Question

A typical pencil has an average length of 15.0 cm and an average mass of 10.0 g. Assume the tip of the pencil does not slip as it falls.
1. What is the angular acceleration of the pencil when it makes an angle of 10.0 degrees with the vertical? Express your answer numerically in radians per second squared to three significant figures.

Answer 1

Answer:

α = 17.0 rad/s²

Explanation:

For this problem let's use the torque expression

         τ = I α

Torque is the product of force by the distance that is half the length of the pencil

        τ = F x

Let's use trigonometry

        sin 10 = x / (l / 2)

        x = l / 2 sin 10

        τ = mg l /2  sin 10

The moment of inertia of pencil that we approximate as a rod with the axis of rotation at one end

          I = 1/3 m L²

 We replace

          mg l / 2 sin 10 = 1/3 m l² α

           α = 3/2 g/l  sin 10  

Let's reduce the magnitudes to the SI system

            l = 15.0 cm = 0.150 m

           m = 10.0 g = 0.0100 kg

Let's calculate

            α = 3/2 9.8 / 0.150 sin 10

            α = 17.0 rad/s²