Question

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.

Answer 1

Concepts and reason

The concepts required to solve the given question are moment of inertia, torque, angular acceleration and the rotational kinematics equations.

Initially, find an expression for the moment of inertia of the pencil using the concept for moment of inertia of a thin rod about its one end. Then find an expression for torque acting on the pencil.

Using the relation between torque and the moment of inertia, find an expression for angular acceleration. Then use the equation of the rotational kinematics, calculate the time taken for the pencil to reach the ground.

Fundamentals

Moment of inertia is defined as “a quantity expressing a body’s tendency to resist angular acceleration, which is the sum of the products of the mass of each particle in the body with the square of its distance from the axis of rotation”. It is the quantity that describes the mass distribution of the object.

The expression for the moment of inertia of a thin rod about its end,

Here, is the moment of inertia of a thin rod about its end, M is the mass of the rod, and L is the length of the rod.

Torque is the “twisting force that causes rotation of the object”.

Mathematical expression is given by,

$t=rxF $

Here, is the torque, r is the perpendicular distance, and F is the force acting on the object.

Torque in rotational motion is the quantity that requires producing the angular acceleration of an object.

The torque on a given axis acting on the pencil is,

$t=la $

Here, is the torque acting on the object, is the moment of inertia, and is the angular acceleration.

In rotational motion, constant angular acceleration gives the relation between the angular velocity, angle of rotation and time.

According to the rotational kinematics equation,

Here, is the angular displacement, is the initial angular displacement, is the initial angular velocity, is the angular acceleration, and t is the time.

Angular acceleration of the pencil:

The expression for moment of inertia of the pencil about its end,

Since the torque acting on the pencil,

…… (1)

Since the torque on a given axis,

…… (2)

Compare the equations (1) and (2) to find the angular acceleration.

Applying the kinematics equation,

Substitute for and for .

Rearrange the above equation,

Use the expression for to rewrite the expression for t.

Substitute for , for g , for , and for L to find t.

Ans:

The time taken for the pencil to hit the ground is.