Post on 07-Feb-2020
transcript
April 9, Week 12
Today: Chapter 9, Rotational Energy
Homework #9 - Due April 16 at 11:59pmMastering Physics: 7 questions from chapter 9. WrittenQuestion: 10.80
Rotational Energy – p. 1/10
Review
The rate at which an objects spins is given by its angularvelocity, −→ω , and angular acceleration −→
α .
ω = dθ
dt, α = dω
dt
b
Rotational Energy – p. 2/10
Review
The rate at which an objects spins is given by its angularvelocity, −→ω , and angular acceleration −→
α .
ω = dθ
dt, α = dω
dt
RHR I - Curl the fingers of your righthand in the “sense" of the rotation.Your extended thumb, points in directionof −→ω .
b
Rotational Energy – p. 2/10
Review
The rate at which an objects spins is given by its angularvelocity, −→ω , and angular acceleration −→
α .
ω = dθ
dt, α = dω
dt
RHR I - Curl the fingers of your righthand in the “sense" of the rotation.Your extended thumb, points in directionof −→ω .
−→ω
⊗
b
Rotational Energy – p. 2/10
Review
The rate at which an objects spins is given by its angularvelocity, −→ω , and angular acceleration −→
α .
ω = dθ
dt, α = dω
dt
RHR I - Curl the fingers of your righthand in the “sense" of the rotation.Your extended thumb, points in directionof −→ω .
−→ω
⊗
b
b
Rotational Energy – p. 2/10
Review
The rate at which an objects spins is given by its angularvelocity, −→ω , and angular acceleration −→
α .
ω = dθ
dt, α = dω
dt
RHR I - Curl the fingers of your righthand in the “sense" of the rotation.Your extended thumb, points in directionof −→ω .
−→ω
⊗
b
−→r
b
Rotational Energy – p. 2/10
Review
The rate at which an objects spins is given by its angularvelocity, −→ω , and angular acceleration −→
α .
ω = dθ
dt, α = dω
dt
RHR I - Curl the fingers of your righthand in the “sense" of the rotation.Your extended thumb, points in directionof −→ω .
−→ω
⊗
b−→v −→
r
b
Rotational Energy – p. 2/10
Review
The rate at which an objects spins is given by its angularvelocity, −→ω , and angular acceleration −→
α .
ω = dθ
dt, α = dω
dt
RHR I - Curl the fingers of your righthand in the “sense" of the rotation.Your extended thumb, points in directionof −→ω .
−→ω
⊗
b−→v −→
r
−→v = −→ω ×
−→r
b
Rotational Energy – p. 2/10
Review
The rate at which an objects spins is given by its angularvelocity, −→ω , and angular acceleration −→
α .
ω = dθ
dt, α = dω
dt
RHR I - Curl the fingers of your righthand in the “sense" of the rotation.Your extended thumb, points in directionof −→ω .
−→ω
⊗
b−→v −→
r
−→v = −→ω ×
−→r
Take the fingers of the right hand and “sweep"−→
A into−→
B ,extended thumb points in direction of cross product
Rotational Energy – p. 2/10
Connected Rotating Objects
r1
r2
Rotational Energy – p. 3/10
Connected Rotating Objects
r1
r2
Rotational Energy – p. 3/10
Connected Rotating Objects
r1
r2
Rotational Energy – p. 3/10
Connected Rotating Objects
r1
r2
−→v
−→v
When connected by a non-slippingchain or belt, the two rotating objectsmust have the same linear velocity
Rotational Energy – p. 3/10
Connected Rotating Objects
r1
r2
−→v
−→v
When connected by a non-slippingchain or belt, the two rotating objectsmust have the same linear velocity
v1 = v2
Rotational Energy – p. 3/10
Connected Rotating Objects
r1
r2
−→v
−→v
When connected by a non-slippingchain or belt, the two rotating objectsmust have the same linear velocity
v1 = v2
ω1r1 = ω2r2
Rotational Energy – p. 3/10
Connected Rotating Objects
r1
r2
−→v
−→v
When connected by a non-slippingchain or belt, the two rotating objectsmust have the same linear velocity
v1 = v2
ω1r1 = ω2r2
Different Angular Velocities
Rotational Energy – p. 3/10
Linear Accelerations
Every point on a rotating object has two accelerationcomponents.
b
Rotational Energy – p. 4/10
Linear Accelerations
Every point on a rotating object has two accelerationcomponents.
b
b
Rotational Energy – p. 4/10
Linear Accelerations
Every point on a rotating object has two accelerationcomponents.
