Answer :
Given that the object is traveling around a circle with a radius of 12 feet. And in 20 seconds a central angle of 1/4 radian is swept out. We are supposed to find out the linear and angular speeds of the object.Let's start by calculating the circumference of the circle.
Circumference of a circle,
C = 2πr
Where r is the radius.C = 2 × π × 12C = 24π feet
Therefore, we can say that the circumference of the circle is 24π feet
In 20 seconds, an object sweeps out 1/4 of the total angle.
Total angle,
θ = 2π∴ 1/4 of θ
= (1/4) × 2π
= π/2 radians
Therefore, we can say that the angular velocity of the object, ω = π/2/20ω = π/40 rad/s
Now, let's find the linear speed. We know that,C = 24π feetThe time taken to complete one revolution is,
t = 2π/ωt
= 2π/(π/40)t
= 80 seconds
We can find out the distance traveled by the object in 80 seconds by,Distance = Speed × time
We know that,Speed = Distance/Time
Thus,Distance = Speed × time24π
= Speed × 80
Therefore,
Speed = 24π/80Speed
= 3π/10 feet/second
Therefore, we can say that the angular speed of the object is π/40 rad/s and the linear speed is 3π/10 feet/second.
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