Angular Velocity
1. The propellers on an average freighter have a radius of 4 feet. At full speed ahead, the
propellers turn at 150 revolutions per minute.
a. What is the angular velocity in radians per minute at the tip of the blades? at the
center of the propeller?
b. What is the linear velocity in feet per minute at the tip of the blades? at the center of
the propeller?
2. Dan Druff and Ella Funt are riding on a Ferris wheel. Dan observes that it takes 20 seconds to make a complete
revolution. Their seat is 25 feet from the axle of the wheel.
a. What is their angular velocity in revolutions per minute? Degrees per minute? Radians per minute?
b. What is their linear velocity?
3. David puts a rock in his sling and starts whirling it around. He realizes that in order for the rock to reach Goliath , it
must leave the sling at a speed of 60 feet per second. So he swings the sling in a circular path of radius 4 feet. What
must the angular velocity be in order for David to achieve his objective?
4. In order for a lawn mower blade to cut grass, it must strike the grass with a
speed of at least 900 inches per second.
a. If the innermost part of the cutting edge is 6 inches from the 19" center of
the blade, how many radians per second must the blade turn? How many
revolutions per minute is this?
b. The blade has a diameter of 19 inches. If the outermost tip of the blade
hits a rock while turning as in part (a), how fast could the rock be hurled
from the mower?
5. Yank Hardy pulls the cord on his power mower. In order for the engine to start, the
pulley must turn at 180 revolutions per minute. The pulley has a radius of 0.2 feet.
a. How many radians per second must the pulley turn?
b. How fast must Yank pull the cord to start the mower?
c. When Yank pulls this hard, what is the angular velocity of the center of the pulley?
6. A train wheel has a diameter of 30 inches to the rim, which rests on the
track. The flange, which keeps the wheel from slipping off the track,
projects 1 inch beyond the rim. When the train is traveling 60 mph,
what is the linear velocity of a point on the outer edge of the flange?
7. A small pulley 6 centimeters in diameter is connected by a belt to a larger pulley I
5 centimeters in diameter. The small pulley is turning at 120 rpm.
a. Find the angular velocity of the small pulley in radians per second.
b. Find the linear velocity of the rim of the small pulley.
c. What is the linear velocity of the rim of the large pulley? Explain.
d. Find the angular velocity of the large pulley in radians/ second.
e. How many rpm is the large pulley turning?
8
A large pulley 20 cm in diameter drives a small pulley 6 cm in diameter by a pulley
belt that goes over the rim of each. The large pulley has an angular velocity of 150
radians per minute.
a. What is the linear velocity of the large pulley's rim?
b. What is the linear velocity of the small pulley's rim?
c. What is the angular velocity of the small pulley?
9. A small gear of radius 5 cm is turning with an angular velocity of 20 radians per second. It
drives a large gear of radius 15 cm.
a. What is the linear velocity of the teeth on the large gear?
b. What is the angular velocity of the teeth on the large gear?
c. What is the angular velocity of a point at the center of the large gear?
10. A large gear of diameter 30 centimeters is revolving at 45 rpm. It drives a small gear of diameter 8 centimeters.
a. How many radians per minute is the large gear turning?
b. What is the linear velocity of the teeth on the large gear?
c. What is the linear velocity of the teeth on the small gear?
d. How many radians per minute is the small gear turning?
e. How many revolutions per minute is the small gear turning?
11. A cockroach is sitting 4 cm from the center of a lazy Susan. Unaware of its presence, Phoebe Small spins the lazy
Susan through an angle of 120 degrees.
a. Through how many radians did the roach turn?
b. What distance did it travel?
c. If Phoebe turned the lazy Susan 120 degrees in 1/2 second, what was the roach's angular velocity?
d. What was its linear velocity?
12. Stan Dupp and his brother Ben play on a see saw. Stan sits at a point 8 feet from the pivot. Ben, being heavier, sits
just 5 feet from the pivot. As Ben goes down and Stan goes up, the see saw rotates through an angle of 37 degrees in
0.7 seconds.
a. What are Ben's angular velocity in radians per second, and his linear velocity in feet per second?
b. What are Stan's angular and linear velocities?
13. Della Casee is riding a racing bike at a speed of 50.4 kilometers per hour. The wheels have a diameter of 70
centimeters. Find the angular velocity of the wheels in radians per second.
14. The rear wheels of a tractor are 4 feet in diameter, and turn at 20 rpm.
a. How fast is the tractor going (feet per second)?
b. The front wheels have a diameter of only 1.8 feet. What is the linear velocity of a
point on their tire treads?
c. What is the angular velocity of the front wheels in rpm?
