overview: rotational kinetic energy
Post on 05-Apr-2022
3 Views
Preview:
TRANSCRIPT
11/16/2021
1
Rotational Motion Overview (Chapter 8, 9, 10)Chapters
[Pure] Rotational
Motion (8.1-4, 8.6-7):
Rotation about a fixed
Axis
Angular Kinematic Variables ( ิฆ๐, ๐, ิฆ๐ผ) โ Axis of Rotation (8.1-2)
Torque ๐ (8.3)
Moment of Inertia ๐ผ (8.4-5) and Rotational Kinetic Energy
Conservation of Energy including rotational motion (8.6)
Conservation of Angular Momentum ๐ฟ (8.7)
Rolling Motion (9.1-9.5)
= rotation + translation
Understanding Rolling Motions (9.1-9.3)
Conservation of Energy: rotation and translation (9.5)
Newtonโs 2nd Law: rotation and translation: acceleration (9.4)
Equilibrium (10.1-10.3) ฮฃ ิฆ๐น = 0 and ฮฃิฆ๐ = 0 (10.1, 10.2)
Equilibrium Lab, Applications (10.3)
Overview: Rotational Kinetic Energy
โข Use conservation of energy to solve rotational motion problems.โข This requires knowing the Kinetic Energy in rotational motion.โข Moment of Inertia.โข Moment of Inertia depends on the axis of rotation.
โข Moment of Inertial for rigid bodies.โข Moment of inertia for a rodโข Moment of inertial for a rectangleโข Moment of inertia for a cylinderโข Moment of inertia for a sphere
โข Example Problems along the way
1
4
11/16/2021
2
๐ฟ = 1.0 ๐, ๐ = 1 ๐๐.
Rotational KE for rod of length ๐ฟ, mass ๐.
From the Rotational KE, we can find the Moment of Inertia about the axis of rotation.
5
7
11/16/2021
3
๐ฟ
๐1
๐2
๐2
๐ฃ = ?
โข A rod with mass ๐1 = 4 ๐๐ and length ๐ฟ = 1.0 ๐ is hung horizontally on the wall with the pivot at the centre of the rod.
โข A second mass ๐2 = 0.2 ๐๐ is attached to one end.
โข Starting horizontally at rest, the rod rotates due to the added mass. What is the speed of the mass when it reaches the bottom?
๐1
Overview: Rotational Kinetic Energy
โข Use conservation of energy to solve rotational motion problems.โข This requires knowing the Kinetic Energy in rotational motion.โข Moment of Inertia.โข Moment of Inertia depends on the axis of rotation.
โข Moment of Inertial for rigid bodies.โข Moment of inertia for a rodโข Moment of inertial for a rectangleโข Moment of inertia for a cylinderโข Moment of inertia for a sphere
โข Example Problems along the way
9
11
11/16/2021
4
Moment of inertia not in table.
โข A rectangle has sides ๐ = 0.25 ๐,๐ = 0.75 ๐ with mass ๐ = 2.0 ๐๐.
โข Given that the moment of inertia of a
rectangle is 1
12๐(๐2 + ๐2) about the
centre of a rectangle, what are the moment of inertia at point A?
ร ร A
๐
๐
12
13
11/16/2021
5
Overview: Rotational Kinetic Energy
โข Use conservation of energy to solve rotational motion problems.โข This requires knowing the Kinetic Energy in rotational motion.โข Moment of Inertia.โข Moment of Inertia depends on the axis of rotation.
โข Moment of Inertial for rigid bodies.โข Moment of inertia for a rodโข Moment of inertial for a rectangleโข Moment of inertia for a cylinderโข Moment of inertia for a sphere
โข Example Problems along the way
What is the moment of inertial of this thin walled hollow cylinder (mass ๐, radius ๐ , length ๐ฟ)?
๐ผ๐๐ฅ๐๐ = ๐1๐12 +๐2๐2
2 +๐3๐32 + โฆ
A. ๐๐ 2
B.1
2๐๐ 2
C.1
4๐๐ 2
D.1
8๐๐ 2
E. 1
12๐๐ 2
14
16
11/16/2021
6
๐1
๐2
Rotational KE example problem
โข Two masses, ๐1 = 2 ๐๐,๐2 = 1 ๐๐ are connected by a string with negligible weight.
