Download Section 2.3 – Linear and Angular Velocities and more Lecture notes Physics in PDF only on Docsity!
Section 2.3 – Linear and Angular Velocities
The most intuitive measure of the rate at which the rider is traveling around the wheel is what we call linear velocity.
Another way to specify how fast the rider is traveling around the wheel is with what we call angular velocity.
Linear Speed
Definition
If P is a point on a circle of radius r , and P moves a distance s on
the circumference of the circle in an amount of time t , then the
linear velocity , v , of P is given by the formula
speed distance time
v s
t
Example
A point on a circle travels 5 cm in 2 sec. Find the linear velocity of the point.
Solution
Given: s = 5 cm
t = 2 sec
5 2
cm sec v s (^) t 2.5 cm / sec
Angular Speed
Definition
If P is a point moving with uniform circular motion on a circle of
radius r , and the line from the center of the circle through P sweeps
out a central angle in an amount of time t , then the angular velocity ,
(omega), of P is given by the formula
^ t where is measured in radians
Example
A point on a circle rotates through^3 4 radians in 3 sec. Give the angular velocity of the point.
Solution
Given: = 34 rad
t = 3 sec
3 4 3
rad sec
^ 4 rad / sec
Example
A bicycle wheel with a radius of 13.0 in. turns with an angular velocity of 3 radians per seconds. Find the distance traveled by a point on the bicycle tire in 1 minute.
Solution
Given: r = 13.0 in.
= 3rad/ sec
t = 1 min = 60 sec.
t ^
t s r sr
t s r
s tr
2,340 inches 2, or (^) 12 195 ft
Example
Suppose that P is on a circle with radius 10 cm, and ray O P is rotating with angular speed s
rad / ec.
a) Find the angle generated by P in 6 seconds
b) Find the distance traveled by P along the circle in 6 seconds.
c) Find the linear speed of P in cm per sec.
Solution
a) t
18 .6^3 rad
^ ^
b) s r
s 10 3 ^103 cm
c) v st
10 3 6 v
^
(^5) / c 9 ^ cm se
Example
A belt runs a pulley of radius 6 cm at 8 0 rev / min. a) Find the angular speed of the pulley in radians per sec. b) Find the linear speed of the belt in cm per sec.
Solution
a) 80 min rev^ 1min 6 0sec 12 rev^
(^8) / c 3
^ rad se
b) v r
^
50 cm / s ce
Example
The diameter of the Ferris wheel is 250 ft , the distance from the ground to the bottom of the wheel is 14 ft , and one complete revolution takes 20 minutes, find
a. The linear velocity, in miles per hour, of a person riding on the wheel.
b. The height of the rider in terms of the time t , where t is measured in minutes.
Solution
Given : = 1 rev = 2 rad
t = 20 min.
250 2 2 125 r D ft
t a. ^ r or v t ^
2 20
^
10 rad / min
v r
(125 ft ) 10 rad / min 39.27 ft / min
min
60min 1 39.27 (^1) 5, ft (^) mile v hr ft
0.45 mi / hr
1
b. cos OPOP
125
^ OP
OP 125cos H PP 0 14
OP 0 OP 14
125 125cos 14 139 125cos
t ^ t
10 t
^
H 139 125cos 10 t ^
H
P 0
P 1
14 ft
O
P
7. A Ferris wheel has a radius 50.0 ft. A person takes a seat and then the wheel turns 23 rad.
a) How far is the person above the ground?
b) If it takes 30 sec for the wheel to turn 23 rad , what is the angular speed of the wheel?
8. A fire truck parked on the shoulder of a freeway next to a long block wall. The red light on the top of the truck is 10 feet from the wall and rotates through a complete revolution every 2 seconds. Find the equations that give the lengths d and in terms of time.
9. Suppose that point P is on a circle with radius 60 cm, and ray OP is rotating with angular speed 12
radian per sec.
a) Find the angle generated by P in 8 sec.
b) Find the distance traveled by P along the circle in 8 sec.
c) Find the linear speed of P in 8 sec.
10. Tires of a bicycle have radius 13 in. and are turning at the rate of 215 revolutions per min. How fast is the bicycle traveling in miles per hour? ( Hint : 1 mi = 5280 ft.)
11. Earth travels about the sun in an orbit that is almost circular. Assume that the orbit is a circle with radius 93,000,000 mi. Its angular and linear speeds are used in designing solar-power facilities. a) Assume that a year is 365 days, and find the angle formed by Earth’s movement in one day. b) Give the angular speed in radians per hour. c) Find the linear speed of Earth in miles per hour. 12. Earth revolves on its axis once every 24 hr. Assuming that earth’s radius is 6400 km, find the following. a) Angular speed of Earth in radians per day and radians per hour. b) Linear speed at the North Pole or South Pole c) Linear speed ar a city on the equator 13. The pulley has a radius of 12.96 cm. Suppose it takes 18 sec for 56 cm of belt to go around the pulley. a) Find the linear speed of the belt in cm per sec. b) Find the angular speed of the pulley in rad per sec. 14. The two pulleys have radii of 15 cm and 8 cm, respectively. The larger pulley rotates 25 times in 36 sec. Find the angular speed of each pulley in rad per sec. 15. A thread is being pulled off a spool at the rate of 59.4 cm per sec. Find the radius of the spool if it makes 152 revolutions per min.