15. Three gears are shown.
a. Gear 1 turns at 200 rpm. What is its angular velocity in radians per second?
b. What is the linear velocity of the teeth on gear I , 13 millimeters from its axle?
c. Gear 1 's teeth mesh with gear 2's teeth. Gear 2 has a radius of 3 millimeters. What is
the linear velocity of gear 2's teeth?
d. What is gear 2's angular velocity in radians per second?
e. Gear 2 and gear 3 are connected to the same axle. What is the linear velocity of gear
3's teeth? (Its radius is 10 millimeters.)
16. A waterwheel of diameter 12 feet turns at 0.3 radian per second .
a. What is the linear velocity of the rim?
b. The wheel is connected by an axle to a grindstone of diameter 3 feet. What is the
angular velocity of a point on the rim Grindstone of the grindstone?
17. Old-fashioned trucks used a chain to transmit power from the engine to the
wheels. Suppose that the drive sprocket had a diameter of 6 inches and the wheel
sprocket had a diameter of 20 inches. If the drive sprocket goes 300 rpm, find
a. The angular velocity of the drive sprocket in radians per minute.
b. The linear velocity of the 20-inch wheel sprocket in inches per minute.
c. The angular velocity of the wheel in radians per minute.
d. The speed of the truck to the nearest mile per hour.
18. A car's wheel turns at 200 revolutions per minute. The radius of each wheel is 1.3 feet.
a. To the nearest radian per minute, what is the angular velocity of a point
i. on the tire tread?
ii. on the hubcap, 0.4 feet from the center?
iii. right at the center?
b. To the nearest foot per minute, what is the linear velocity of a point
i. on the tire tread?
ii. on the hubcap, 0.4 feet from the center?
iii. right at the center?
c. To the nearest mile per hour, how fast is the car going?
19. If the minute hand of a clock is long enough, the human eye can perceive the motion of its tip. The shortest minute
hand you can see moving is about 10 inches long. What is the slowest linear motion the human eye can perceive
(inches per minute)?
20. An electric clock transmits rotation from its motor to the clock hands through
a series of small gears driving larger gears. The second hand must make one
revolution every minute (obviously!)
a. What is the angular velocity of the second hand in radians per minute?
b. The second hand is fastened to gear 1, whose diameter is 3.8 centimeters.
What is gear 1's angular velocity?
c. Gear 2, of diameter 0.6 centimeter, meshes with gear 1's teeth. What is the
linear velocity of gear 2's teeth?
d. Gear 3, of diameter 4 centimeters, is connected to the same axle as gear 2.
What is the linear velocity of gear 3's teeth?
Answers
1. a) 300 π rad/min
b) Tip: 1200 π ft/min
Center: 0 ft/min
2. a) 3 rpm; 1080 degrees/min; 6 π rad/min
b) 150 π ft/min
3. 15 rad/sec
4. a) 150 rad/sec; 1500 rpm
b) 1425 in/sec
5. a) 6 π rad/sec
b) 1.2 π rad/sec
c) 6 π rad/sec
6. 64 mph
7. a) 4 π rad/sec
b) 12 π cm/sec
c) 12 π cm/sec
d) 1.6 π rad/sec
e) 48 rpm
8. a) 1500 cm/min
b) 1500 cm/min
c) 500 rad/min
9. a) 100 cm/sec
b) 20/3 rad/sec
c) 20/3 rad/sec
10. a) 90 π rad/min
b) 1350 π cm/min
c) 1350 π cm/min
d) 337.5 π rad/min
e) 168.75 rpm
11. a) 2 π /3 radians
b) 8 π /3 cm
c) 4 π /3 rad/sec
d) 16 π /3 cm/sec
12. a) 0.92 rad/sec
b) 7.38 ft/sec
13. 40 rad/sec
14. a) 4 π /3 ft/sec
b) 4 π /3 ft/sec
c) 44.444 rpm
15. a) 20 π /3 rad/sec
b) 260 π /3 mm/sec
c) 260 π /3 mm/sec
d) 260 π /9 rad/sec
e) 2600 π /9 mm/sec
16. a) 1.8 ft/sec
b) 0.3 rad/sec
17. a) 600 π rad/min
b) 1800 π in/min
c) 180 π rad/min
d) 10 mph
18. a) i)400 π rad/min
ii) 400 π rad/min
iii) 400 π rad/min
b) i) 520 π ft/min
ii) 160 π ft/min
iii) 0 ft/min
c) 19 mph
19. π /3 in/min
20. a) 2 π rad/min
b) 2 π rad/min
c) 3.8 π cm/min
d) 76 π /3 cm/min