โข The string is then put around a pulley with mass ๐ = 6 ๐๐ and radius ๐ = 0.5 ๐.
โข The pulley is made like a bicycle wheel where the mass is concentrated on the rim only.
โข Initially, mass ๐1is at rest at height โ = 2 ๐while ๐2 is resting on the ground.
โข When released, what is the speed of ๐1 just before it hits the ground?
๐ = 6 ๐๐๐ = 0.5 ๐
๐1 = 2 ๐๐
๐2
= 1 ๐๐
๐1
๐2
Massless wheel for the moment (๐ = 0)
โข ฮ๐พ๐ธ =1
2๐1๐ฃ
2 +1
2๐2๐ฃ
2
โข What is โฮ๐๐ธ = ๐๐ธ๐ โ ๐๐ธ๐?
A. ๐1๐โ
B. ๐2๐โ
C. ๐1๐โ +๐2๐โ
D. ๐1๐โ โ๐2๐โ
Conservation of Energy ฮ๐พ๐ธ = โฮ๐๐ธ๐ = 0
๐1 = 2 ๐๐
๐2
= 1 ๐๐
18
19
11/16/2021
7
๐1
๐2
Massless wheel for the moment (๐ = 0)
โข ฮ๐พ๐ธ =1
2๐1๐ฃ
2 +1
2๐2๐ฃ
2
โข โฮ๐๐ธ = ๐๐ธ๐ โ ๐๐ธ๐ = ๐1๐โ โ๐2๐โ
Conservation of Energy ฮ๐พ๐ธ = โฮ๐๐ธ
1
2๐1๐ฃ
2 +1
2๐2๐ฃ
2 =๐1๐โ โ๐2๐โ
๐ = 0
๐1 = 2 ๐๐
๐2
= 1 ๐๐
๐1
๐2
Massless wheel for the moment (๐ = 6 ๐๐)
โข โฮ๐๐ธ = ๐๐ธ๐ โ ๐๐ธ๐ = ๐1๐โ โ๐2๐โ
โข ฮ๐พ๐ธ =1
2๐1๐ฃ
2 +1
2๐2๐ฃ
2 + ฮ๐พ๐ธ๐๐๐ก๐๐ก๐๐๐
Conservation of Energy ฮ๐พ๐ธ = โฮ๐๐ธ๐ = 6 ๐๐๐ = 0.5 ๐
๐1 = 2 ๐๐
๐2
= 1 ๐๐
20
21
11/16/2021
8
What is the moment of inertial of this solid cylinder (mass ๐, radius ๐ , length ๐ฟ)?
๐ผ = ๐๐ 2
๐ผ = ?
Replace the pulley in the previous problem with a solid disk. What is the moment of inertial ๐ผ of a solid disk (mass ๐, radius ๐ ) ?
๐1
๐2
๐1
๐2
Previously ๐ผ =๐๐ 2
23
25
11/16/2021
9
Overview: Rotational Kinetic Energy
โข Use conservation of energy to solve rotational motion problems.โข This requires knowing the Kinetic Energy in rotational motion.โข Moment of Inertia.โข Moment of Inertia depends on the axis of rotation.
โข Moment of Inertial for rigid bodies.โข Moment of inertia for a rodโข Moment of inertial for a rectangleโข Moment of inertia for a cylinderโข Moment of inertia for a sphere
โข Example Problems along the way
28
29
11/16/2021
11
1. Torque โ Concepts (magnitudes only)
We need a torque to start rotating the objectLinear Angular
KE 1
2๐๐ฃ2
1
2๐ผ๐2
displacement ๐ฅ ๐
velocity ๐ฃ ๐
acceleration ๐ ๐ผ
Inertia ๐ ๐ผ
Force Torque
๐น ๐
32
33
11/16/2021
12
Definition of torque ๐
๐๐
๐๐
Same force ิฆ๐น applied at two different locations yield different torques
Easier to rotate the nut when the force is applied at ๐๐ than at ๐๐
Applied force here is perpendicular to ๐๐ and ๐๐.
Magnitudes only:
Definition of torque ๐
ิฆ๐
๐ิฆ๐น
The radial component ๐น cos ๐ does not rotate the wrench/nut. Only the tangential component of the force ๐น sin ๐ does rotate it.
