 β 0 = 5.0 β 10β3 + 1.0 β π£πππ β π£πππ = 1.5 ππ β1
Newtonβs laws
Vectors in mechanics 28
For each separate object:
Between objects:
Ξ£ πΉ =Ξ π
Ξπ‘Ξ£ πΉ = π
Ξ π£
Ξπ‘β Ξ£ πΉ = π π
Constant mass2
If the resultant force on an object is zero, the object is in βtranslational equilibriumβ1
3 β ππ΄ = ββ ππ΅ ββ ππ΄βπ‘=β ππ΅βπ‘
β πΉπ΄π΅ = β πΉπ΅π΄
πΉπ΅π΄ πΉπ΄π΅
A B
βForce by A on Bβ
Wikimedia.org
Newtonβs laws β Example (1/5)
Vectors in mechanics 29
Neglect air resistance in this problem.
A caravan (820 kg) is pulled by a car. The combination accelerates with π = 0.60 ππ β2.The caravan is subject to a rolling resistance of πΉππππ = 1.2 β 10
2π
Question 1: Calculated the pulling force by the car on the caravan.
The car with passengers (total 1580 kg) is subject to a rolling resistance of 1.6 β 102π during the acceleration.
Question 2: Calculate the forward force by the road on the car.
Newtonβs laws β Example (2/5)
Vectors in mechanics 30
A
B
ππ΄ = 1580 ππ
ππ΅ = 820 ππ
π = 0.60ππ β2
+
Preparation: Make sketch of the connected car & caravan. Designate them βAβ & βBβ.List the information given in the text: mass, acceleration and define a PLUS direction!
Newtonβs laws β Example (3/5)
Vectors in mechanics 31
A
B
ππ΄ = 1580 ππ
ππ΅ = 820 ππ
π = 0.60ππ β2
+
πΉπ΄π΅
πΉππππ,π΅ = β1.2 β 102π
1: Draw the relevant force vectors on the caravan
Question 1
2: Apply 2nd law on B
Ξ£ πΉπ΅ = ππ΅ π β πΉπ΄π΅ + πΉππππ,π΅ = ππ΅ π β πΉπ΄π΅ + β1.2 β 102 = 820 β +0.60 β πΉπ΄π΅ = 612π
Round result to 6.1 β 102π
Newtonβs laws β Example (4/5)
Vectors in mechanics 32
A
B
ππ΄ = 1580 ππ
ππ΅ = 820 ππ
π = 0.60ππ β2
+
πΉπ΄π΅ = 612π
πΉππππ,π΅ = β1.2 β 102π
1: Add the relevant force vectors on the car
Question 2
3: Apply 2nd law on A
Round result to 1.7 β 103π
πΉπ΅π΄
πΉππππ,π΄ = β1.6 β 102π
πΉππππ,π΄
2: Apply 3rd law on A: πΉπ΅π΄ = β πΉπ΅π΄ = β612π
Ξ£ πΉπ΄ = ππ΄ π β πΉππππ,π΄ + πΉπ΅π΄ + πΉππππ,π΄ = ππ΅ π β πΉππππ,π΄ β 612 β 1.6 β 102 = 1580 β 0.60 β πΉππππ,π΄ = 1720π
Newtonβs laws β Example (5/5)
Vectors in mechanics 33
A
B
ππ΄ = 1580 ππ
ππ΅ = 820 ππ
π = 0.60ππ β2
+
πΉπ΄π΅
πΉππππ,π΅ = β1.2 β 102π
1: Add the relevant force vectors on the car
Question 2: Alternative solution: treat A & B as combination AB
3: Apply 2nd law on A and B together
Round result to 1.7 β 103π
πΉπ΅π΄
πΉππππ,π΄ = β1.6 β 102π
πΉππππ,π΄
2: Regard the combination as a whole. Now πΉπ΄π΅ and πΉπ΅π΄ are internal forces and cancel out !
Ξ£ πΉπ΄π΅ = ππ΄ +ππ΅ π β πΉππππ,π΄ + πΉππππ,π΄ + πΉππππ,π΅ = ππ΄ +ππ΅ π β
β πΉππππ,π΄ β 1.2 β 102 β 1.6 β 102 = 1580 + 820 β 0.60 β πΉππππ,π΄ = 1720π
AB
Impulse
Vectors in mechanics 34
πΉ =β π
βπ‘β β π = πΉ β βπ‘
π β β π£ = πΉ β βπ‘
π β π£2 β π£1 = π‘0
π‘1 πΉ π‘ β ππ‘
Constant mass
πΉ
π‘π‘1 π‘2
Constant force
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Impulse β Example (1/2)
Vectors in mechanics 35
πΉ π
π‘
1000
A baseball (145 g) is thrown and flies with 155 m/s. The hitterβs club hits the ball frontally. Assume that the ballFlies back in the opposite direction. The force by the club on the ball is drawn in the diagram below.
Question: Determine the speed after the hit.
105ππ
Impulse β Example (2/2)
Vectors in mechanics 36
πΉ π
π‘
1000
95ππ
3: β π = πΉ β βπ‘ β π π£2 β π£1 = β42.3
2: The impulse πΉ β βπ‘ equals the area under the graph. It can be approximated by the dashed triangle
πΉ β βπ‘ =1
2Γ 0.095 Γ β900 = β42.3ππ
1: make sketch and choose PLUS direction π£1 = +155ππ
β1
π£2
β π = πΉ β βπ‘
Tock!
+ πΉ
0.145 β π£2 β 155 = β42.3
π£2 = β140ππ β1
4: Round to β1.4 β 102ππ β1
(graph area reading uncertainty)
END
Vectors in mechanics 37
DisclaimerThis document is meant to be apprehended through professional teacher mediation (βlive in classβ) together with a physics text book, preferably on IB level.