# ANGULAR SPEED AND LINEAR SPEED WORKSHEET

Problem 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.)

Problem 2 :

The circular blade on a saw has a diameter of 7.25 inches and rotates at 4800 revolutions per minute.

(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.)

Problem 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.)

Problem 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.)

(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

Formula to find the angular speed :

ω = θ/t

Substitute θ = 16π and t = 20.

ω = 16π/20

ω = 4π/5

(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

(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

Formula to find the angular speed :

ω = θ/t

Substitute θ = 9600π and t = 1.

ω = 9600π/1

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.

Therefore,

(ii) Linear Speed :

Diameter = 7.25 inches

= 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

(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

Formula to find the angular speed :

ω = θ/t

Substitute θ = 20,000π and t = 1.

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.

Therefore,

(ii) Linear Speed :

Diameter = 12 cm

= 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

(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

Formula to find the angular speed :

ω = θ/t

Substitute θ = 960π and t = 1.

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.

60 x 1 minute ----> 60 x 960π radians

Therefore,

(ii) Linear Speed :

Diameter = 25 inches

= 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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