Transcript Chapter 5 and 8 PowerPoint, Mr. Mardit

Chapter 5: Circular Motion;
Gravitation
Chapter 8: Rotational Motion
5-1: Kinematics of Circular Motion
• Uniform circular motion: movement in a circle at constant speed
1- Is the object accelerating? YES/NO ? WHY?
HINT: 𝑎 =
∆𝑣
∆𝑡
2- Direction of 𝑎 ?
HINT: What is the direction of ∆𝑣? (What is the direction of 𝑣2 and 𝑣1 ?)
Remember vectors add tip to tail!
𝑣2
3- 𝑎𝑐𝑒𝑛𝑡𝑟𝑖𝑝𝑒𝑡𝑎𝑙 =
; proof see text page 107
𝑟
5-1: Kinematics of Circular Motion
• 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = 𝑓 =
# 𝑜𝑓 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠/𝑐𝑦𝑐𝑙𝑒𝑠
𝑠𝑒𝑐𝑜𝑛𝑑
• 𝑝𝑒𝑟𝑖𝑜𝑑 = 𝑇 = time required for one revolution/cycle
•𝑇 =
1
𝑓
• Examples: Earth, moon, heart rate, waves breaking on the shore,
….Can you name some more….?
5-2: Applying N2L to Uniform Circular Motion
• Remember Newton’s 2nd Law:
•
𝐹𝑜𝑟𝑐𝑒𝑠 = 𝐹𝑛𝑒𝑡 = 𝑚𝑎
• Let’s apply N2L to an object moving in a circle at constant speed?
• 𝐹𝑜𝑟𝑐𝑒𝑠 = 𝐹𝑛𝑒𝑡 = 𝑚𝑎𝑐𝑒𝑛𝑡𝑟𝑖𝑝𝑒𝑡𝑎𝑙
• 1-What is the direction of 𝐹𝑛𝑒𝑡 ?
• 2-Brainstorm some possible sources of 𝐹𝑛𝑒𝑡 :
5-6: Newton’s Law of Universal Gravitation
•𝐹𝑔𝑟𝑎𝑣𝑖𝑡𝑦 =
𝐺𝑚1 𝑚2
𝑟2
• The above gives only the magnitude of the force of gravity NOT the direction, gravity is always
an attractive force that acts along the line joining 𝑚1 and 𝑚2
• G=6.67 x 10-11 units??? What are the units of G?
5-7: Applying N2L to Gravity; g
• Remember Newton’s 2nd Law:
•
𝐹𝑜𝑟𝑐𝑒𝑠 = 𝐹𝑛𝑒𝑡 = 𝑚𝑎
• Let’s apply N2L to two masses 𝑚1 𝑎𝑛𝑑 𝑚2
•
𝐹𝑜𝑟𝑐𝑒𝑠 = 𝐹𝑛𝑒𝑡 = 𝐹𝑔𝑟𝑎𝑣𝑖𝑡𝑦 =
𝐺𝑚1 𝑚2
=𝑚2 𝑎
𝑟2
• Now let’s say 𝑚1 is the Earth and 𝑚2 is a person standing on the surface of the Earth:
𝐺𝑚𝐸𝑎𝑟𝑡ℎ 𝑚𝑝𝑒𝑟𝑠𝑜𝑛
𝑚𝑝𝑒𝑟𝑠𝑜𝑛 𝑎=
; 𝑚2 cancels; substitute
2
𝑟
𝐺𝑚𝐸𝑎𝑟𝑡ℎ
=
𝑟2
• 𝑎=
2
Nm
6.67×10−11 kg2
×(5.98×1024 𝑘𝑔)
(6.371×106 𝑚𝑒𝑡𝑒𝑟𝑠)
≅ 9.8 𝑚
𝑠2
≡𝑔
Polar Plot: (r,θ)
Good news:
We will only be studying rotational motion of CONSTANT radius.
Why do we care?
Imagine you are walking in a circle, riding a bicycle, twisting a wrench,
torqueing a screw driver, rolling a ball, or doing a cart wheel. All of these
consist of motion in both the x and y– which must be resolved into two—one
dimensional problems before they can be analyzed.
However, using polar coordinates—when the radius is held constant—
only one variable changes: θ
YESSSSS! Much Simpler!
2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 360 𝑑𝑒𝑔𝑟𝑒𝑒𝑠
Example:
The hockey puck spins 185 times between defending zone and
the attacking zone. How many radians? How many degrees?
