The large blade of a helicopter is rotating in a horizontal circle. The length of the blade is 6.07 m, measured from its tip to the center of the circle. Find the ratio of the centripetal acceleration at the end of the blade to that which exists at a point located 2.56 m from the center of the circle.
To find the ratio of the centripetal acceleration at the end of the blade to that which exists 2.56 m from the center of the circle, we need to understand that the centripetal acceleration is directly related to the radius of the circular path.
Centripetal acceleration is given by the formula: a = (v^2) / r
where:
a is the centripetal acceleration,
v is the linear velocity of the object, and
r is the radius of the circular path.
In this case, the radius at the end of the blade is 6.07 m (measured from the tip to the center of the circle), and the radius 2.56 m is given.
To compare the centripetal accelerations at the two points, we need to find the linear velocities at those points.
The linear velocity is given by the formula: v = ω * r
where:
v is the linear velocity,
ω is the angular velocity in radians per second, and
r is the radius of the circular path.
Since we know the radius of each point, we will focus on finding the angular velocity at each point.
The angular velocity can be obtained through the relationship between the linear velocity and the angular velocity:
v = ω * r
ω = v / r
Now, we can use the angular velocities to find the centripetal accelerations at each point.
At the end of the blade:
r1 = 6.07 m
v1 = ω1 * r1
At point 2.56 m from the center:
r2 = 2.56 m
v2 = ω2 * r2
The ratio of the centripetal accelerations is:
(a1 / a2) = (v1^2 / r1) / (v2^2 / r2)
Let's solve using this approach.