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 Figure 5.4 is 3.1 m and the angle is is 25°. What is the distance OA?
This question is hmmmm....Where is point P. What is the relation of P to O and A?
Point P is touching the circle at 0 degrees. Point C is at the center of the circle. Point O is also touching the circle. PCO forms an angle 25 degrees. The object is released from O and goes in a straight line away from the center of the circle to point A. A is close to P outside the circle.
To find the distance OA, we can use the information given in the problem and apply some basic geometry principles.
First, we can observe that the angle formed by POC is given as 25°. Since P is on the circle and O is the starting point, the object would have moved along the arc OP if it had stayed on the circular path.
The length of an arc on a circle is given by the formula:
arc length = radius * angle (in radians)
To find the length of the arc OP, we need to convert the angle from degrees to radians. Since there are 2π radians in a full circle (360°), we can use the following proportion:
2π radians = 360°
angle (in radians) = (angle in degrees * 2π) / 360
Let's calculate the angle in radians:
angle (in radians) = (25° * 2π) / 360 = 25π / 180 radians
Now we can find the length of the arc OP:
arc OP = 3.1 m * (25π / 180 radians) = (31π / 18) m ≈ 5.44 m (rounded to two decimal places)
The object takes the same time to travel from O to P on the circular path as it does to travel the distance OA in a straight line. Therefore, the length of OA is equal to the length of the arc OP.
Hence, the distance OA is approximately 5.44 meters.