For every linear (straight line) physical quantity, there is a rotational analogue:
To review the linear physical quantities to date:
x (distance), v, a, m, F, KE, p,
Position  Velocity  Acceleration
ω = angular velocity =
θ2 −θ1
𝑡2 −𝑡1
compare with linear velocity=
α = angular acceleration =
compare with
=
∆θ
∆𝑡
𝑥2 −𝑥1
𝑡2 −𝑡1
ω2 −ω1
𝑡2 −𝑡1
≡ ω (omega)
=
∆𝑥
∆𝑡
=
≡𝑣
∆ω
∆𝑡
≡ α (𝑎𝑙𝑝ℎ𝑎)
𝑥2 −𝑥1
linear acceleration=
𝑡2 −𝑡1
=
∆𝑥
∆𝑡
≡𝑣
Examples: θ, ω, α
1- A small spider walks an arc length of constant radius
around a sun dial from 1 o’clock to 8 o’clock. What is the ∆θ of the
ant? ( in “hours”, in degrees, in radians) The journey takes 2.6
minutes what is the ω of the spider? (radians/second)
2- Timmy hops on his bicycle and starts pedaling. He
completes 4.2 revolutions of the rear tire in the first 3 seconds.
What is the ω𝑖𝑛𝑖𝑡𝑖𝑎𝑙 of the tire? One minute later the rear tire
completes 15.6 revolutions in 0.6 seconds. What is the ω𝑓𝑖𝑛𝑎𝑙 of the
tire? What is the angular acceleration α of the tire?
Remember this equation from middle school:
Circumference of a circle = 2PI *r
…well that was just a special case of the following:
Manipulating l=rθ to connect linear and angular quantities; a little bit of calculus (PRODUCT RULE)
𝑑
𝑓 𝑡 ×𝑔 𝑡
𝑑𝑡
𝑑𝑓
𝑑𝑔
=
× 𝑔 𝑡 + 𝑓(𝑡) ×
𝑑𝑡
𝑑𝑡
l=r θ
𝑑
𝑑
𝑙 =
𝑟θ
𝑑𝑡
𝑑𝑡
𝑑𝑙
𝑑𝑟
𝑑θ
= •θ+𝑟•
𝑑𝑡
𝑑𝑙
𝑑𝑡
𝑑𝑡
𝑑θ
= 𝑟 • 𝑑𝑡
v=rω
𝑑𝑡
v=rω
Example: A cyclist with a 24 inch diameter tire
glances down at his speedometer and notices his
speed is 28 miles/hour . What is the angular
velocity of the wheel in radians/second?
Applying the PRODUCT RULE a second time
𝑑
𝑓 𝑡 ×𝑔 𝑡
𝑑𝑡
𝑑𝑓
𝑑𝑔
=
× 𝑔 𝑡 + 𝑓(𝑡) ×
𝑑𝑡
𝑑𝑡
v=r ω
𝑑
𝑑
𝑣 =
rω
𝑑𝑡
𝑑𝑡
𝑑𝑣
𝑑𝑟
𝑑ω
= •ω+𝑟•
𝑑𝑡
𝑑𝑣
𝑑𝑡
𝑑𝑡
= 𝑟•
a=rα
𝑑ω
𝑑𝑡
𝑑𝑡
atangetial = r α
Example: At the Indianapolis 6000 a x-wing
fighter banks a turn around the 4th moon of
Xarnex at 18000 m/𝑠 2 . The radius of curvature is
200 meters. What is the angular acceleration?
• 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 ω = 2π ∗ 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = 2π𝑓 =
• 𝑝𝑒𝑟𝑖𝑜𝑑 = 𝑇 = time required for one revolution
•𝑇 =
1
𝑓
# 𝑜𝑓 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠
2π
𝑠𝑒𝑐𝑜𝑛𝑑
1 Dimensional kinematic equation also in cases of constant angular acceleration with appropriate change in variable
𝑣𝑓 = 𝑣0 + 𝑎𝑡
→
ω𝑓 = ω0 + α𝑡
Δ𝑥 = 𝑣0 𝑡 + 1 2 𝑎𝑡 2
→
Δθ = ω0 𝑡 + 1 2 α𝑡 2
𝑣𝑓2 = 𝑣02 + 2𝑎Δ𝑥
→
ω𝑓 = ω0 + α𝑡
𝑣=
𝑣𝑓 + 𝑣0
2
→
ω=
ω𝑓 + ω0
2
Example: A cyclist with a 24 inch diameter tire glances down at
his speedometer and notices his speed is 28 miles/hour . What
is the angular velocity of the wheel in radians/second? He
comes to a stop in 6 seconds. How many revolutions does his
wheel turn during this time?
Torque: the rotational analogue of force
suppose we have a rigid, massive body:
 if a force F is applied a distance
r away from a fixed axis of rotation,
then the rigid body will experience a torque
τ
when a force is directed in a straight line the result is a push or a pull; a linear force;
however when that force causes a rotation it is called a torque
It is only when the force is applied perpendicular to the lever arm r, that a torque is generated.
What if the force is at an angle?
Torque Example: Susan is changing her tire and applies a 20
Newton force perpendicular to the end of a wrench that is 20
centimeters long. What is the magnitude of the torque
generated?
