Angle Measure Worksheet With Answers - Section 6.1 Page 9

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EXAMPLE: A disk with a 12-inch diameter spins at the rate of 45 revolutions per minute.
Find the angular and linear velocities of a point at the edge of the disk in radians per second
and inches per second, respectively.
Solution: In 60 s, the angle θ changes by 45 · 2π = 90π radians. So the angular velocity is
θ
90π rad
3
ω =
=
=
π rad/s
t
60 s
2
The distance traveled by the point in 60 s is s = 45 · πd = 45 · π · 12 = 540π in. So the linear
speed of the point is
s
540π in
v =
=
= 9π in/s
t
60 s
EXAMPLE: The second hand of a clock is 10.2 centimeters long, as
shown in the Figure on the right. Find the linear speed of the tip of this
second hand as it passes around the clock face.
Solution: In one revolution, the arc length traveled is
s = 2πr = 2π(10.2) = 20.4π cm
The time required for the second hand to travel this distance is 60 seconds. So, the linear speed
of the tip of the second hand is
s
20.4π cm
≈ 1.068 cm/s
v =
=
t
60 s
EXAMPLE: A Ferris wheel with a 50-foot radius (see the Figure on the
right) makes 1.5 revolutions per minute.
(a) Find the angular speed of the Ferris wheel in radians per minute.
(b) Find the linear speed of the Ferris wheel.
Solution:
(a) Because each revolution generates 2π radians, it follows that the wheel turns (1.5)(2π) = 3π
radians per minute. In other words, the angular speed is
θ
3π rad
ω =
=
= 3π rad/min
t
1 min
(b) The linear speed is
s
50(3π) ft
≈ 471.2 ft/min
v =
=
=
t
t
1 min
REMARK: Notice that angular speed does not depend on the radius of the circle, but only
on the angle θ. However, if we know the angular speed ω and the radius r, we can find linear
speed as follows:
(
)
s
θ
v =
=
= r
= rω
t
t
t
9

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