Circular motion & relative velocity

Circular motion & relative velocity
•  Prelectures from
smartphysics are now
being counted.
•  Tutorials tomorrow –
pages 13-17 in red book.
•  CAPA due Friday at 10pm
Web page:
Clicker question 1
Set frequency to BA
A flare is dropped from an airplane flying at uniform velocity
(constant speed in a straight line). Neglecting air resistance, the
flare will
A: quickly lag behind the plane
B: remain vertically under the plane
C: move ahead of the plane
D: it depends how fast the plane is flying.
The horizontal velocity of the flare will remain constant,
and equal to what it began with, namely, the airplane's
forward velocity. It will fall and remain under the plane all
the time.
Centripetal acceleration
A particle goes around a circle of radius R
with constant speed so
are perpendicular to their
respective radials and have the same length
as do the two radials, we identify two similar
triangles so
and average acceleration is
Instantaneous acceleration:
Centripetal acceleration
In order to maintain constant speed, the
acceleration vector for uniform circular
motion must always be perpendicular to
the velocity vector, i.e. pointing to the
center of the circle.
If you're in car going around a curve, you feel as if you're
being thrown to the outside. Most people conclude since you
feel this way, the acceleration must be outward. You're
trying to go in a straight line, it's the car that's turning. Thus
the accel. on you is not throwing you out. What you feel is
the door or the seat belt on you, pulling you in. This
acceleration prevents you from flying out of the car in the 4
straight line you'd like to go in. It's called centripetal
Clicker question 2
Set frequency to BA
Q. An object is moving along a circular
path and is slowing down, as shown.
Which arrow best represents the
object’s acceleration vector at point X?
the acceleration vector
points in the same direction as
Nonuniform circular motion
What is the acceleration for circular motion with
varying speed (nonuniform circular motion)?
Can divide acceleration vector into two parts
Tangential acceleration is related to
change in speed and is parallel to the
velocity vector with magnitude
Radial acceleration is perpendicular to the velocity vector and
points to the center of the circle with magnitude
Since they are perpendicular:
Circular motion problem
A child is on the outside of a merry-go-round with a diameter of 3
m. Starting from rest, another kid spins the merry-go-round up to
60 revolutions per minute in 3 seconds at a constant linear
(tangential) acceleration. What is the acceleration 1 second into
the spin up? The final frequency is
The final speed is
Tangential acceleration during spin up:
Speed at 1 second:
The radial acceleration
at 1 second is
Clicker question 3
Set frequency to BA
Q. A race car travels around the track shown at constant
speed. Over which portion of the track is the magnitude of
the acceleration the smallest?
From 1 to 2
From 3 to 4
Different portion of track
Impossible to tell
Car at constant speed so no linear (tangential) acceleration
Centripetal acceleration is
and v is constant so
minimum acceleration occurs when R is largest. This is in
the straight sections where R is infinite and so a=0.
Relative velocity
Roger Clemens is on a moving walkway
which moves at 2 m/s and throws his
fastest pitch which he knows is 45 m/s.
What speed is measured by an observer
with a radar gun on the walkway? 45 m/s
How about an observer with a radar
gun off the end of the walkway? 47 m/s
This is the principle behind relative velocity.
It just comes down to vector addition
Relative velocity
Reference frame defines a coordinate system and a velocity
Define reference frame E as motionless relative to the Earth
Define reference frame W as motionless relative to the walkway
Velocity of Clemens’ pitch relative to the walkway is
Velocity of walkway reference frame with respect to the Earth
reference frame is
Velocity of Clemens’ pitch with respect to the Earth reference
frame is
Relative velocity
In vector form we have
A row boat in still water has a speed of vb
It heads directly east across a river of width
w which is flowing south at a speed of vr
If vb=1.0 m/s, vr=0.5 m/s, and w=50 m,
where and when will the boat land?
Easy solution:
so the trip takes 50 seconds and the boat lands 25 m
downstream of the point directly across from the starting point
Relative velocity
A row boat with speed of vb heads directly east
across a river of width d flowing south at vr. For
vb = 1.0 m/s, vr = 0.5 m/s, and w = 50 m, when
and where will the boat land?
Distance traveled is
Trip takes
Clicker question 4
Set frequency to BA
Q. To cross straight across a river (that is, end up at a place
directly across from the starting point), will the presence of a
current cause the trip to take a longer, shorter, or same
amount of time?
A.  Trip will take a longer time if there is a current
B.  Trip will take a shorter time if there is a current
C.  Trip will take the same amount of time
D.  Not enough information to solve the problem
Relative velocity
A row boat with speed of vb wants to reach a
point directly across a river with a current of vr.
For vb = 1.0 m/s, vr = 0.5 m/s, and river width of
w = 50 m, when will the boat land?
To go straight across we need a vertical velocity of zero so
and so the crossing time is
, longer than the no-current time of 50 s
New subject: Forces
•  Colloquially the idea of a force might be
the thing that makes other things move
–  Although this is not entirely correct it
summarizes the basic idea
•  Might also think of a force that surrounds and
permeates us and binds the galaxies together
–  This also has some relevance
Types of forces
Contact forces
Examples are pushing a block, pulling a rope, hitting a ball
as well as the force of friction
Long range forces
Examples are gravity and the force that causes a
magnet to attract some metal objects