The large blade of a helicopter is rotating in a horizontal circle. The length of the blade is 6.7m, 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 3.0m from the center of the circle.
i think its 3.7 meters but that's a lil advanced for me so idk
cent acceleration= w^2 r
w is constant, so acceleration is proportional to r.
centacceleration(3)=wtip(3/6.7)
To find the ratio of the centripetal acceleration at the end of the blade to that at a point located 3.0m from the center of the circle, we need to use the formula for centripetal acceleration:
ac = v^2 / r
Where ac is the centripetal acceleration, v is the linear velocity, and r is the radius of the circular path.
First, let's find the linear velocity at the end of the blade. We know that the length of the blade is 6.7m, so the radius of the circular path is 6.7m. To find the linear velocity, we need to know the time it takes for the blade to complete one rotation.
Let's assume that the blade takes t seconds to complete one rotation. The linear velocity can be calculated as:
v = distance / time
Since the distance covered by the blade is equal to the circumference of the circular path, which can be calculated as 2πr, the equation becomes:
v = 2πr / t
Now, let's find the linear velocity when the point is located 3.0m from the center. The radius of the circular path at this point is 3.0m. Using the same formula:
v' = 2πr' / t
Where r' = 3.0m.
To find the ratio of the centripetal accelerations, we need to find:
ac / ac'
To calculate ac and ac', we can substitute the expressions for v and v' in the formula for centripetal acceleration:
ac = v^2 / r
ac' = (v')^2 / r'
By substituting the expressions for v and v', we get:
ac = (2πr / t)^2 / r
ac' = (2πr' / t)^2 / r'
Now, let's calculate the ratio:
(ac / ac') = (v^2 / r) / ((v')^2 / r')
= (4π^2r^2 / t^2) / (4π^2r'^2 / t^2)
The t^2 and t^2 terms cancel out, so the ratio simplifies to:
(ac / ac') = (r^2 / r'^2)
Substituting the given values:
(ac / ac') = (6.7m)^2 / (3.0m)^2
Calculating:
(ac / ac') = 44.89 / 9.0
Therefore, the ratio of the centripetal acceleration at the end of the blade to that at a point located 3.0m from the center of the circle is approximately 4.99.
To find the ratio of the centripetal acceleration at the end of the blade to that which exists at a point located 3.0m from the center of the circle, we need to use the formula for centripetal acceleration.
The formula for centripetal acceleration is a = (v^2)/r, where:
- a represents the centripetal acceleration,
- v represents the linear velocity of the object, and
- r represents the radius of the circle.
In this case, we are given the length of the blade (6.7m) and the length from the center of the circle to a point (3.0m) on the blade.
Since the helicopter blade is rotating horizontally in a circle, the linear velocity at any point on the blade is the same, as they all complete one revolution in the same amount of time.
Therefore, we can equate the linear velocities at the end of the blade and the 3.0m point:
v_end = v_3
(2πr_end)/T = (2πr_3)/T
r_end = r_3
Thus, we can conclude that the radii at the end of the blade (r_end) and at the 3.0m point (r_3) are equal.
Since the radius of the circle does not change, the ratio of the centripetal acceleration at the end of the blade to that at the 3.0m point remains the same.
Therefore, the ratio of the centripetal acceleration at the end of the blade to that at the point located 3.0m from the center of the circle is 1:1 or simply 1.