A fan blade is rotating with a constant angular acceleration of +13.7 rad/s2. At what point on the blade, as measured from the axis of rotation, does the magnitude of the tangential acceleration equal that of the acceleration due to gravity?
The magnitude of the tangential acceleration is ra=13.7rad/s^2
It equals the gravitational acceleration, which is 9.8m/s^2
From there, just set the two equal to one another: 13.7r=9.8
r=.7313m
To find the point on the blade where the magnitude of the tangential acceleration equals that of the acceleration due to gravity, we can start by determining the expressions for both accelerations.
Let's assume the length from the axis of rotation to the point on the blade is 'r', the angular acceleration is 'α', and the tangential acceleration is 'at'.
The relation between angular acceleration, tangential acceleration, and radius is given by:
at = α * r
The acceleration due to gravity, denoted as 'g', is approximately 9.8 m/s^2.
Using this information, we can equate the magnitude of the tangential acceleration to the acceleration due to gravity and solve for 'r':
at = g
α * r = g
r = g / α
Substituting the given values:
r = 9.8 m/s^2 / 13.7 rad/s^2
Calculating the result:
r ≈ 0.715 m
Therefore, the distance from the axis of rotation to the point on the blade where the magnitudes of the tangential acceleration and acceleration due to gravity are equal is approximately 0.715 meters.