−→ω⊙
b
b
Rotational Energy – p. 4/10
Linear Accelerations
Every point on a rotating object has two accelerationcomponents.
−→ω⊙
b
−→r
b
Rotational Energy – p. 4/10
Linear Accelerations
Every point on a rotating object has two accelerationcomponents.
−→ω⊙
b
−→r
−→arad - changes in direction
b
Rotational Energy – p. 4/10
Linear Accelerations
Every point on a rotating object has two accelerationcomponents.
−→ω⊙
b
−→r −→
arad
−→arad - changes in direction
b
Rotational Energy – p. 4/10
Linear Accelerations
Every point on a rotating object has two accelerationcomponents.
−→ω⊙
b
−→r −→
arad
−→arad - changes in direction
arad = v2
r= ω2r
b
Rotational Energy – p. 4/10
Linear Accelerations
Every point on a rotating object has two accelerationcomponents.
−→ω⊙
b
−→r −→
arad
−→arad - changes in direction
arad = v2
r= ω2r
−→arad = −ω2−→
r
b
Rotational Energy – p. 4/10
Linear Accelerations
Every point on a rotating object has two accelerationcomponents.
−→ω⊙
b
−→r −→
arad
−→arad - changes in direction
−→atan - changes in speed
arad = v2
r= ω2r
−→arad = −ω2−→
r
b
Rotational Energy – p. 4/10
Linear Accelerations
Every point on a rotating object has two accelerationcomponents.
−→ω⊙
b
−→r −→
arad
−→atan
−→arad - changes in direction
−→atan - changes in speed
arad = v2
r= ω2r
−→arad = −ω2−→
r
b
Rotational Energy – p. 4/10
Linear Accelerations
Every point on a rotating object has two accelerationcomponents.
−→ω⊙
b
−→r −→
arad
−→atan
−→arad - changes in direction
−→atan - changes in speed
arad = v2
r= ω2r
−→arad = −ω2−→
r
atan = αr
b
Rotational Energy – p. 4/10
Linear Accelerations
Every point on a rotating object has two accelerationcomponents.
−→ω⊙
b
−→r −→
arad
−→atan
−→arad - changes in direction
−→atan - changes in speed
arad = v2
r= ω2r
−→arad = −ω2−→
r
atan = αr −→atan = −→
α ×−→r
b
Rotational Energy – p. 4/10
Linear Accelerations
Every point on a rotating object has two accelerationcomponents.
−→ω⊙
b
−→r −→
arad
−→atan
−→arad - changes in direction
−→atan - changes in speed
arad = v2
r= ω2r
−→arad = −ω2−→
r
atan = αr −→atan = −→
α ×−→r
−→a = −→
arad +−→atan
b
Rotational Energy – p. 4/10
Linear Accelerations
Every point on a rotating object has two accelerationcomponents.
−→ω⊙
b
−→r
−→atan
−→arad
−→arad - changes in direction
−→atan - changes in speed
arad = v2
r= ω2r
−→arad = −ω2−→
r
atan = αr −→atan = −→
α ×−→r
−→a = −→
arad +−→atan a =
√
a2rad
+ a2tan
Rotational Energy – p. 4/10
Non-Circular Objects
Putting the origin of the coordinate system at the axis ofrotation allows us to use all of the equations for circularobjects.
b
b
Rotational Energy – p. 5/10
Non-Circular Objects
Putting the origin of the coordinate system at the axis ofrotation allows us to use all of the equations for circularobjects.
b
b
Rotational Energy – p. 5/10
Non-Circular Objects
Putting the origin of the coordinate system at the axis ofrotation allows us to use all of the equations for circularobjects.
b
b
Rotational Energy – p. 5/10
Non-Circular Objects
Putting the origin of the coordinate system at the axis ofrotation allows us to use all of the equations for circularobjects.
b
b
Rotational Energy – p. 5/10
Non-Circular Objects
Putting the origin of the coordinate system at the axis ofrotation allows us to use all of the equations for circularobjects.
b
b
Rotational Energy – p. 5/10
Non-Circular Objects
Putting the origin of the coordinate system at the axis ofrotation allows us to use all of the equations for circularobjects.
b
b
Rotational Energy – p. 5/10
Non-Circular Objects
Putting the origin of the coordinate system at the axis ofrotation allows us to use all of the equations for circularobjects.
