h) You are testing a decorative clock, to possibly be manufactured by your Consumer Electronics division, and attach a mass \( (\mathrm{m}) \) to a string of length \( (l) \) to form a simple pendulum. Assuming that the acceleration due to gravity \( (\mathrm{g}) \) of the earth may have an influence on the period \( (\mathrm{t}) \) of the pendulum swing, use dimensional analysis to find a formula for twhich could possibly involve \( \mathrm{m}, l \), and g .

Using dimensional analysis, we need to find a relationship for the period \( t \) of a simple pendulum in terms of mass \( m \), length \( l \), and the acceleration due to gravity \( g \). The period \( t \) has the dimension of time \([T]\), while the length \( l \) has a dimension of \([L]\) and acceleration due to gravity \( g \) has a dimension of \([L][T]^{-2}\). By equating dimensions, we can formulate: \[ t = k \cdot l^{a} \cdot g^{b} \cdot m^{c} \] where \( k \) is a dimensionless constant. Assuming \( t \) is proportional to the square root of length and inversely proportional to gravity, we can deduce \( a = \frac{1}{2} \) and \( b = -\frac{1}{2} \). Given that mass does not influence the period of a simple pendulum, we set \( c = 0 \). Thus, the resulting formula simplifies to: \[ t = k \sqrt{\frac{l}{g}} \] and for a simple pendulum the constant \( k \) is often taken to be \( 2\pi \), leading to: \[ t = 2\pi \sqrt{\frac{l}{g}} \] The most astonishing fact is that the weight of the pendulum does not affect its period, just its length and the local acceleration due to gravity. Pendulums have been used throughout history not only in clocks but also in sciences, such as physics and engineering, to study harmonic motion. This design dates back to Galileo in the late 1500s, who was truly captivated by the idea of pendular motion after observing the swinging of a chandelier. It became the foundation for accurate timekeeping, forever revolutionizing how we measure time!