Suppose that you are holding a pencil balanced on its point. if you release the pencil and it begins to fall, what will be the angular acceleration when it has an angle of 10.0 degrees from the vertical? a typical pencil has an average length of 15.0 cm and an average mass of 10.0g. calculate the time for the pencil to hit the ground, assuming that it falls from standing perfectly vertical and maintains this angular acceleration.

  α  = 17 rad / s² ,   t = 0.4299 s

Explanation:

Let's use Newton's second angular law or torque to find angular acceleration

     τ = I α

     W r = I α

The weight is applied in the middle of the pencil,

    sin 10 = r / (L/2)

    r = L/2  sin 10

The pencil can be approximated by a bar that rotates on one end, in this case its moment of inertia is

     I = 1/3 M L²

Let's calculate

    mg L / 2 sin 10 = (1/3 m L²) α  

    α f = 3/2 g / L sin 10

    α  = 3/2 9.8 / 0.150 sin 10

    α  = 17 rad / s²

If the pencil has a constant acceleration we can use the angular kinematics relationships, as part of the rest wo = 0

     θ = w₀ t + ½ α  t²

     t = RA (2θ / α )

The angle from the vertical to the ground is

    θ = π / 2

    t = √ (2 π / (2 α ))

    t = √ (π / α )

    t = √ (π / 17)

     t = 0.4299 s