b
b
Rotational Energy – p. 5/10
Non-Circular Objects
Putting the origin of the coordinate system at the axis ofrotation allows us to use all of the equations for circularobjects.
b
b
Rotational Energy – p. 5/10
Non-Circular Objects
Putting the origin of the coordinate system at the axis ofrotation allows us to use all of the equations for circularobjects.
b
b
Rotational Energy – p. 5/10
Non-Circular Objects
Putting the origin of the coordinate system at the axis ofrotation allows us to use all of the equations for circularobjects.
b
b
Rotational Energy – p. 5/10
Non-Circular Objects
Putting the origin of the coordinate system at the axis ofrotation allows us to use all of the equations for circularobjects.
b
b
Rotational Energy – p. 5/10
Non-Circular Objects
Putting the origin of the coordinate system at the axis ofrotation allows us to use all of the equations for circularobjects.
b
b
Rotational Energy – p. 5/10
Non-Circular Objects
Putting the origin of the coordinate system at the axis ofrotation allows us to use all of the equations for circularobjects.
b
b
Rotational Energy – p. 5/10
Non-Circular Objects
Putting the origin of the coordinate system at the axis ofrotation allows us to use all of the equations for circularobjects.
br
b
Rotational Energy – p. 5/10
Non-Circular Objects
Putting the origin of the coordinate system at the axis ofrotation allows us to use all of the equations for circularobjects.
br
r = distance from axis of rotation
Rotational Energy – p. 5/10
Clicker Quiz
The following object is rotated about one end as shown.What is the linear speed of the point A?
b A
b
Rotational Energy – p. 6/10
Clicker Quiz
The following object is rotated about one end as shown.What is the linear speed of the point A?
d d/2b A
b
Rotational Energy – p. 6/10
Clicker Quiz
The following object is rotated about one end as shown.What is the linear speed of the point A?
d
d/2 b A
b
Rotational Energy – p. 6/10
Clicker Quiz
The following object is rotated about one end as shown.What is the linear speed of the point A?
dd/2
b A
b
Rotational Energy – p. 6/10
Clicker Quiz
The following object is rotated about one end as shown.What is the linear speed of the point A?
d
d/2
bA
b
Rotational Energy – p. 6/10
Clicker Quiz
The following object is rotated about one end as shown.What is the linear speed of the point A?
dd/2
bA
b
Rotational Energy – p. 6/10
Clicker Quiz
The following object is rotated about one end as shown.What is the linear speed of the point A?
dd/2bA
b
Rotational Energy – p. 6/10
Clicker Quiz
The following object is rotated about one end as shown.What is the linear speed of the point A?
d
d/2bA
b
Rotational Energy – p. 6/10
Clicker Quiz
The following object is rotated about one end as shown.What is the linear speed of the point A?
dd/2
bA
b
Rotational Energy – p. 6/10
Clicker Quiz
The following object is rotated about one end as shown.What is the linear speed of the point A?
d
d/2bA
b
Rotational Energy – p. 6/10
Clicker Quiz
The following object is rotated about one end as shown.What is the linear speed of the point A?
d d/2b A
b
Rotational Energy – p. 6/10
Clicker Quiz
The following object is rotated about one end as shown.What is the linear speed of the point A?
d d/2b A
(a) dω
b
Rotational Energy – p. 6/10
Clicker Quiz
The following object is rotated about one end as shown.What is the linear speed of the point A?
d d/2b A
(a) dω
(b) d
2ω
b
Rotational Energy – p. 6/10
Clicker Quiz
The following object is rotated about one end as shown.What is the linear speed of the point A?
d d/2b A
(a) dω
(b) d
2ω
(c) 3d
2ω
b
Rotational Energy – p. 6/10
Clicker Quiz
The following object is rotated about one end as shown.What is the linear speed of the point A?
d d/2b A
(a) dω
(b) d
2ω
(c) 3d
2ω
(d) 2dω
b
Rotational Energy – p. 6/10
Clicker Quiz
The following object is rotated about one end as shown.What is the linear speed of the point A?
