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
Formula to find the angular speed ω :
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.
Example 1 :
A circular wheel with a 20-cm radius makes 8 revolutions in 20 seconds. Find the linear speed of the wheel in centimeters per second. (Round your answer to two decimal places.)
Solution :
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 radians/sec
ω = 4π/5 radians/sec
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 :
A carousel with a 50-foot diameter makes 4 revolutions per minute. Find the linear speed of the carousel in feet per minute. (Round your answer to two decimal places.)
Solution :
Let θ be the angular distance (in radians) covered by the carousel in 4 revolutions.
The angular distance (in radians) covered in one revolution is equal to 2π.
Then, the angular distance covered in 4 revolutions :
θ = 4 x 2π
θ = 8π radians
Formula to find the angular speed :
ω = θ/t
Substitute θ = 8π and t = 1.
ω = 8π/1
ω = 8π radians/sec
Given : Diameter of the carousel is 50 feet.
Then,
radius = 50/2
= 25 feet
Formula to find the linear speed :
v = rω
Substitute r = 25 and ω = 8π.
v = 25(8π)
v = 200π
v = 628.32 feet/min
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. 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 :
Let θ be the angular distance (in radians) covered by the Blue-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
According to the question, the linear speed has to be found in meters per sec. 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 20,000π radians.
1 minute ----> 20,000π radians
60 seconds ----> 20,000π radians
(60 seconds)/60 ----> (20,000π/60) radians
1 second ----> 1000π/3 radians
Therefore,
ω = 1000π/3 radians/sec
Given : Diameter of the disc is approximately 12 cm.
Then,
radius = 12/2
= 6 centimeters
= 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 road speed (in miles per hour) at which the tire is being balanced. (Round your answer to two decimal places.)
(ii) At what rate should the spin balance machine be set so that the tire is being tested for 70 miles per hour? (Round your answer to two decimal places.)
Solution :
Part (i) :
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
Given : Diameter of the tire is 25 inch.
Then,
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
Part (ii) :
From part 1, when the number of revolutions is 480 per minute, the linear speed is 35.70 miles per hour.
Here, we have to find the number of revolutions per minute, when the linear speed is 70 miles per hour.
Let x be the number of revolutions per minute required when the linear speed is 70 miles per hour.
Now, we can form a proportion and solve for x.
x : 70 = 480 : 35.70
x/70 = 480/35.70
x = 941.18 revolutions/min
The spin balance machine should be set at the rate of 941.18 revolutions per minute so that the tire is being tested for 70 miles per hour.
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