Let θ be the angular distance (in radians) covered by a circular shaped object in n number of revolutions in t seconds.
If ω represents the angular speed, then the formula to find the angular speed is
Note :
There are 360 degrees or 2π radians in full rotation (one complete circle around). So, the angular distance covered in full rotation (one revolution) is equal to 2π.
Let ω be the angular speed in radians per second, r be the radius of the circular path in meters and v be the linear speed in meters per second.
To find the linear speed, we have to multiply the angular speed ω and the radius of the circular path r.
Then the formula to find the linear speed :
v = ωr meters/second
Example 1 :
A circular wheel with a 20-cm radius makes 8 revolutions in 20 seconds.
(i) Find the angular speed of the wheel in radians per second. (Round your answer to two decimal places.)
(ii) Find the linear speed of the wheel in centimeters per second. (Round your answer to two decimal places.)
Solution :
(i) Angular Speed :
Let θ be the angular distance (in radians) covered in 8 revolutions.
The angular distance (in radians) covered in one revolution is equal to 2π.
Then, the angular distance covered in 8 revolutions :
θ = 8 x 2π
θ = 16π radians
Formula to find the angular speed :
ω = θ/t
Substitute θ = 16π and t = 20.
ω = 16π/20
ω = 4π/5
ω ≈ 2.51 radians/sec
(ii) Linear Speed :
Formula to find the linear speed :
v = rω
Substitute r = 20 and ω = 4π/5.
v = 20(4π/5)
v = 80π/5
v = 50.27 cm/sec
Example 2 :
The circular blade on a saw has a diameter of 7.25 inches and rotates at 4800 revolutions per minute.
(i) Find the angular speed of the blade in radians per second. (Round your answer to two decimal places.)
(ii) Find the linear speed of the saw teeth (in feet per second) as they contact the wood being cut. (Round your answer to two decimal places.)
Solution :
(i) Angular Speed :
Let θ be the angular distance (in radians) covered in 4800 revolutions.
The angular distance (in radians) covered in one revolution is equal to 2π.
Then, the angular distance covered in 4800 revolutions :
θ = 4800 x 2π
θ = 9600π radians
Formula to find the angular speed :
ω = θ/t
Substitute θ = 9600π and t = 1.
ω = 9600π/1
ω = 9600π radians/min
We have to convert the above speed from radians per minute to radians per second.
From the above result, it is clear that the angular distance covered in 1 minute is equal to 9600π radians.
1 minute ----> 9600π radians
60 seconds ----> 9600π radians
(60 seconds)/60 ----> (9600π/60) radians
1 second ----> 160π radians
Therefore,
ω = 160π radians/sec
ω ≈ 502.65 radians/sec
(ii) Linear Speed :
Diameter = 7.25 inches
Radius = 7.25/2
= 3.625 inches
= 3.625/12 feet
Formula to find the linear speed :
v = rω
Substitute r = 3.625/12 and ω = 160π.
v = (3.625/12)(160π)
v = 151.84 feet/sec
Example 3 :
A Blu-ray disc is approximately 12 centimeters in diameter. The drive motor of a Blu-ray player is able to rotate a Blue-ray disc up to 10,000 revolutions per minute.
(i) Find the angular speed (in radians per second) of the Blu-ray disc as it rotates. (Round your answer to two decimal places.)
(ii) Find the linear speed (in meters per second) of a point on the outermost track as the disc rotates. (Round your answer to two decimal places.)
Solution :
(i) Angular Speed :
Let θ be the angular distance (in radians) covered by the Blu-ray disc in 10,000 revolutions.
The angular distance (in radians) covered in one revolution is equal to 2π.
Then, the angular distance covered in 10,000 revolutions in 1 minute :
θ = 10,000 x 2π
θ = 20,000π radians
Formula to find the angular speed :
ω = θ/t
Substitute θ = 20,000π and t = 1.
ω = 20,000π/1 radians/min
ω = 20,000π radians/min
We have to convert the above speed from radians per minute to radians per second.
From the above result, it is clear that the angular distance covered in 1 minute is equal to 20,000π radians.
1 minute ----> 20,000π radians
60 seconds ----> 20,000π radians
(60 seconds)/60 ----> (20,000π/60) radians
1 second ----> (20,000π/60) radians
1 second ----> (1000π/3) radians
Therefore,
ω = (1000π/3) radians/sec
ω ≈ 1047.20 radians/sec
(ii) Linear Speed :
Diameter = 12 cm
Radius = 12/2
= 6 cm
= 6/100 meters
= 0.06 meters
Formula to find the linear speed :
v = rω
Substitute r = 0.06 and ω = 1000π/3.
v = 0.06(1000π/3)
v = 20π
v = 62.83 meters/sec
Example 4 :
A computerized spin balance machine rotates a 25-inch diameter tire at 480 revolutions per minute.
(i) Find the angular speed (in radians per hour) of the tire. (Round your answer to two decimal places.)
(i) Find the linear speed (in miles per hour) at which the tire is being balanced. (Round your answer to two decimal places.)
Solution :
(i) Angular Speed :
Let θ be the angular distance (in radians) covered by the computerized spin balance in 480 revolutions.
The angular distance (in radians) covered in one revolution is equal to 2π.
Then, the angular distance covered in 480 revolutions :
θ = 480 x 2π
θ = 960π radians
Formula to find the angular speed :
ω = θ/t
Substitute θ = 960π and t = 1.
ω = 960π/1 radians/min
ω = 960π radians/min
According to the question, the road speed has to be found in miles per hour. So, we have to convert the above angular speed from radians per minute to radians per second.
From the above result, it is clear that the angular distance covered in 1 minute is equal to 980π radians.
1 minute ----> 960π radians
60 x 1 minute ----> 60 x 960π radians
60 minutes ----> 57,600π radians
1 hour ----> 57,600π radians
Therefore,
ω = 57,600π radians/hr
ω = 180,955.74 radians/hr
(ii) Linear Speed :
Diameter = 25 inches
Radius = 25/2
= 12.5 inches
(1 mile = 63360 inches)
= 12.5/63360 miles
Formula to find the linear speed (road speed) :
v = rω
Substitute r = 12.5/63360 and ω = 57,600π.
v = (12.5/63360)(57,600π)
v = 35.70 miles/hour
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