MATH 130 · Monday, October 26 · Linear and angular velocity

How fast is the tip of a propeller?

A Cessna 310 propeller is 81 inches across and turns at 2200 rpm.

Guess: how fast is the tip moving, in miles per hour?

About 530 mph.

By the end of class you can get that number yourself, and do it on Quiz 9 (Wednesday) and Test 2 (Friday).

Same turn, different speed

Both dots turn through the same angle in the same time: the same angular velocity ω.
The outer dot travels a bigger circle in that time, so it moves faster: a bigger linear velocity v.
ω = how fast it turns (radians per unit of time).
v = how fast a point moves (distance per unit of time).

Predict

Ana sits 2 m from the center of a merry-go-round. Ben sits 6 m from the center. It turns once every 10 s.

Question 1Who has the larger angular velocity?

Neither. Both turn 2π rad in 10 s: ω = 2π/10 ≈ 0.628 rad/s.

Question 2Who has the larger linear velocity, and how many times larger?

Ben, 3 times larger: 6 m is 3 × 2 m. (Ana ≈ 1.26 m/s, Ben ≈ 3.77 m/s.)

Where v = rω comes from

v = ^s⁄_t

On a circle, the distance traveled is an arc: s = rθ (θ in radians).

v = ^rθ⁄_t = r · ^θ⁄_t = rω

Linear velocity = radius × angular velocity, with ω in radians per unit of time. Radians have no units, so (ft)(rad/min) = ft/min.

The unit chain: five lines, every time

1

Radius. Given a diameter? Divide by 2.

2

Angular velocity in radians. rpm × 2π rad per revolution.

3

Multiply. v = rω

4

Convert units. 12 in = 1 ft · 5280 ft = 1 mi · 60 min = 1 h · 1000 m = 1 km · 3600 s = 1 h

5

Round once, at the end, and write the units.

Shortcut for rpm: v = (rpm)(2πr) gives distance per minute. It is the same as lines 2 and 3 together.

Worked example

Riverboat paddle wheel

The paddles have a radius of 8.50 ft and turn at 20.0 rpm. How fast is a paddle tip moving, in ft/min and mi/h?

1 · Radiusr = 8.50 ft

2 · Radians per minuteω = 20.0 ^rev⁄_min · ^{2π rad}⁄_rev = 40π rad/min

3 · Multiplyv = rω = 8.50(40π) = 340π ≈ 1068.1 ft/min

4–5 · Convert, round1068.1 ^ft⁄_min · ^{60 min}⁄_{1 h} · ^{1 mi}⁄_{5280 ft} ≈ 12.1 mi/h

Worked example · say the next line before I click

Hang glider

A hang glider flies a circular arc of radius 90.0 m with angular velocity 0.125 rad/s. Find its speed in m/s and km/h.

1 · Radiusr = 90.0 m

2 · Radians per secondω = 0.125 rad/s (already in radians)

3 · Multiplyv = rω = 90.0(0.125) = 11.25 m/s

4–5 · Convert, round11.25 ^m⁄_s · ^{3600 s}⁄_{1 h} · ^{1 km}⁄_{1000 m} = 40.5 km/h Round 11.25 to 11.3 first and you get 40.7, which is wrong.

Your turn, with a head start · work with a partner

Back to the propeller

Diameter 81 in, turning at 2200 rpm. Find the tip speed in ft/min and mi/h.

1 · Radius (done for you)r = 81 ÷ 2 = 40.5 in

2 · Radians per minuteω = = 2200 · 2π = 4400π rad/min

3 · Multiplyv = in/min = 40.5(4400π) ≈ 559,832 in/min

4–5 · Convert, round ft/min and mi/h ≈ 46,652.7 ft/min ≈ 530.1 mi/h

Spot the error

Three wrong answers, one right one

A wheel 2.0 ft across turns at 300 rpm. Find the rim speed in ft/min.

Answer Av = 2.0(300)(2π) ≈ 3769.9 ft/min

Used the diameter. The answer is doubled.

Answer Bv = 1.0(300)(360) = 108,000 ft/min

Used degrees. ω must be in radians.

Answer Cv = 1.0(300) = 300 ft/min

Left ω in rpm (forgot 2π per revolution).

Answer Dv = 1.0(300)(2π) = 600π ≈ 1885.0 ft/min

Correct: radius, radians, multiply.

You try · on your own

Watch second hand (worksheet 12a)

The sweep second hand of a watch is 21.0 mm long. It makes one turn per minute.

Find the linear velocity of its tip in mm/s and in cm/min. Use the five lines.

1

Radius

2

Radians per second

3

Multiply

4

Convert

5

Round and label

Check your work, line by line

Watch second hand: answer

1 · Radiusr = 21.0 mm

2 · Radians per secondω = ^{2π rad}⁄_{60 s} = ^π⁄₃₀ rad/s

3 · Multiplyv = 21.0 · ^π⁄₃₀ = 0.7π ≈ 2.2 mm/s

4–5 · Convert, round0.7π ^mm⁄_s · ^{60 s}⁄_{1 min} · ^{1 cm}⁄_{10 mm} ≈ 13.2 cm/min

Stretch · Test 2 style

Wind turbine

The blades of a GE 1.5s wind turbine are 116 ft long and turn at 22.0 rpm.

a) Tip speed in ft/min and mi/h

r = 116 ft (the blade is the radius)
v = 116(22.0)(2π) = 5104π ≈ 16,035 ft/min
≈ 182.2 mi/h

b) Area swept by the blades, in acres (1 acre = 43,560 ft²)

A = πr² = π(116)² ≈ 42,273 ft²
42,273 ÷ 43,560 ≈ 0.970 acre

This week

Wednesday

Quiz 9: circles, in class on paper

degrees to radians
circumference, arc length, sector area
central and inscribed angles
a volume word problem

Practice: Quiz 9 practice (PDF) on the course page

Friday

Test 2, in class on paper

Printed: compound and continuous interest, Heron's formula
From memory: v = rω, rpm × 2π, arc length, sector area, circle formulas

Practice: Test 2 practice (PDF) and the Test 2 review page

Exit ticket · hand in before you leave

Box fan

A box fan's blades are 24 in across and turn at 1500 rpm.

a) What is the radius, in feet?

b) What is the linear speed of a blade tip, in ft/min?

c) What is that speed in mi/h?

Show all five lines of the unit chain. Round only at the end.