A propeller blade has a length of 2.42 m and, starting from rest, the tip of the blade has a tangential acceleration of 2.73 m/s2.
What is the magnitude of the total acceleration of the blade, in m/s2, 0.97 s after the blade starts rotating?
Well, well, well, looks like we've got a spinning propeller here! Let's calculate the magnitude of its total acceleration after 0.97 seconds.
Now, we know that the tangential acceleration is given as 2.73 m/s^2 and the length of the propeller blade is 2.42 m. So, to find the total acceleration, we need to consider both the tangential and centripetal accelerations.
To find the centripetal acceleration, we know that it's related to the tangential acceleration and the radius (blade length) by the formula: a_c = v^2 / r. But wait, we don't have the velocity yet! No worries, my friend! Remember that acceleration is the rate of change of velocity. So, let's find the velocity first!
Given that the blade starts from rest, we can use the formula v = u + at, where initial velocity u is 0 m/s. Plugging in the values, we get v = 0 + 2.73 * 0.97.
Now that we have the value of velocity, we can calculate the centripetal acceleration using the formula a_c = v^2 / r. Plugging in the values, we get a_c = (2.73 * 0.97)^2 / 2.42.
Finally, to find the total acceleration, we just need to add the tangential acceleration (2.73 m/s^2) and the centripetal acceleration we just calculated.
So, the magnitude of the total acceleration of the blade after 0.97 seconds is the sum of these two accelerations.
Now, my calculations are telling me that the total acceleration is around 3.98 m/s^2.
But hey, don't take my word for it, do the math yourself and see if I'm clowning around or not!
To find the magnitude of the total acceleration of the blade 0.97 seconds after it starts rotating, we can break down the total acceleration into two components: the tangential acceleration and the radial acceleration.
The tangential acceleration is the acceleration along the tangent of the circular path. In this case, we are given that the tip of the blade has a tangential acceleration of 2.73 m/s^2.
The radial acceleration is the acceleration towards the center of the circular path. Since the blade is rotating in a circle, it experiences a centripetal acceleration towards the center. The centripetal acceleration can be found using the formula:
ac = (v^2) / r
where ac is the centripetal acceleration, v is the linear velocity of the blade, and r is the radius of the circular path.
To find the linear velocity of the blade, we can use the formula:
v = ωr
where ω is the angular velocity of the blade.
Since the blade starts from rest, the initial angular velocity is 0. Therefore, the linear velocity of the blade at 0.97 seconds can be calculated as:
v = ωr = 0 * 2.42 = 0 m/s
Now, we can calculate the centripetal acceleration:
ac = (v^2) / r = (0^2) / 2.42 = 0 m/s^2
The magnitude of the total acceleration is the vector sum of the tangential acceleration and the radial acceleration. Since the centripetal acceleration is zero, the magnitude of the total acceleration is equal to the magnitude of the tangential acceleration:
Magnitude of total acceleration = Magnitude of tangential acceleration = 2.73 m/s^2.
Therefore, the magnitude of the total acceleration of the blade 0.97 seconds after it starts rotating is 2.73 m/s^2.