GEOGRAPHY, MECHANICAL SHOP THEORY & PRACTICE, and MATHEMATICS: Arc length, radius, central angle, Angular displacement, velocity

S = rt


where:

S = arc length

r = radius

t = central angle, radians



Applications: Latitude & Longitude


LATITUDE

Latitude is the angle measured at the Earth's center between a point on the Earth and the equator


LONGITUDE

Longitude is the angle between the meridian passing through a point on the Earth, and the prime meridian passing through Greenwich, England


MERIDIAN

A Meridian is an imaginary circle passing through any point on the Earth's surface and through the North and South poles



Example Problem:

A city has a latitude of 45 N. The Earth has a radius of 4000 miles. Determine how many miles north of the equator the city is.


find:

S = distance of the city North from the equator


given:

r = 4000 miles

t = 45 degrees


solution:

S = rt

S = 4000 * 45 * pi/180

S = 4000 * 45 * 3.1416/180

S = 3,142 miles




Applications: Gears, Pulleys, Rack & Pinion


Example Problem:

A rack and pinion assembly rotates 240 degrees. How far will the rack move if the pitch diameter is 24 in.


find:

S = distance the rack moved


given:

r = 24/2

r = 12 in

t = 240 degrees


solution:

t = 240 * pi/180

t = 240 * 3.1416/180

t = 4.19 radians


S = rt

S = 12 * 4.19

S = 50.3 in




Example Problem:

Three pulleys of the same diameters of 6 in are arranged such that a right triangle with sides 5, 12, 13 ft is formed. Calculate the length of the belt around the pulleys.


find:

L = length of the belt around the pulleys


given:

d = 6 in

d = 6 in * 1 ft/12 in

d = 0.5 ft


solution:

The total curved portion of the belt is equal to the circumference of one pulley, Lc

Lc = pi * d

Lc = 3.1416 * 0.5

Lc = 1.57 ft


the (total) length of the belt around the pulleys

L = Lc + 5 + 12 + 13

L = 1.57 + 5 + 12 + 13

L = 31.57 ft




---------------------------------------------------------------
ANGULAR DISPLACEMENT & ANGULAR VELOCITY
---------------------------------------------------------------


Sa = wt


where:

Sa = angular displacement

w = angular velocity in degrees, radians, revolutions per unit time

t = time



Example Problem:

A wheel is rotating with angular velocity of 2400 rpm (revolutions per minute). Find how many revs does it make in 4 seconds.


find:

Sa = angular displacement


given:

w = 2400 rev/min * 1 min/60 sec

w = 40 rps (rev/sec)

t = 4 sec


solution:

Sa = wt

Sa = 40 rev/sec * 4 sec

Sa = 160 revs