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![Page 1: Mechanics - Department of Physics and Astronomy - Home 298 summer...Mechanics Chapter 2 of Essential University Physics, Richard Wolfson, 3rd Edition. Summer 2018 Prof. Sergio B. Mendes](https://reader035.vdocument.in/reader035/viewer/2022081514/5f0495677e708231d40eb0eb/html5/thumbnails/1.jpg)
1Prof. Sergio B. MendesSummer 2018
Mechanics
Chapter 2 of Essential University Physics, Richard Wolfson, 3rd Edition
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2Prof. Sergio B. MendesSummer 2018
Mechanics
DynamicsKinematicsβ’ How something moves ?? β’ Why something moves ??
β’ Geometrical description
β’ Mathematics
β’ Physical cause
β’ Physics
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3Prof. Sergio B. MendesSummer 2018
Kinematics
β’ Object in motion: a small particle (a point)
β’ 1D: along a straight direction
π₯π₯, π‘π‘ π₯π₯Γ
π₯π₯ = 0
ππ
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4Prof. Sergio B. MendesSummer 2018
Average Velocity
οΏ½Μ οΏ½π£ β‘π₯π₯2 β π₯π₯1π‘π‘2 β π‘π‘1
Between two well-defined events.
π₯π₯1, π‘π‘1 π₯π₯2, π‘π‘2
Event β1β Event β2β
Average between WHAT ??
π₯π₯Γ
π₯π₯ = 0
ππ
Average Velocity
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5Prof. Sergio B. MendesSummer 2018
Examples
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6Prof. Sergio B. MendesSummer 2018
Displaying the Two Events in a Plot
π₯π₯
π‘π‘
π₯π₯1, π‘π‘1Event β1β
π₯π₯2, π‘π‘2Event β2β
π‘π‘2π‘π‘1
π₯π₯2
π₯π₯1π‘π‘2 β π‘π‘1
π₯π₯2 β π₯π₯1
ππ
οΏ½Μ οΏ½π£ β‘π₯π₯2 β π₯π₯1π‘π‘2 β π‘π‘1
= π‘π‘π‘π‘π‘π‘ ππ
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7Prof. Sergio B. MendesSummer 2018
Making a Plot to Display the Particle Motion: π₯π₯ π‘π‘
π‘π‘
π₯π₯1, π‘π‘1Event β1β
π₯π₯2, π‘π‘2Event β2β
π‘π‘2π‘π‘1
π₯π₯2
π₯π₯1π‘π‘2 β π‘π‘1
π₯π₯2 β π₯π₯1
οΏ½Μ οΏ½π£ = οΏ½Μ οΏ½π£ = οΏ½Μ οΏ½π£ = οΏ½Μ οΏ½π£Regardless of the different types of motion:
π₯π₯ π‘π‘
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8Prof. Sergio B. MendesSummer 2018
Instantaneous Velocity
π‘π‘
π₯π₯1, π‘π‘1Event β1β
Event β2β
π‘π‘1
π₯π₯1ππ1
π₯π₯2 = π₯π₯1 + βπ₯π₯, π‘π‘2 = π‘π‘1 + βπ‘π‘
= limβπ‘π‘β0
βπ₯π₯βπ‘π‘
π£π£ π‘π‘1 β‘ limβπ‘π‘β0
π₯π₯2 β π₯π₯1π‘π‘2 β π‘π‘1
= π‘π‘π‘π‘π‘π‘ ππ1 =πππ₯π₯ π‘π‘ = π‘π‘1
πππ‘π‘
π‘π‘2 β π‘π‘1 = βπ‘π‘
π₯π₯2 β π₯π₯1 = βπ₯π₯
π₯π₯ π‘π‘
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9Prof. Sergio B. MendesSummer 2018
Instantaneous Velocityπ₯π₯ π‘π‘
π‘π‘
