In the figure below, an object, after being released from its circular path, travels the distance OA in the same time it would have moved from O to P on the circle. The speed of the object on and off the circle remains constant at the same value. Suppose that the radius of the circle in the figure below is 3.7 m and the angle θ is 25°. What is the distance OA?
To find the distance OA, we can use the concept of arc length in a circle.
The formula for the arc length is given by L = rθ, where L is the arc length, r is the radius of the circle, and θ is the central angle in radians.
In this case, we are given the radius of the circle (r = 3.7 m) and the central angle (θ = 25°).
First, we need to convert the angle from degrees to radians. Recall that 1 radian is equal to π/180 degrees. Therefore, we can convert 25 degrees to radians as follows:
θ (in radians) = 25° * (π/180°) = 25π/180 radians
Now we can use the formula for arc length to calculate the distance OA:
L = rθ
L = 3.7 m * (25π/180)
L = 205π/180 m
L ≈ 3.59 m
Therefore, the distance OA is approximately 3.59 meters.
To find the distance OA, we can use the arc length formula for a circle:
Arc length = θ * radius
In this case, the radius of the circle is given as 3.7 m, and the angle θ is given as 25°.
So, the arc length from O to P on the circle is:
Arc length = 25° * 3.7 m
Arc length = 92.5 m°
Since the object travels the same distance OA in the same time, OA is also equal to 92.5 meters.