Final answer:
The magnitude of the angular acceleration of the stick at the instant shown is 75.67 cm²
Step-by-step explanation:
To find the magnitude of the angular acceleration of the stick, we first need to determine the length of the stick. In the given scenario, a bug collides and sticks to the end of the stick, causing it to swing out to a maximum angle of 5.0° from the vertical. Since the mass of the stick is 10 times the mass of the bug, we can use the conservation of angular momentum to calculate the length of the stick.
The equation for conservation of angular momentum is: I₁ω₁ = I₂ω₂
Where I is the moment of inertia and ω is the angular velocity.
Since the bug sticks to the stick, the initial angular velocity ω₁ of the system is 0.
The moment of inertia I of the stick is given by: I = mL²/3
Where m is the mass of the stick and L is the length of the stick.
Substituting the given values into the equation, we have:
m₁₀ * 0 = m₁₀ * L²/3 * ω₂
Simplifying the equation, we get: L = √(3 * ω₂)
Therefore, the length of the stick is equal to √(3 times the magnitude of the angular acceleration). Substituting the given values, we have: 15.1 cm = √(3 * α)
Squaring both sides of the equation, we get: 227.01 cm² = 3 * α
Dividing both sides of the equation by 3, we get: 75.67 cm²= α
Therefore, the magnitude of the angular acceleration of the stick at the instant shown is 75.67 cm².