a certain type of propeller blade can be modeled as a thin uniform bar 2.5 m long and of mass 24 kg that is free to rotate about a frictionless axle perpendicular to the bar at its midpoint. if a technician strikes this blade witha mallet 1.15 m fromthe center with 35 N force perpendicular to the blade, find the max angular acceleration the blade could achieve.
To find the maximum angular acceleration of the propeller blade, we can use the concept of torque. Torque is the measure of the force that can cause an object to rotate about an axis. The torque (τ) can be calculated using the equation:
τ = r * F * sin(θ)
Where:
- τ is the torque
- r is the perpendicular distance from the axis of rotation to the point of force application (in this case, 1.15 m)
- F is the force applied (35 N)
- θ is the angle between the force vector and the perpendicular line from the axis of rotation to the point of force application (which is 90 degrees in this case)
Now, since the propeller blade is free to rotate about a frictionless axle, we can apply torque (τ) to find the angular acceleration (α) using the equation:
τ = I * α
Where:
- I is the moment of inertia of the propeller blade
- α is the angular acceleration
For a thin uniform bar rotating about its center perpendicular to its length, the moment of inertia (I) is given by:
I = (1/12) * m * L^2
Where:
- m is the mass of the propeller blade (24 kg)
- L is the length of the propeller blade (2.5 m)
Substituting the given values into the equations, we can solve for the maximum angular acceleration (α):
τ = r * F * sin(θ)
τ = (1.15 m) * (35 N) * sin(90°)
τ = 40.25 N·m
I = (1/12) * m * L^2
I = (1/12) * (24 kg) * (2.5 m)^2
I = 5 kg·m^2
τ = I * α
40.25 N·m = (5 kg·m^2) * α
Solving for α:
α = 40.25 N·m / 5 kg·m^2
α = 8.05 rad/s^2
Therefore, the maximum angular acceleration the blade could achieve is 8.05 rad/s^2.