mechanics - department of physics and astronomy - home 298 summer...mechanics chapter 2 of essential...
TRANSCRIPT
1Prof. Sergio B. MendesSummer 2018
Mechanics
Chapter 2 of Essential University Physics, Richard Wolfson, 3rd Edition
2Prof. Sergio B. MendesSummer 2018
Mechanics
DynamicsKinematicsβ’ How something moves ?? β’ Why something moves ??
β’ Geometrical description
β’ Mathematics
β’ Physical cause
β’ Physics
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
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
5Prof. Sergio B. MendesSummer 2018
Examples
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
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‘
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‘
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‘
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‘
11Prof. Sergio B. MendesSummer 2018
Acceleration
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‘
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£
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‘
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
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‘
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
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
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
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‘
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
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
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
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
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
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