Question 1.4.2: Use the energy method to determine the equation of motion of......

Several assumptions must first be made to ensure simple behavior (a more complicated version is considered in Example 1.4.6). Using the same assumptions given in Example 1.1.1 (massless rod, no friction in the hinge), the mass moment of inertia about point 0 is

J = ml²

The angular displacement θ(t) is measured from the static equilibrium or rest position of the pendulum. The kinetic energy of the system is

T = \frac{1}{2} J \dot{θ}²  =  \frac{1}{2}ml² \dot{θ}²

The potential energy of the system is determined by the distance h in the figure so that

U = m\mathtt{g}l(1 – \cos  θ)

since h = l(1 – \cos  θ) is the geometric change in elevation of the pendulum mass. Substitution of these expressions for the kinetic and potential energy into equation (1.51) and differentiating yields

T + U = constant          (1.51)

\frac{d}{dt}\left[\frac{1}{2}ml² \dot{θ}²  +  m\mathtt{g}l(1  –  \cos  θ)\right] = 0

or

ml² \dot{θ} \ddot{θ}  +  m\mathtt{g}l(\sin  θ)\dot{θ} = 0

Factoring out \dot{θ} yields

\dot{θ}(ml² \ddot{θ}  +  m\mathtt{g}l \sin  θ) = 0

Since \dot{θ}(t) cannot be zero for all time, this becomes

ml²\ddot{θ} + m\mathtt{g}l \sin  θ = 0

or

\ddot{θ}  +  \frac{\mathtt{g}}{l} \sin  θ = 0

This is a nonlinear equation in θ and is discussed in Section 1.10 and is derived from summing moments on a free-body diagram in Example 1.1.1. However, since sin θ can be approximated by θ for small angles, the linear equation of motion for the pendulum becomes

\ddot{θ}  +  \frac{\mathtt{g}}{l} θ = 0

This corresponds to an oscillation with natural frequency w_{n}  =  \sqrt{\mathtt{g}/l} for initial conditions such that θ remains small, as defined by the approximation sin θ ≈ θ, as discussed in Example 1.1.1.

In Example 1.4.2, it is important to not invoke the small-angle approximation before the final equation of motion is derived. For instance, if the small-angle approximation  is used in the potential energy term, then U = m\mathtt{g}l(1 – cos θ) = 0, since the small-angle approximation for cos θ is 1. This would yield an incorrect equation of motion.