The large blade of a helicopter is rotating in a horizontal circle. The length of the blade is 5.2 m, measured from its tip to the center of the circle. Find the ratio of the centripetal acceleration a1 at the end of the blade to the centripetal acceleration a2 which exists at a point located 1.8 m from the center of the circle.

Question

The large blade of a helicopter is rotating in a horizontal circle. The length of the blade is 5.2 m, measured from its tip to the center of the circle. Find the ratio of the centripetal acceleration a1 at the end of the blade to the centripetal acceleration a2 which exists at a point located 1.8 m from the center of the circle.

a1/a2=

Answer 1

To find the ratio of the centripetal acceleration a1 at the end of the blade to the centripetal acceleration a2 at a point located 1.8 m from the center of the circle, we need to use the concept of the centripetal acceleration formula.

The formula for centripetal acceleration is:

a = (v^2) / r

Where:
- a is the centripetal acceleration
- v is the linear velocity of the object in circular motion
- r is the radius of the circular path

In this case, we are given the length of the blade (5.2 m), which is the radius of the circular path. We need to find the linear velocity (v) at the end of the blade and at the point located 1.8 m from the center of the circle to calculate the respective centripetal accelerations.

First, let's find the linear velocity (v1) at the end of the blade:

Since we know the length of the blade (5.2 m) and it is rotating in a horizontal circle, we can calculate the circumference of the circle covered by the blade:

C = 2πr = 2π(5.2 m)

Now, to find the linear velocity at the end of the blade, we divide the circumference (C) by the time taken for one complete rotation (T1):

v1 = C / T1

Next, let's find the linear velocity (v2) at the point located 1.8 m from the center of the circle:

Since we know the radius of the circle at that point (1.8 m), we can find the circumference covered by that smaller radius:

C2 = 2π(1.8 m)

Now, to find the linear velocity at the point located 1.8 m from the center of the circle, we divide the circumference (C2) by the time taken for one complete rotation (T2):

v2 = C2 / T2

Finally, we can calculate the ratio of the centripetal accelerations a1 to a2:

(a1 / a2) = (v1^2 / r) / (v2^2 / r)

Since the radius (r) cancels out, the ratio simplifies to:

a1 / a2 = (v1^2 / v2^2)

Using the values of v1 and v2 that we found earlier, we can substitute them into the formula above and calculate the ratio of the centripetal accelerations.