Tilting Pencils

If a pencil is carefully balanced on its point, how long will it take to fall over? Neglecting all external disturbances and non-ideal effects, this can be modeled as an inverted pendulum of length L and mass m, released from an initial position making an angle of θ₀ relative to vertical, as illustrated below.

The simple equation of motion for this tilting pencil can be found by either equating the torque with the product of the moment of inertia and the angular acceleration, or by an energy balance. If the mass per unit length is ρ = m/L, then the kinetic energy of an incremental segment dr of the rod is dK = (1/2) ρ dr v² where v = ωr. Hence the total kinetic energy of the rod is

The potential energy P can be expressed as the product of the acceleration of gravity times the total mass of the pencil times the height of the mid-point of the pencil, so we have P = ρLg(L/2)cos(θ) where g is the acceleration of gravity. Setting the sum of the kinetic and potential energies to a constant C gives the equation of motion

Dividing through by ρL³/6, we arrive at

This implies

so we have

Consequently the time for the pencil to fall to the horizontal position from an initial angle of θ₀ beginning with an initial angular speed of  is given by the integral

There is no simple closed-form expression for this integral, although it can be converted to the standard form of an elliptic integral. However, for extremely small initial angles θ₀ (e.g., 10^–10 degrees) the integral is very ill-conditioned and not easily evaluated.

To arrive at a more easily evaluated expression, we can first consider only the initial interval of time, during which the pencil just begins to tip from nearly a vertical position. Differentiating (1) and dividing through by 2(dθ/dt) gives

This is the same equation of motion we would derive directly from equating the torque to the product of angular acceleration and moment of inertia mL²/3 of a slender rod about its end point. Note that as long as the angle θ is extremely small, the value of sin(θ) is essentially equal to θ itself. (This is the same "small angle approximation" that we commonly use for an ordinary non-inverted pendulum to predict simple harmonic motion.) Therefore, as long as the pencil is very near vertical its motion essentially satisfies the linear equation

This simple second-order linear equation has the solution

where k is the square root of 3g/(2L), and the constants A,B are determined by the initial conditions. If the pencil is initially at rest at an angle θ₀ from vertical, then A = B = θ₀/2, and the solution is

Solving for the time t₁ when the pencil has reached an angle of θ₁, we find that

The angular velocity of the pencil at this instant is given by differentiating the expression for θ(t) and substituting this value for t. The result is

Now we can use equation (2) to give the time required to tip from θ₁ to θ₂ = π/2, and we can add this to the time t₁ required to tip from θ₀ to θ₁ to give the total time. By equation (2) we have

For very small values of θ₀ we can neglect the θ₀² in this expression, and we can expand the expression under the square root as

Since we are taking a small value of θ₁, the higher order terms can be neglected, and the argument of the square root is simply 1 – cos(θ) = 2sin(θ/2)². Making this substitution, the integral can be evaluated in closed form as

Returning to the expression for t₁, notice that for arguments much greater than 1 the inverse of the hyperbolic cosine is essentially equal to the natural log of twice the argument, so we have

Adding this to the previous expression gives a very close approximation to the total time for the pencil to tip from θ₀ to π/2

Recall that θ₁ is really just an intermediate small angle, which can be chosen small enough so that sin(θ₁/2) is essentially equal to θ₁/2, and so that 1 + cos(θ₁/2) is essentially equal to 2. Therefore the above expression can be simplified to

As an example, if the pencil is L = 0.2 m long, and if we take g = 9.8 m/sec², then the total time for the pencil to tip over from an initial stationary position of θ₀ = 10^–10 degrees (which equals (π/180)10^–10 radians) is 3.29773... seconds. On the other hand, if we imagine that the initial pencil position is just θ₀ = 10^–100 degrees, then the time to tip over is 27.469838205... seconds. In general, the time to tip over from an initial stationary position of 10^–n degrees is

This idealized pencil would take 1 year = 31557600 seconds to tip over (theoretically) if its initial stationary position is θ₀ = 10^–117498396 degrees from vertical.

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