Download Physics 207: Lecture 2 Notes

Lecture 6
Goals:
 Discuss circular motion
Chapters 5 & 6
 Recognize different types of forces and know how they act
on an object in a particle representation

 Identify forces and draw a Free Body Diagram
 Begin to solve 1D and 2D problems with forces in
equilibrium and non-equilibrium (i.e., acceleration) using
Newton’s 1st and 2nd laws.
Assignment: HW3, (Chapters 4 & 5, due 2/10, Wednesday)
Finish reading Chapter 6
Exam 1 Wed, Feb. 17 from 7:15-8:45 PM Chapters 1-7
Physics 207: Lecture 6, Pg 1
Concept Check
Q1. You drop a ball from rest, how much of the acceleration
from gravity goes to changing its speed?
A. All of it
B. Most of it
C. Some of it
D. None of it
Q2. A hockey puck slide off the edge of the table, at the
instant it leaves the table, how much of the acceleration
from gravity goes to changing its speed?
A. All of it
B. Most of it
C. Some of it
D. None of it
Physics 207: Lecture 6, Pg 2
Uniform Circular Motion (UCM)
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Arc traversed
Tangential speed
Period
Frequency
Angular position
Angular velocity
s=qr
| vt | = Ds/Dt or (in the limit) ds/dt = r dq /dt
T = 2p r / | vt |
s
f=1/T
vt
q
r q
w = dq /dt = | vt | / r
Period (T): The time required to do one
full revolution, 360° or 2p radians
Frequency (f): 1/T, number of cycles per unit time
Angular velocity or speed w = 2pf = 2p/T, number of
radians traced out per unit time (in UCM average
and instantaneous will be the same)
Physics 207: Lecture 6, Pg 3
Example
A horizontally mounted disk 2 meters in diameter
spins at constant angular speed such that it first
undergoes
(1) 10 counter clockwise revolutions in 5 seconds
and then, again at constant angular speed,
(2) 2 counter clockwise revolutions in 5 seconds.
 1 What is T the period of the initial rotation?

T = time for 1 revolution = 5 sec / 10 rev = 0.5 s
also
T = 2p r / | vt |
( just like x = x0 + v Dt  Dt = (x- x0 ) / v )
Physics 207: Lecture 6, Pg 5
Example

A horizontally mounted disk 2 meters in diameter spins at
constant angular speed such that it first undergoes 10 counter
clockwise revolutions in 5 seconds and then, again at
constant angular speed, 2 counter clockwise revolutions in 5
seconds.

1 What is T the period of the initial rotation?

2 What is w the initial angular velocity?
w = dq /dt = Dq /Dt
w = 10 • 2π radians / 5 seconds
= 12.6 rad / s ( also 2 p f = 2 p / T )
Physics 207: Lecture 6, Pg 6
Example

A horizontally mounted disk 2 meters in diameter spins at
constant angular speed such that it first undergoes 10 counter
clockwise revolutions in 5 seconds and then, again at
constant angular speed, 2 counter clockwise revolutions in 5
seconds.

1 What is T the period of the initial rotation?

2 What is w the initial angular velocity?
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3 What is the tangential speed of a point on the rim
during this initial period?
| vt | = ds/dt = (r dq) /dt = r w
| vt | = r w = 1 m • 12.6 rad/ s = 12.6 m/s
Physics 207: Lecture 6, Pg 7
Angular displacement and velocity