π£π£ π‘π‘ = π‘π‘π‘π‘π‘π‘ ππ π‘π‘ =πππ₯π₯ π‘π‘πππ‘π‘
ππ π‘π‘
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10Prof. Sergio B. MendesSummer 2018
Evaluating Instantaneous Velocity
π‘π‘
π₯π₯1, π‘π‘1Event β1β
π₯π₯2, π‘π‘2Event β2β
π‘π‘2π‘π‘1
π₯π₯1
π₯π₯ π‘π‘
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11Prof. Sergio B. MendesSummer 2018
Acceleration
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12Prof. Sergio B. MendesSummer 2018
π‘π‘
π£π£1, π‘π‘1
π£π£2, π‘π‘2
π‘π‘2π‘π‘1
π£π£2
π£π£1π‘π‘2 β π‘π‘1
π£π£2 β π£π£1
ππ
= π‘π‘π‘π‘π‘π‘ ππ
Average Acceleration
οΏ½π‘π‘ β‘π£π£2 β π£π£1π‘π‘2 β π‘π‘1
π£π£ π‘π‘
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13Prof. Sergio B. MendesSummer 2018
Instantaneous Accelerationπ£π£
π‘π‘
π£π£1, π‘π‘1
π‘π‘1
π£π£1ππ1
π£π£2 = π£π£1 + βπ£π£, π‘π‘2 = π‘π‘1 + βπ‘π‘
= limβπ‘π‘β0
βπ£π£βπ‘π‘
π‘π‘ π‘π‘1 β‘ limβπ‘π‘β0
π£π£2 β π£π£1π‘π‘2 β π‘π‘1
= π‘π‘π‘π‘π‘π‘ ππ1 =πππ£π£ π‘π‘ = π‘π‘1
πππ‘π‘
π‘π‘2 β π‘π‘1 = βπ‘π‘
π£π£2 β π£π£1 = βπ£π£
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14Prof. Sergio B. MendesSummer 2018
Instantaneous Accelerationπ£π£ π‘π‘
π‘π‘
π‘π‘ π‘π‘ = π‘π‘π‘π‘π‘π‘ ππ π‘π‘ =πππ£π£ π‘π‘πππ‘π‘
ππ π‘π‘
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15Prof. Sergio B. MendesSummer 2018
π£π£ π‘π‘ =πππ₯π₯ π‘π‘πππ‘π‘
π₯π₯ π‘π‘
π‘π‘ π‘π‘ =πππ£π£ π‘π‘πππ‘π‘ =
πππππ‘π‘
πππ₯π₯ π‘π‘πππ‘π‘
=ππ2π₯π₯ π‘π‘πππ‘π‘2
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16Prof. Sergio B. MendesSummer 2018
Constant Acceleration
π‘π‘ π‘π‘ = π‘π‘
οΏ½π‘π‘ = π‘π‘ =π£π£ π‘π‘ β π£π£πππ‘π‘ β 0
οΏ½π‘π‘ = π‘π‘
π£π£ π‘π‘ = π£π£ππ + π‘π‘ π‘π‘
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17Prof. Sergio B. MendesSummer 2018
π£π£ π‘π‘ = π£π£ππ + π‘π‘ π‘π‘
π£π£ π‘π‘
π‘π‘0
π£π£ππ
οΏ½Μ οΏ½π£ = π£π£ππ +12π‘π‘ π‘π‘ =
π₯π₯ π‘π‘ β π₯π₯πππ‘π‘ β 0
π£π£ππ + π‘π‘ π‘π‘
π₯π₯ π‘π‘ = π₯π₯ππ + π£π£ππ π‘π‘ +12π‘π‘ π‘π‘ 2
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18Prof. Sergio B. MendesSummer 2018
π₯π₯ π‘π‘ = π₯π₯ππ + π£π£ππ π‘π‘ +12π‘π‘ π‘π‘ 2
π£π£ππ = 0
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19Prof. Sergio B. MendesSummer 2018
π₯π₯ π‘π‘ = π₯π₯ππ + π£π£ππ π‘π‘ +12π‘π‘ π‘π‘ 2
π£π£ π‘π‘ = π£π£ππ + π‘π‘ π‘π‘
π‘π‘ π‘π‘ = π‘π‘
Constant Acceleration
π£π£2 π‘π‘ = π£π£ππ2 + 2 π‘π‘ π₯π₯ π‘π‘ β π₯π₯ππ
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20Prof. Sergio B. MendesSummer 2018
0 = π£π£ππ + π‘π‘ π‘π‘π‘
π‘π‘π‘ = βπ£π£πππ‘π‘
When does the velocity go to zero ?