๐น sin ๐
๐น cos ๐
When ิฆ๐น is not perpendicular to ิฆ๐
34
35
11/16/2021
13
Definition of torque ๐
ิฆ๐
๐ิฆ๐น
๐
Keep the full magnitude of
vector ิฆ๐น, but take the perpendicular component (๐โฅ) of the displacement vector ิฆ๐.
We need a torque to start rotating the objectLinear Angular
KE 1
2๐๐ฃ2
1
2๐ผ๐2
๐ฅ ๐
๐ฃ ๐
๐ ๐ผ
๐ ๐ผ
Force Torque
๐น
Newtonโs Law
๐น = ๐๐
ิฆ๐
๐ ิฆ๐น
๐นโฅ = ๐น sin ๐
36
37
11/16/2021
14
We need a torque to start rotating the object
1. Tangential acceleration ๐โฅ
๐โฅ = ๐๐ผ
๐ผ is the angular accel.
2. ๐นโฅ = ๐๐โฅ
Starting with ๐ = ๐๐นโฅ which is correct?
A. ๐ = ๐๐ผ
B. ๐ = ๐๐๐ผ
C. ๐ = ๐๐2๐ผ
ิฆ๐
๐ิฆ๐น
๐นโฅ = ๐น sin ๐
๐โฅ
๐ถ
๐
We need a torque to start rotating the objectLinear Angular
KE 1
2๐๐ฃ2
1
2๐ผ๐2
๐ฅ ๐
๐ฃ ๐
๐ ๐ผ
๐ ๐ผ
Force Torque
๐น
Newtonโs Law
๐น = ๐๐
ิฆ๐
๐ ิฆ๐น
๐นโฅ = ๐น sin ๐
๐ถ
39
41
11/16/2021
15
2. Torque โ Vector Math
From ๐ = ๐ผ๐ผ to ิฆ๐ = ๐ผ ิฆ๐ผ
We need a torque to start rotating the objectLinear Angular
KE 1
2๐๐ฃ2
1
2๐ผ๐2
๐ฅ ๐
๐ฃ ๐
๐ ๐ผ
๐ ๐ผ
Force Torque
๐น
Newtonโs Law
๐น = ๐๐
ิฆ๐
๐ ิฆ๐น
๐นโฅ = ๐น sin ๐
๐ถ
42
44
11/16/2021
16
๐
๐ฅ
๐ฆ
๐ง
โข Consider 1D rotation
โข Displacement ๐
โข Ang. Velocity ฯ = ฮ๐/ฮ๐ก
โข Ang. Accel. ๐ผ = ฮ๐/ฮ๐ก
โข object rotates about
Rotational Motion
โข Consider 1D rotation
โข Displacement ๐
โข Ang. Velocity ฯ = ฮ๐/ฮ๐ก
โข Ang. Accel. ๐ผ =ฮ๐
ฮ๐ก
โข object rotates about +๐ง axis
โข They are all in the direction of the axis of rotation: in this example,
โข ิฆ๐ = ๐๐
โข ๐ = ๐๐
โข ิฆ๐ผ = ๐ผ๐
Rotational Motion
๐
๐ฅ
๐ฆ
๐ง
Vector ิฆ๐ = ๐ผ ิฆ๐ผMagnitude: ๐ = ๐ผ๐ผ
48
50
11/16/2021
17
We need a torque to start rotating the objectLinear Angular
KE 1
2๐๐ฃ2
1
2๐ผ๐2
๐ฅ ๐
๐ฃ ๐
๐ ๐ผ
๐ ๐ผ
Force Torque
๐น ๐ = ๐๐น sin๐
Newtonโs Law
ิฆ๐น = ๐ ิฆ๐
ิฆ๐
๐ ิฆ๐น
๐นโฅ = ๐น sin ๐
๐ถ
We need a torque to start rotating the object
ิฆ๐๐ ิฆ๐น
๐นโฅ = ๐น sin ๐
Recall
Axis of rotation:+๐
Torque, angular accel, velocity, displacement are all in the + ๐ direction.
๐ = ๐๐ญ ๐ฌ๐ข๐ง๐ฝ can be rewritten as
๐ฅ
๐ฆ
๐ง๐ = ๐๐นโฅ = ๐๐ญ๐ฌ๐ข๐ง๐ฝ
52
53
top related