Solution Section 2.3 – Linear and Angular Velocities
Exercise
Find the linear velocity of a point moving with uniform circular motion, if s = 12 cm and t = 2 sec.
Solution
v s t
2 sec ^ cm
6 cm / es c
Exercise
Find the distance s covered by a point moving with linear velocity v = 55 mi/hr and t = 0.5 hr.
Solution
s vt 55 mihr 0.5 hr 27.5 miles
Exercise
Point P sweeps out central angle = 12 as it rotates on a circle of radius r with t = 5 sec. Find the
angular velocity of point P.
Solution
t
^
sec
^ rad
2.4 rad / sec
Exercise
Find an equation that expresses l in terms of time t. Find l when t is 0.5 sec, 1.0 sec, and 1.5 sec. (assume
the light goes through one rotation every 4 seconds.)
Solution
/sec 4 2
t sec^ rad
^ rad ^
/sec 2 rad t
^
t 2
^
l t^100 2 cos
cos
l ^ t
l t t
100 sec 2 cos 2
100
For t = 0.5 sec
^
100 100 100 cos 1 cos^1 2 2 (^4 )
l 100 2 141 ft
For t = 1.0 sec
100 100 cos^0 2
l (^) Undefined
For t = 1.5 sec
^
100 100 100 cos 3 cos^31 2 2 (^4 )
l (^) 100 2 141 ft
Exercise
Find the angular velocity, in radians per minute, associated with given 7.2 rpm.
Solution
- (^2) min 2 14. 4 45. (^2) min rad rev rev^ radians
Exercise
A fire truck parked on the shoulder of a freeway next to a long block wall. The red light on the top of the truck is 10 feet from the wall and rotates through a complete revolution every 2 seconds. Find the equations that give the lengths d and in terms of time.
Solution
t
^
2 2
^
rad / sec
tan 10^ d
d 10tan 10tan t
sec 10^ l
l 10sec 10sec t
Exercise
Suppose that point P is on a circle with radius 60 cm, and ray OP is rotating with angular speed 12
radian per sec.
a) Find the angle generated by P in 8 sec.
b) Find the distance traveled by P along the circle in 8 sec.
c) Find the linear speed of P in 8 sec.
Solution
a) t
2
2 1 .8^3 rad
^ ^
b) s r s (^60) 23 40 cm
c) v s t
v 408 5 cm / sec
Exercise
A Ferris wheel has a radius 50.0 ft. A person takes a seat and then the wheel turns 23 rad.
a) How far is the person above the ground?
b) If it takes 30 sec for the wheel to turn 23 rad , what is the angular speed of the wheel?
Solution
a) 23 2 6
cos h^1 r
h 1 (^) r cos
h 1 (^) 50cos 6 43.3 ft Person is 50 43.3 93.3 ft above the ground
b) t ^
2 3 30
rad sec
/ 45 ^ rad sec
Exercise
Tires of a bicycle have radius 13 in. and are turning at the rate of 215 revolutions per min. How fast is the bicycle traveling in miles per hour? ( Hint : 1 mi = 5280 ft.)
Solution
215 rev^21 revrad^ 430 rad / min
v r 13 430 (^) 5590 in / min
60min 1 1 (^5590) min 1 12 5280 in ft mi
v hr in ft
16.6 mph
Exercise
The pulley has a radius of 12.96 cm. Suppose it takes 18 sec for 56 cm of belt to go around the pulley. a) Find the linear speed of the belt in cm per sec. b) Find the angular speed of the pulley in rad per sec.
Solution
Given : s = 56 cm in t = 18 sec
r = 12.96 cm
a)^56 18 v s 3.1 cm / sec t
b)^ ^ vr 12.963.1^ .24 rad / sec
Exercise
The two pulleys have radii of 15 cm and 8 cm, respectively. The larger pulley rotates 25 times in 36 sec. Find the angular speed of each pulley in rad per sec.
Solution
Given :^25 / sec 36
times
r 1 (^) 15 cm r 2 8 cm
The angular velocity of the larger pulley is:
^2536^ ti sec^ mes^^21 timerad ^2518 r da / sec
The linear velocity of the larger pulley is:
^ / s ce
v r 6 cm
^
The angular velocity of the smaller pulley is: v (^1) v
r r
^
(^125) / sec 48 ^ rad
Exercise
A thread is being pulled off a spool at the rate of 59.4 cm per sec. Find the radius of the spool if it makes 152 revolutions per min.
Solution
Given : 152 rev / min v 59.4 cm / sec
r v^1 v
sec min
cm rev
^ ^
min sec min sec
rev cm
rev rad
3.7 cm
Exercise
A railroad track is laid along the arc of a circle of radius 1800 ft. The circular part of the track subtends a central angle of 40. How long (in seconds) will it take a point on the front of a train traveling 30 mph to go around this portion of the track?
Solution
Given : r = 1800 ft.
180
^ rad
v 30 mph
The arc length: 1800 2 400
s r (^) 9 ft
v s^ t s (^) t v
400 30
f mi hr
t ^ ^ t
1 3600sec 5280 1
mi ft
hr f tm i hr
^
29 sec