π£π£ π‘π‘
π‘π‘
π£π£ππ
π‘π‘π‘
π£π£ π‘π‘ = π£π£ππ + π‘π‘ π‘π‘
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21Prof. Sergio B. MendesSummer 2018
π‘π‘π‘ = βπ£π£πππ‘π‘
π₯π₯ π‘π‘π‘ = π₯π₯ππ + π£π£ππ π‘π‘π‘ +12π‘π‘ π‘π‘π‘2
= π₯π₯ππ βπ£π£ππ2
2 π‘π‘
π₯π₯ π‘π‘
π‘π‘π‘π‘π‘ = βπ£π£πππ‘π‘
π₯π₯ π‘π‘π‘ = π₯π₯ππ βπ£π£ππ2
2 π‘π‘
π₯π₯ππ
At the time the velocity goes to zero, then
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22Prof. Sergio B. MendesSummer 2018
Constant Acceleration due to Gravityπ₯π₯ β π¦π¦
π¦π¦ π‘π‘ = π¦π¦ππ + π£π£ππ π‘π‘ β12ππ π‘π‘ 2
π£π£ π‘π‘ = π£π£ππ β ππ π‘π‘
π£π£2 π‘π‘ = π£π£ππ2 β 2 ππ π¦π¦ π‘π‘ β π¦π¦ππ
π¦π¦
π‘π‘ π‘π‘ = βππ ππ β 9.8 ππ/π π 2
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23Prof. Sergio B. MendesSummer 2018
π‘π‘π‘ =π£π£ππππ π¦π¦ π‘π‘π‘ = π¦π¦ππ +
π£π£ππ2
2 ππ
π‘π‘π‘
π¦π¦ π‘π‘π‘
π¦π¦ π‘π‘ = π¦π¦ππ + π£π£ππ π‘π‘ β12ππ π‘π‘ 2
π£π£ π‘π‘ = π£π£ππ β ππ π‘π‘
π‘π‘ π‘π‘ = βππ
π‘π‘π‘
π¦π¦ππ
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24Prof. Sergio B. MendesSummer 2018
You toss a ball straight up at 7.3 m/s; it leaves your hand at 1.5 m above the floor.
a) Find its maximum height.
b) Find when it hits the floor.
a) At the maximum height the ball is instantaneously at rest: v = 0
π¦π¦ππ = 1.5 πππ£π£ππ = 7.3 ππ/π π
π£π£2 = π£π£ππ2 β 2 ππ π¦π¦ β π¦π¦ππ
What do we know?
ππ β 9.8 ππ/π π 2
π¦π¦ = π¦π¦ππ +π£π£ππ2
2 ππ= 4.2 ππ
b) When it hits the floor: y = 0
π¦π¦ = π¦π¦ππ + π£π£ππ π‘π‘ β12ππ π‘π‘ 2 π‘π‘ =
βπ£π£ππ Β± π£π£ππ2 + 2π¦π¦ππ ππβππ
β0.18 π π
+1.7 π π
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25Prof. Sergio B. MendesSummer 2018
πππ₯π₯ π‘π‘πππ‘π‘
= π£π£ π‘π‘
π₯π₯ π‘π‘
πππ£π£ π‘π‘πππ‘π‘
= π‘π‘ π‘π‘
οΏ½0
π‘π‘π£π£ π‘π‘ πππ‘π‘ = π₯π₯ π‘π‘ β π₯π₯ππ
οΏ½0
π‘π‘π‘π‘ π‘π‘ πππ‘π‘ = π£π£ π‘π‘ β π£π£ππ
Big Picture of Chapter 2:Kinematics in 1D
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26Prof. Sergio B. MendesSummer 2018
π‘π‘ π‘π‘ = π‘π‘
π₯π₯ π‘π‘ β π₯π₯ππ = π£π£ππ π‘π‘ +12π‘π‘ π‘π‘ 2 = οΏ½
0
π‘π‘π£π£ π‘π‘ πππ‘π‘
οΏ½0
π‘π‘π‘π‘ π‘π‘ πππ‘π‘ = π‘π‘ π‘π‘ = π£π£ π‘π‘ β π£π£ππ
For example, in the case ofConstant Acceleration