More with Arc Length, Angular Speed and Linear Speed
Arc Length: As discussed before, in a circle of radius r, the length s of an arc
that subtends (is created by) a central angle of  radians is
s  r (answer is in radians)
1)
All of the wheels on Bob’s car have a diameter of 30 inches. If his car
starts to roll down the hill, and the tires rotate 17 times before he stops the
car, how many feet has the car traveled? (round to nearest tenth of a foot)
2)
A man wants to measure the length of his fence, but does not have a tape
measure. He does have a disk that is 13 inches in diameter. When he
rolls the disk along the ground from the start of the fence to the end of the
fence the disk rotates 5,000 times (assume the disk never slips). How
long, in yards is his fence. (round to nearest tenth of a yard)
3)
How many rotations does a rock of radius 10 feet make if it rolls down a
hill that is 120 yards. (round to nearest tenths)
Angular and Linear Speed:
As previously stated it is better to think of finding the:
4)
angular speed as:
angular speed 
linear speed as:
linear speed 
dist traveled
given period of time
A car with 40 inch diameter wheels is traveling 40 miles per hour.
a)
How many revolutions per minute do the wheels make?
b)
5)
created angle in radians 
given period of time
Find the angular speed of the wheels in radians/minute
A pinwheel is being used to measure the speed of the wind. If the
pinwheel has a radius of 12 inches and it is observed that the pinwheel
rotates at 200 rpm
a)
what is the angular speed of the pinwheel?
b)
What is the speed of the wind? (in miles per hour)
6)
Two wheels, A and B, are connected by a chain. If wheel A has a
diameter of 20 inches and rotates and 366 times per minute, find:
a)
the angular speed of wheel A
b)
If the diameter of wheel B is 80 inches, find its rpm.
c)
d)
If the diameter of wheel B is three times as big as wheel A, find its
rpm.
e)
f)
what is the angular speed of wheel B in this case in radians
per minute?
If the diameter of wheel B is 10 inches, find its rpm.
g)
h)
what is the angular speed of wheel B in this case in radians
per minute?
what is the angular speed of wheel B in this case in radians
per minute?
If the diameter of wheel B is 15 inches, find its rpm.
i)
what is the angular speed of wheel B in this case in radians
per minute?
Note, rpm’s change when two different sized circles are attached by a
chain, if they are attached by a pole, the rpm’s would remain the same
7)
A 4 inch pulley on a motor makes 1500 revolutions/minute. The 4 inch
pulley is attached to a 2 inch pulley by a belt. The 2 inch pulley and an 8
inch saw blade are attached by a pole.
a)
Find the angular speed of the 4 inch pulley in radians/minute
b)
Find the linear speed of the 4 inch pulley in inches/minute.
c)
Find the angular speed of the saw in radians/ minute
d)
Find the linear speed of the blade in inches/minute.