PreCalculus
ANGULAR VELOCITY WORD PROBLEMS
• Do all work on a separate piece of paper. • Show all work using dimensional analysis.
• Include units for EVERY step. • Exact answers only! NO decimals. • Fully simplify final answers.
EVERY SINGLE PROBLEM NEEDS A CALCULATION OR EXPLANATION TO JUSTIFY THE ANSWER!!!
1. Three Gear Problem Three gears are as shown in the figure.
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 1 at a distance of 13 mm from its axle?
c. Gear 1’s teeth mesh with gear 2’s teeth. Gear 2 has a radius of 3 mm.
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 mm.)
2. Wheel and Grindstone Problem
A waterwheel of diameter 12 ft turns at 0.3 radians 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 ft.
What is the angular velocity of a point on the rim of the grindstone?
3. Truck Problem 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 in. and
the wheel sprocket had a diameter of 20 in. If the drive sprocket goes 300 rpm:
a. Find the angular velocity of the drive sprocket in radians per minute.
b. Find the linear velocity of the 20-in. wheel sprocket in inches per minute.
c. Find the angular velocity of the wheel in radians per minute.
d. Find the speed of the truck to the nearest mile per hour.
4. Car Wheel Problem A car’s wheel turns at 200 rpm. 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. At the center of the wheel
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. At the center of the wheel
c. To the nearest mile per hour, how fast is the car going?
5. Clock Minute Hand Problem 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 in. long. What is the
slowest linear motion the human eye can perceive (inches per minute)?
1. a.
20π rad
260π mm
260π mm
260π rad
2600π mm
b.
c.
d.
e.
3 sec
3 sec
3 sec
9 sec
9 sec
4. a. 400π
2. a.
9 ft
3 rad
b.
5 sec
10 sec
3. a. 600π
rad
in
b. 1800π
min
min
rad
rad
rad
ft
ft
ft
65π mi
mi
, 400π
, 400π
b. 520π
, 160π
,0
c.
≈19
min
min
min
min
min
min
11 hour
hour
5.
π in
3 min
c. 180π
rad
min
d.
285π
mph
88
6. Clock Second Hand Problem 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 cm.
What is gear 1’s angular velocity?
c. Gear 2, of diameter 0.6 cm, meshes with gear 1’s teeth. What is the linear velocity of gear 2’s teeth?
d. Gear 3’s diameter is 4 cm. Gear 3 is connected to the same axle as gear 2. What is the linear velocity
of gear 3’s teeth?
7. Projector Problem When you are showing a movie, the film goes through the projector with a constant
linear velocity of about 30 cm/sec. The film winds off reel 1 and onto reel 2. As the projector runs, the
radius of the film rolled up on reel 1 decreases, and the radius on reel 2 increases.
When the radius of film on reel 1 is 8 cm and that on reel 2 is 18 cm:
a. Find the angular velocity of reel 1 in:
i. Radians per second
ii. Revolutions per second
iii. Degrees per second
b. Find the angular velocity of reel 2.
c. Find the linear velocity of a point on the rim of reel 2
if the diameter of the reel is 40 cm.
d. Find the linear and angular velocities of a point at the
center of reel 2.
8. Earth’s Rotation Problem The earth revolves once on its axis every 24 hours.
a. Find the angular velocity of the earth in radians per hour.
b. Find the linear velocity of a point on the equator 4000 miles from the axis.
c. Find the linear velocity at the North Pole.
9. Earth’s Revolution Problem The earth orbits the sun once every 365 days. It is approximately 93,000,000
miles from the sun to the earth.
a. Assuming the earth’s orbit of the sun is circular, what is the earth’s angular velocity about the sun in
radians per hour?
b. What is the earth’s linear velocity in miles per hour?
The turntable on a record player rotates at 3313 rpm.
10. Record Player Problem
a. What is the linear velocity of a point on the rim of the turntable?
b. A large drive wheel is attached to the turntable’s axle, below the turntable.
What is the linear velocity of its rim?
c. A motor drives a small drive wheel at 600 rpm. What must the radius of the
small drive wheel be so that the turntable will rotate at 3313 rpm with no
slippage between the two drive wheels?
6. a. 2π
8. a.
π rad
12 h
rad
min
b. 2π
rad
min
b. 1000 π ≈ 1047.2 mph
3
c.
19π cm
5 min
c. 0 mph
d.
76π cm
3 min
9. a.
7. a.
rad
π
4380 hour
15 rad 15 rev 675 deg
,
,
4 sec 8π sec
π sec
b.
b. 1550000π mph ≈ 66705 mph
73
5 rad
3 sec
c.
100 cm
3 sec
10. a.
d.
1000π in
3
min
5 rad
cm
,0
3 sec
sec
b. 200π
in
min
c.
1
in
6