A uniform rectangular sign of width 47.6 cm, height 25.0 cm, and negligible thickness hangs vertically from supporting hinges attached at its upper edge. Find the period of small-amplitude oscillations of the sign.
To find the period of small-amplitude oscillations of the sign, we can use the formula for the period of a simple pendulum-like system.
The period of a small-amplitude oscillation of a simple pendulum is given by the formula:
T = 2π√(L/g),
where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.
In the case of the rectangular sign hanging vertically, we can treat it as a simple pendulum with its length equal to its height. Therefore, for the sign, L = height = 25.0 cm.
Now, we need to determine the value of g. The acceleration due to gravity near the surface of the Earth is approximately 9.8 m/s².
To convert the height from centimeters to meters, we divide by 100:
L = 25.0 cm / 100 = 0.25 m.
Substituting the values into the formula, we get:
T = 2π√(0.25 m / 9.8 m/s²).
Calculating this expression gives us the period of small-amplitude oscillations of the sign.