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, if the tip of the blade has a tangential acceleration of 2.73 m/s², it's safe to say it's in a hurry to go somewhere! Now, remember that the total acceleration of an object consists of two components: the tangential acceleration and the radial acceleration.

To find the magnitude of the total acceleration of the blade, we'll need to calculate both! However, since the length of the propeller blade is given, we also need to calculate the angular acceleration first.

Now, we know that the tangential acceleration is related to the angular acceleration by the equation aₜ = rα, where r is the length of the blade and α is the angular acceleration. Rearranging the equation, we have α = aₜ / r.

Substituting the given values, we have α = 2.73 m/s² / 2.42 m ≈ 1.13 m/s². Great! We found the angular acceleration!

Now, to calculate the total acceleration at 0.97 s, we need to find the angular velocity ω first. We can use the equation ω = ω₀ + αt, where ω₀ is the initial angular velocity (which is zero in this case) and t is the time.

Substituting the values, we have ω = 0 + (1.13 m/s²)(0.97 s) ≈ 1.10 rad/s. Wonderful! We found the angular velocity!

Finally, we can find the magnitude of the total acceleration using the equation a_total = √((aₜ²) + (aᵣ²)), where aₜ is the tangential acceleration and aᵣ is the radial acceleration.

Since the radial acceleration is given by aᵣ = rω², let's calculate it. We have aᵣ = (2.42 m)(1.10 rad/s)² ≈ 2.85 m/s².

Now, plug those values back into the total acceleration equation, we get a_total = √((2.73 m/s²)² + (2.85 m/s²)²). After crunching the numbers, we find a_total ≈ 3.96 m/s².

So, the magnitude of the total acceleration of the blade, 0.97 s after it starts rotating, is approximately 3.96 m/s². That propeller blade sure is spinning its way to the top!