Notice that if
w ≡ dq / dt and, if w is constant, then
integrating w = dq / dt, we obtain: q = qo + w Dt
( In one dimensional motion if
v = dx/dt = constant then x = x0 + v Dt )
Counter-clockwise is positive, clockwise is negative
s
q = qo + w Dt
vt
r
q
Physics 207: Lecture 6, Pg 9
Example
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A horizontally mounted disk 2 meters in diameter spins at
constant angular speed such that it first undergoes 10 counter
clockwise revolutions in 5 seconds and then, again at
constant angular speed, 2 counter clockwise revolutions in 5
seconds.
1 What is T the period of the initial rotation?
2 What is w the initial angular velocity?
3 What is the tangential speed of a point on the rim during this
initial period?
4 Sketch the q (angular displacement) versus time
plot.
Physics 207: Lecture 6, Pg 10
Sketch of q vs. time
q (radians)
30p
q = qo + w Dt
q = 0 + 4p 5 rad
q = qo + w Dt
q = 20p rad + (4p/5) 5 rad
q = 24 rad
20p
10p
0
5
10
time (seconds)
Physics 207: Lecture 6, Pg 11
Example
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A horizontally mounted disk 2 meters in diameter spins at
constant angular speed such that it first undergoes 10 counter
clockwise revolutions in 5 seconds and then, again at
constant angular speed, 2 counter clockwise revolutions in 5
seconds.
1 What is T the period of the initial rotation?
2 What is w the initial angular velocity?
3 What is the tangential speed of a point on the rim during this
initial period?
4 Sketch the q (angular displacement) versus time plot.
5 What is the average angular velocity
over the 1st 10 seconds?
Physics 207: Lecture 6, Pg 12
Sketch of q vs. time
q (radians)
30p
q = qo + w Dt
q = 0 + 4p 5 rad
q = qo + w Dt
q = 20p rad + (4p/5) 5 rad
q = 24 rad
20p
10p
0
5
10
time (seconds)
5 Avg. angular velocity = Dq / Dt = 24 p /10 rad/s
Physics 207: Lecture 6, Pg 13
Example
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A horizontally mounted disk 2 meters in diameter spins at
constant angular speed such that it first undergoes 10 counter
clockwise revolutions in 5 seconds and then, again at
constant angular speed, 2 counter clockwise revolutions in 5
seconds.
6 If now the turntable starts from rest and uniformly
accelerates throughout and reaches the same
angular displacement in the same time, what must
be the angular acceleration ?
Physics 207: Lecture 6, Pg 14
What if w is linearly increasing …
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Then angular velocity is no longer constant so dw/dt ≠ 0
Define tangential acceleration as at = dvt/dt = r dw/dt
So
s = s0 + (ds/dt)0 Dt + ½ at Dt2 and s = q r
We can relate at to dw/dt
q = qo + wo Dt + 1 at Dt2
2 r
w = wo + at Dt
r
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Many analogies to linear motion but it isn’t one-to-one
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Remember: Even if w is constant, there is always a radial
acceleration.
Physics 207: Lecture 6, Pg 16
Circular motion also has a radial (perpendicular) component
Uniform circular motion involves only changes in the
direction of the velocity vector, thus acceleration is
perpendicular to the trajectory at any point, acceleration
is only in the radial direction. Quantitatively (see text)
Centripetal Acceleration
vt
ar
r
ar = vt2/r
Circular motion involves
continuous radial acceleration
Physics 207: Lecture 6, Pg 17
Tangential acceleration?

6 If now the turntable starts from rest and uniformly
accelerates throughout and reaches the same angular
displacement in the same time, what must the “tangential
acceleration” be?
q = qo + wo Dt +
1 at
2 r
s
Dt2
vt
r
q
(from plot, after 10 seconds)
24 p rad = 0 rad + 0 rad/s Dt + ½ (at/r) Dt2
48 p rad 1m / 100 s2 = at
7
What is the magnitude and direction of the acceleration
after 10 seconds?
Physics 207: Lecture 6, Pg 18
Tangential acceleration?
7
What is the magnitude and direction of the acceleration
after 10 seconds?
s
vt
at = 0.48 p m / s2
and w r = wo r + r
at
r
r
q
Dt = 4.8 p m/s = vt
ar = vt2 / r = 23 p2 m/s2
Tangential acceleration is too small to plot!
Physics 207: Lecture 6, Pg 20
Angular motion, sign convention
 If
angular displacement
velocity
accelerations
are counter clockwise then sign is positive.
 If clockwise then negative
Physics 207: Lecture 6, Pg 21
What causes motion?
(Actually changes in motion)
What are forces ?
What kinds of forces are there ?
How are forces and changes in motion
related ?
Physics 207: Lecture 6, Pg 23
Newton’s First Law and IRFs
An object subject to no external forces moves with constant
velocity if viewed from an inertial reference frame (IRF).
If no net force acting on an object, there is no
acceleration.

The above statement can be used to define inertial
reference frames.
Physics 207: Lecture 6, Pg 24
IRFs
 An IRF is a reference frame that is not
accelerating (or rotating) with respect to the “fixed
stars”.
 If one IRF exists, infinitely many exist since they
are related by any arbitrary constant velocity vector!
 In many cases (i.e., Chapters 5, 6 & 7) the surface
of the Earth may be viewed as an IRF
Physics 207: Lecture 6, Pg 25
Newton’s Second Law
The acceleration of an object is directly proportional to the
net force acting upon it.
The constant of proportionality is the mass.
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This expression is vector expression: Fx, Fy, Fz
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Units
The metric unit of force is kg m/s2 = Newtons (N)
The English unit of force is Pounds (lb)
Physics 207: Lecture 6, Pg 26
Lecture 6
Assignment: HW3, (Chapters 4 & 5, due 2/10, Wednesday)
Read rest of chapter 6
Physics 207: Lecture 6, Pg 27