The large blade of a helicopter is rotating in a horizontal circle. The length of the blade is 7.0 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 3.3 m from the center of the circle.
To find the ratio of the centripetal acceleration at the end of the blade to that at a point located 3.3 m from the center of the circle, we can use the centripetal acceleration formula:
ac = (v^2) / r
Here, "v" is the linear velocity and "r" is the radius of the circle.
Let's assume the angular velocity of the blade is ω and it completes one revolution in time period T.
The linear velocity at the tip of the blade can be found as:
v_tip = ω * r
Since the length of the blade is 7.0 m, the radius of the circle is also 7.0 m.
v_tip = ω * 7.0
The linear velocity at a point located 3.3 m from the center of the circle can be found as:
v_3.3m = ω * 3.3
Now, the ratio of the centripetal acceleration can be found using the formula:
ac_ratio = (ac_tip) / (ac_3.3m)
To simplify the formula, we can replace ac with (v^2) / r:
ac_ratio = (v_tip^2 / 7) / (v_3.3m^2 / 3.3)
Now, substitute the expressions for v_tip and v_3.3m:
ac_ratio = [(ω * 7.0)^2 / 7] / [(ω * 3.3)^2 / 3.3]
Simplifying the expression:
ac_ratio = (49.0 * ω^2) / 7.0 / (10.89 * ω^2) / 3.3
ac_ratio = (49.0 * ω^2 * 3.3) / (7.0 * 10.89 * ω^2)
ω^2 cancels out from the numerator and denominator:
ac_ratio = 49.0 * 3.3 / (7.0 * 10.89)
Simplifying further, we get:
ac_ratio = 16.815 / 76.23
ac_ratio ≈ 0.2207
Therefore, the ratio of the centripetal acceleration at the end of the blade to that at a point located 3.3 m from the center of the circle is approximately 0.2207.
To find the ratio of the centripetal acceleration at the end of the blade to that which exists at a point located 3.3 m from the center of the circle, we can use the formula for centripetal acceleration:
a = ω^2 * r
where:
a is the centripetal acceleration,
ω is the angular velocity, and
r is the radius of the circle.
In this case, the radius at the end of the blade is 7.0 m, and the radius at a point located 3.3 m from the center is 3.3 m.
The angular velocity (ω) is the same for both points since they are located on the same horizontal circle and rotating at the same rate.
Therefore, the ratio of the centripetal acceleration at the end of the blade to that at the point 3.3 m from the center can be calculated as:
a_end / a_point = (ω^2 * r_end) / (ω^2 * r_point)
Since the angular velocity cancels out, we can simplify the equation:
a_end / a_point = (r_end) / (r_point)
Substituting the given values:
a_end / a_point = (7.0 m) / (3.3 m)
Finally, we can calculate the ratio:
a_end / a_point ≈ 2.12
Therefore, the ratio of the centripetal acceleration at the end of the blade to that which exists at a point located 3.3 m from the center of the circle is approximately 2.12.