Example 2.1.3 : Derive the equation of motion of a pendulum of length l mass m which is attached to a mass less moving support as shown in Figure 2.1.5.

Figure 2.1.5: (a) Simple pendulum attached to a periodically translating support, (b) free body diagram of the mass.

Solution: Considering the free body diagram as shown in Fig. 2.1.5 (b), the body is under dynamic equilibrium under the action of tension, apparent weight and inertia force. Fixing unit vector and as shown in Figure 2.1.5(b) and applying Newton's 2nd Law one can write

................................ (2.1.23)

Separating the ith and jth component of the forces and equating them to 0 one obtains the expression for the tension and the governing equation of motion as given below.

..........................................................................................(2.1.24)

...............................................................................................(2.1.25)

Or, .............................................................................................(2.1.26)

Taking the oscillation to be very small, and hence Eq.(2.1.26) reduces to

...................................................................................................(2.1.27)