d d/2b A
(a) dω
(b) d
2ω
(c) 3d
2ω
(d) 2dω
Rotational Energy – p. 6/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
ri
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
ri
vi = riω
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
ri
vi = riω All pieces have same ω
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
ri
vi = riω All pieces have same ω
Ki =1
2Mi (riω)
2 = 1
2Mir
2
iω2
Rotational Energy – p. 7/10
Rotational Kinetic Energy
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
Look at the i-th piece
It has kinetic energy, Ki =1
2Miv
2
i
vi = riω All pieces have same ω
ri
Ki =1
2Mi (riω)
2 = 1
2Mir
2
iω2
K ≈
∑
i
Ki =∑
i
1
2Mir
2
iω2
Rotational Energy – p. 7/10
Rotational Kinetic Energy II
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
ri
K ≈1
2
(
∑
i
Mir2
i
)
ω2
Rotational Energy – p. 8/10
Rotational Kinetic Energy II
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
ri
K ≈1
2
(
∑
i
Mir2
i
)
ω2
This expression becomes exactin the limit as the number of piecesapproaches infinity (and so the sizeof each piece approaches zero)
Rotational Energy – p. 8/10
Rotational Kinetic Energy II
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
ri
K ≈1
2
(
∑
i
Mir2
i
)
ω2
This expression becomes exactin the limit as the number of piecesapproaches infinity (and so the sizeof each piece approaches zero)
Rotational Energy – p. 8/10
Rotational Kinetic Energy II
Any rotating object has a kinetic energy due to its motion.
We have to imagine splitting the rotating object up intomany small pieces.
r
K ≈1
2
(
∑
i
Mir2
i
)
ω2
This expression becomes exactin the limit as the number of piecesapproaches infinity (and so the sizeof each piece approaches zero)
Rotational Energy – p. 8/10
Moment of Inertia
In the limit, the sum becomes the Moment of Inertia, I, forthe rotating object.
Rotational Energy – p. 9/10
Moment of Inertia
In the limit, the sum becomes the Moment of Inertia, I, forthe rotating object.
ri
I ≈
∑
i
Mir2
i
Rotational Energy – p. 9/10
Moment of Inertia
In the limit, the sum becomes the Moment of Inertia, I, forthe rotating object.
ri
I ≈
∑
i
Mir2
i
I = limi→∞
∑
i
Mir2
i
Rotational Energy – p. 9/10
Moment of Inertia
In the limit, the sum becomes the Moment of Inertia, I, forthe rotating object.
r
I ≈
∑
i
Mir2
i
I = limi→∞
∑
i
Mir2
i
I =∫
r2dM =∫
r2ρ dV
(ρ = density)
Rotational Energy – p. 9/10
Moment of Inertia
In the limit, the sum becomes the Moment of Inertia, I, forthe rotating object.
r
I ≈
∑
i
Mir2
i
I = limi→∞
∑
i
Mir2
i
I =∫
r2dM =∫
r2ρ dV
(ρ = density)
K = 1
2Iω2
Rotational Energy – p. 9/10
Moment of Inertia II
The moment of inertia, I, is the rotational counterpart tomass, i.e., it plays the same role in rotation as mass does inlinear motion.
Rotational Energy – p. 10/10
Moment of Inertia II
The moment of inertia, I, is the rotational counterpart tomass, i.e., it plays the same role in rotation as mass does inlinear motion.
The moment of inertia tells us how “hard" it is to make anobject rotate.
Rotational Energy – p. 10/10
Moment of Inertia II
The moment of inertia, I, is the rotational counterpart tomass, i.e., it plays the same role in rotation as mass does inlinear motion.
The moment of inertia tells us how “hard" it is to make anobject rotate.
The moment of inertia depends on:
Rotational Energy – p. 10/10
Moment of Inertia II
The moment of inertia, I, is the rotational counterpart tomass, i.e., it plays the same role in rotation as mass does inlinear motion.
The moment of inertia tells us how “hard" it is to make anobject rotate.
The moment of inertia depends on:
(a) The object’s shape.
Rotational Energy – p. 10/10
Moment of Inertia II
The moment of inertia, I, is the rotational counterpart tomass, i.e., it plays the same role in rotation as mass does inlinear motion.
The moment of inertia tells us how “hard" it is to make anobject rotate.
The moment of inertia depends on:
(a) The object’s shape.(b) The axis of rotation.
Rotational Energy – p. 10/10
Moment of Inertia II
The moment of inertia, I, is the rotational counterpart tomass, i.e., it plays the same role in rotation as mass does inlinear motion.
The moment of inertia tells us how “hard" it is to make anobject rotate.
The moment of inertia depends on:
(a) The object’s shape.(b) The axis of rotation.(c) The total mass of the object.
Rotational Energy – p. 10/10