High School

An object is traveling around a circle with a radius of 12 feet. If in 20 seconds a central angle of [tex]\frac{1}{4}[/tex] radian is swept out, what are the linear and angular speeds of the object?

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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