The fan blades suddenly experience an angular acceleration of 2 rad/s2. If the blades are rotating with an initial angular velocity of 4 rad/s, determine the speed of point P when the blades have turned 2 revolutions (when ω = 8.14 rad/s).

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BlackTom AI · Accepted Answer

The question asks for the speed of point P on a fan blade given an angular acceleration α = 2 rad/s^2, initial angular velocity ω0 = 4 rad/s, and that the blades have turned 2 revolutions (which corresponds to ω = 8.14 rad/s as stated). First, we can verify the angular motion using rotational kinematics: the relation ω^2 = ω0^2 + 2 α Δθ, where Δθ is the angular displacement in radians. Two revolutions equal 4π radians, so Δθ = 4π. Substituting, ω^2 = 4^2 + 2*(2)*(4π) = 16 + 16π ≈ 66.265, hence ω ≈ 8.14 rad/s, which is consistent with the given final angular velocity. However, to determine the linear speed of point P, v = r ω, we need the radius r (the distance from the axis of rotation to point P). The problem statement does not provide r, so we cannot compute a numerical value for v without that information.

If r were known, the steps would be:
- Confirm the final angular velocity using ω^2 = ω0^2 + 2 α Δθ or equivalently ω = ω0 + α t after finding the time from Δθ if needed.
- Compute the tangential speed at point P as v = r ω with the final ω (8.14 rad/s in this case).

Potential pitfalls or misconceptions:
- Some might try to use only ω without considering r, leading to an incomplete answer. Remember that tangential speed depends on both angular velocity and radius.
- Assuming a constant radius or a special position for P without given geometry can mislead you; the correct approach is to use v = r ω with the true r.

In summary, with the information provided, we can determine the angular velocity at the specified instant, but the numeric speed of point P cannot be found without the radius r.

41056 Machines and Mechanisms A - Autumn 2025 Week 1 Quiz - Unlock Module Week 2

The fan blades suddenly experience an angular acceleration of 2 rad/s2. If the blades are rotating with an initial angular velocity of 4 rad/s, determine the speed of point P when the blades have turned 2 revolutions (when ω = 8.14 rad/s).

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