Physics 207: Lecture 2 Notes

1
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 Newtons 1st and 2nd laws.

Assignment HW3, (Chapters 4 5, due 2/10,
Wednesday) Finish reading Chapter 6 Exam 1 Wed,
Feb. 17 from 715-845 PM Chapters 1-7
2
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

3
Uniform Circular Motion (UCM)

• Arc traversed s q r
• Tangential speed vt Ds/Dt or (in the
  limit) ds/dt r dq /dt
• Period T 2p r / vt
• Frequency f 1 / T
• Angular position q
• Angular velocity w dq /dt vt / r

s
vt
r
q
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)
4
Example Question

• 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?
• 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 angular acceleration be?
• 7 What is the magnitude and direction of the
  acceleration after 10 seconds?

5
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 )

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?
• w dq /dt Dq /Dt
• w 10 2p radians / 5 seconds
• 12.6 rad / s ( also 2 p f 2 p / T )

7
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?
• 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

8
Example

• A horizontal turntable 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 the period of the turntable during the
  initial rotation
• T (time for one revolution) Dt / of
  revolutions/ time 5 sec / 10 rev 0.5 s
• 2 What is initial angular velocity?
• w angular displacement / time 2 p f 2 p / T
  12.6 rad / s
• 3 What is the tangential speed of a point on the
  rim during this initial period?
• vt r dq /dt r w 1.0 m x 12.6 rad / s
  12.6 m/s

9
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

q qo w Dt
10
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?
• 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.

11
Sketch of q vs. time
q qo w Dt q 20p rad (4p/5) 5 rad q 24
rad
30p
q qo w Dt q 0 4p 5 rad
20p
q (radians)
10p
0
10
5
time (seconds)
12
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?
• 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?

13
Sketch of q vs. time
q qo w Dt q 20p rad (4p/5) 5 rad q 24
rad
30p
q qo w Dt q 0 4p 5 rad
20p
q (radians)
10p
0
10
5
time (seconds)
5 Avg. angular velocity Dq / Dt 24 p /10
rad/s
14
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.
• 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 ?

15
Key point ..

• Angular acceleration is associated with
  tangential acceleration.

16
What if w is linearly increasing

• 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 Dt2
• w wo Dt
• Many analogies to linear motion but it isnt
  one-to-one
• Remember Even if w is constant, there is always
  a radial acceleration.

17
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 ar
vt2/r Circular motion involves continuous
radial acceleration
18
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 Dt2
• (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?

19
Non-uniform Circular Motion
For an object moving along a curved trajectory,
with non-uniform speed a ar at (radial and
tangential)
at
ar
20
Tangential acceleration?

• 7 What is the magnitude and direction of the
  acceleration after 10 seconds?

• at 0.48 p m / s2
• and w r wo r r Dt 4.8 p m/s
  vt
• ar vt2 / r 23 p2 m/s2

Tangential acceleration is too small to plot!
21
Angular motion, sign convention

• If
• angular displacement
• velocity
• accelerations
• are counter clockwise then sign is positive.
• If clockwise then negative

22
What is the path?

• Circular motion demo.

23
What causes motion?(Actually changes in
motion)What are forces ?What kinds of forces
are there ?How are forces and changes in
motion related ?
24
Newtons 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.

25
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

26
Newtons Second Law

• The acceleration of an object is directly
  proportional to the net force acting upon it.
• The constant of proportionality is the mass.

• This expression is vector expression Fx, Fy, Fz
• Units
• The metric unit of force is kg m/s2 Newtons (N)
• The English unit of force is Pounds (lb)

27
Lecture 6
Assignment HW3, (Chapters 4 5, due 2/10,
Wednesday) Read rest of chapter 6