Tangential and Centripetal
Acceleration
TANGENTIAL SPEED
TANGENTIAL SPEED
vt= rω
tangential speed = distance from axis x angular speed
Note that ω is the instantaneous angular speed, rather
than the average angular speed, because the time
interval is so short. This equation is valid only when co
is measured in radians per unit of time. Other
measures of angular speed, such as degrees per second
and revolutions per second, must not be used in this
equation.
Tangential speed
PROBLEM
The radius of a CD in a computer is 0.0600 m. If a microbe riding on
the disc's rim has a tangential speed of 1.88 m/s, what is the disc's
angular speed?
SOLUTION
Given: r= 0.0600 m
Unknown: ω =?
vt = 1.88 m/s
Use the tangential speed equation on this page to solve for angular
speed.
vt= rω
Problems
1. A woman passes through a revolving door
with a tangential speed of 1.8 m/s. If she is
0.80 rn from the center of the door, what is
the door's angular speed?
2. An athlete spins in a circle before releasing a
discus with a tangential speed of 9.0 m/s.
What is the angular speed of the spinning
athlete? Assume the discus is 0.75 m from the
athlete's axis of rotation.
Practice/Assignment
4. A cylinder with a diameter of 0.457 m rotates
in a lathe at a constant angular speed of 35
rad/s. What is the tangential speed of the
surface of the cylinder?
5. A wheel with a radius of 2.0 m rotates at a
constant angular speed of 15.5 rad/s. What is
the tangential speed of a point 0.55 m from
the wheel's axis?
Assignment
2. A softball pitcher throws a ball with a tangential
speed of 6.93 m/s. If the pitcher's arm is 0.660 rn long,
what is the angular speed of the ball before the pitcher
releases it?
4. Fill in the unknown quantities in the following table:
vt
ω
r
a
?
121.5 rad/s
0.030 m
b
0.75 m/s
?
0.050 m
c
?
1.2 turns/s
3.8 m
1.5π rad/s
?
d
2.0π m/s
Tangential Acceleration
If the time interval t is very small, then the left
side of this relationship gives the tangential
acceleration of the point.
The angular speed divided by the time interval
on the right side is the angular acceleration.
TANGENTIAL ACCELERATION
at = rα
tangential acceleration = distance from axis x
angular acceleration
Again, the angular acceleration in this equation
refers to the instantaneous angular
acceleration. This equation must use the unit
radians to be valid. In SI, angular acceleration
is expressed as radians per second per
second.
Tangential acceleration
PROBLEM
A spinning ride at a carnival has an angular acceleration of
0.50 rad/s2. How far from the center is a rider who has a
tangential acceleration of 3.3 m/s2?
SOLUTION
Given: α = 0.50 rad/s2
Unknown: r=?
at=3.3 m/s
Use the tangential acceleration equation on this page.
Rearrange to solve for r.
at= rα
Practice problems
1. A dog on a merry-go-round undergoes a 1.5 m/s2
linear acceleration. If the merry-go-round's
angular acceleration is 1.0 rad/s2 , how far is the
dog from the axis of rotation?
2. A young boy swings a yo-yo horizontally above
his head at an angular acceleration of 0.35 rad/s2.
If tangential acceleration of the yo-yo at the end
of the string is 0.18 m/s2, how long is the string?
3. What is a tire's angular acceleration if the
tangential acceleration at a radius of 0.15 m is 9.4
X 10-2 m/s2?
Centripetal Acceleration
PROBLE
A test car moves at a constant speed around a
circular track. If the car is 48.2 m from the track's
center and has a centripetal acceleration of 8.05
m/s2, what is its tangential speed?
SOLUTION
Given: r= 48.2m
Unknown: vt
ac= 8.05 m/s2
Practice problems
1. A girl sits on a tire that is attached to an overhanging
tree limb by a rope. The girl's father pushes her so that
her centripetal acceleration is 3.0 m/s . If the length of
the rope is 2.1 m, what is the girl's tangential speed?
2. A dog sits 1.5 m from the center of a merry-go-round.
If the dog undergoes a 1.5 m/s centripetal
acceleration, what is the dog's linear speed? What is
the angular speed of the merry-go-round?
3. A piece of clay sits 0.20 m from the center of a potter's
wheel. If the potter spins the wheel at an angular
speed of 20.5 rad/s, what is the magnitude of the
centripetal acceleration of the piece of clay on the
wheel?