Degrees, Minutes, Seconds

In the DMS (degree minute second) system of angular measure, each degree is subdivided into 60 minutes (denoted by ‘ ) and each minute is subdivided into 60 seconds (denoted by “)

Measuring in degrees, minutes, seconds is a common practice in surveying and navigation.

Since there are 60 ‘ in 1 degree we can convert degrees to minutes by multiplying by the conversion ratio

0

'

1

60

Convert 34.80 to DMS

We need to convert the fractional part to minutes

'48608.

'00 48348.34

Convert 112.420 to DMS

Convert the fractional part

'2.256042. Convert the fractional part of the minutes into seconds

''12602. '''00 122511242.112

Convert 42024’36’’ to degrees

This is the reverse of the last example. Instead if multiplying by 60, we need to divide by 60

000

0'''0 41.426060

36

60

2442362442

Arc Length• If θ is the radian measure of a central

angle in a circle with a radius of r, then the length, s, of the arc intercepted by θ is s = rθ.

Example 1

A central angle in a circle with a diameter of 30m measures π/3 radians. Find the length of the arc intercepted by this angle.

Solution: s = rθ

= 15(π/3)

= 5π m

Try This

A central angle in a circle with a radius of 12.5ft measures 0.6 radian. Find the length of the arc intercepted by this angle.

7.5 ft

Sector Area

• To find the area of part of a circle, multiply the shaded fraction by the area of the whole circle.

• The fraction that is shaded is given by

or , depending on your units. 2

360

Example

• Find the area of a sector with a central angle of π/3 radians in a circle with a radius of 10 inches.

Solution:

2

2A r

2

2r

23 10

2

100

6

503

10 in

3

Linear and Angular Speed

• Question 1:

A bicycle traveled a distance of 100 meters. The diameter of the wheel of this bicycle is 40 cm. Find the number of rotations of the wheel.

• Solution to Question 1:

For every one rotation of the wheel, the bicycle moves a distance equal to the circumference of the wheel. The circumference C of the wheel is given by C = 40 Pi cm

• The number of rotations N of the wheel is obtained by dividing the total distance traveled, 100 m = 10000 cm, by the circumference.

N = 10000 cm / 40 Pi cm = 80 rotations (rounded to the nearest unit)

• Questions 2: The wheel of a car made 100 rotations. What distance has the car traveled if the diameter of the wheel is 60 cm?

• Solution to Question 2: The circumference C of the wheel is given by

C = 60 Pi cm • For each rotation of the wheel, the car travel a

distance equal to the circumference of the wheel. 100 rotations correspond a distance d traveled by the car where d is given by

d = 100 * 60 Pi cm = 18850 cm (rounded to the nearest cm)

• Questions 3:

The wheel of a machine rotates at the rate of 300 rpm (rotation per minute). If the diameter of the wheel is 80 cm, what are the angular (in radian per second) and linear speed (in cm per second) of a point on the wheel?

• Solution to Question 3: Each rotation corresponds to 2 Pi radians. Hence 300

rotations per minute correspond to an angular speed a given by

a = 300 * 2 Pi radians / minute We now substitute 1 minute by 60 seconds above

a = 300 * 2 Pi radians / 60 seconds

= 10 Pi rad/sec

= 31.41 rad/sec (rounded to 2 decimal places)

• The linear speed is obtained by noting that one rotation corresponds to the circumference of the wheel. Hence the linear speed s is given by

s = 300 * (80 Pi) cm / minute

• = 300 * 80 Pi / 60 cm/sec

= 1257 cm/sec

• Questions 4:

The Earth rotates about its axis once every 24 hours (approximately). The radius R of the equator is approximately 4000 miles. Find the angular (radians / second) and linear (feet / second) speed of a point on the equator.

Solution to Question 4:

One rotation every 24 hours (or 24 *3600 seconds) gives an angular speed a equal to

a = 2 Pi / (24*3600) = 0.0000727 rad/sec

We first convert the radius R in feet

R = 4000 * 5280 = 21,120,000 feet

One rotation every 24 hours (or 24 *3600 seconds) gives a linear speed s equal to

s = 2 Pi R / (24*3600)

= 2 * Pi * 21,120,000 / 86,400

= 1,536 feet / sec

HOMEWORK

• Pg. 257, 258 # 65-97 odd

• Check your answers with the back of the book.