𝜏 = 𝑟 × 𝐹 = 𝑟 𝐹 sin 𝜃
Torque examples everyday life:
Center of Mass (cm) or Center of Gravity: average of the masses factored by their distances from a reference
point
If you take out your cell phone and balance it on the tip of your finger why does it balance and not
fall over?
Example:
𝑥𝑐𝑚
𝑚1 𝑥1 + 𝑚2 𝑥2 + 𝑚3 𝑥3 +
=
=
𝑚1 + 𝑚2 + 𝑚3 +
𝑚𝑖 𝑥𝑖
𝑚𝑖 𝑥𝑖
=
𝑚𝑖
𝑀𝑇𝑜𝑡𝑎𝑙
𝑦𝑐𝑚
𝑚1 𝑦1 + 𝑚2 𝑦2 + 𝑚3 𝑦3 +
=
=
𝑚1 + 𝑚2 + 𝑚3 +
𝑚𝑖 𝑦𝑖
𝑚𝑖 𝑦𝑖
=
𝑚𝑖
𝑀𝑇𝑜𝑡𝑎𝑙
Continuous Mass Distribution
Moment of Inertia: the rotational analogue of mass
When a force is applied to a mass that then moves in a straight line, we consider as if all of the mass were located
at the center of mass; the balance point of mass distribution and only consider the total mass.
However, when we consider torques we need to know the position of where the mass is located. It takes more
force to rotate a mass around an axis of symmetry when the mass is located 100 meters away then when it is 4
meters away.
Moment of intertia=
2
𝑚𝑖 𝑟𝑖
Example: Determine the moment of inertia of a system of 4 point masses located at
coordinates (1,2), (-2,3), (-3,4), and (-5,-7) each of mass 2 kilograms when the axis of
rotation is about the z-axis.
Moment of intertia=
𝑚𝑖 𝑟𝑖2
Continuous mass distributions
Newtons’ 2nd Law:
𝐹𝑜𝑟𝑐𝑒𝑠 = 𝐹𝑛𝑒𝑡 = 𝑚𝑎 →
𝑇𝑜𝑟𝑞𝑢𝑒𝑠 = τ𝑛𝑒𝑡 = 𝐼α
Example: If each of the three rotor blades is 8cm long and has a mass of 95 gram, calculate the amount of
torque that the motor must supply to bring the blades up to a speed of 5 rev/sec in 8.0 seconds.
Linear Kinetic Energy  Rotational Kinetic Energy
1
2
m𝑣
2
→
1
2
Iω
2
Example: A pitcher throws a fastball with all translational kinetic energy (ZERO spin)
at 44 meters/second. If the pitcher throws the same baseball with a curveball pitch
with the same total kinetic energy at 33 meters/second determine how many
rotations/second the baseball spins at. NOTE: mass_baseball=143 grams;
diameter_baseball=75 millimeters;
Angular Momentum
Linear Momentum: p=mv  Angular Momentum: L=Iω
What are the units of angular momentum?
Example: Determine the angular momentum of the baseball in the previous example.
Newton’s 2nd Law revisited:
We initially saw N2L like this:
𝑑
𝑓 𝑡 ×𝑔 𝑡
𝑑𝑡
𝑑𝑓
𝑑𝑔
=
× 𝑔 𝑡 + 𝑓(𝑡) ×
𝑑𝑡
𝑑𝑡
𝐹 = 𝑚𝑎
∆𝑝
𝑑𝑝 𝑑
𝑑
𝑑𝑚
𝑑𝑣
𝑑𝑣
however, Issac originally wrote it like: 𝐹 = = = (𝑝)= (𝑚𝑣)= 𝑣 + 𝑚 =𝑚 =𝑚𝑎
∆𝑡
𝑑𝑡 𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡
SO…use this and our new definition of angular momentum to derive N2L for rotational
motion:
𝜏=
∆𝐿
∆𝑡
=
𝑑𝐿 𝑑
𝑑
𝑑𝐼
= (𝐿)= (𝐼𝜔)= 𝜔
𝑑𝑡 𝑑𝑡
𝑑𝑡
𝑑𝑡
+
𝑑𝜔
𝐼
𝑑𝑡
=
𝑑𝜔
𝐼 =𝐼
𝑑𝑡
𝛼
Conservation of angular momentum: L=I𝜔 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Explain:
Comparison chart:
Translation
Rotation
Connection
x
𝜃
x=rθ
v
𝜔
v=rω
a
𝛼
a=rα
m
𝐼
I=Σmr2
F
𝜏
Τ=rFsinθ
KE=½mv2
KE=½Iω2
p=mv
L=Iω
W=Fd
W=τα
ΣF=ma
Στ=Iα
ΣF=∆𝑝/∆𝑡
Στ=∆𝐿/∆𝑡
Add example problems after each slide where a new concept is introduced that might be unclear
Comparison chart at the end; with linear and rotational values