Final answer:
To achieve a beat of 105 beats per minute, first determine the period for one swing, which is about 0.5714 seconds. Then, using the pendulum period formula and the acceleration due to gravity, calculate the pendulum length required, which is approximately 0.8 meters.
Step-by-step explanation:
To determine the length of a pendulum needed to achieve a beat of 105 beats per minute, first calculate the period for one swing. Given that there are 60 seconds in a minute, a beat of 105 beats per minute translates to a frequency of 105/60 Hz, which is 1.75 Hz. The period T is the reciprocal of the frequency f, so T = 1/f.
Therefore:
T = 1/1.75 s ≈ 0.5714 s
The formula for the period T of a pendulum is given by T = 2π√(L/g), where L is the length of the pendulum, and g is the acceleration due to gravity, approximately 9.81 m/s².
To find the length L:
L = gT²/(4π²)
Plugging in the numbers:
L = (9.81 m/s² * (0.5714 s)²)/(4π²)
L ≈ 0.8 m
So, the pendulum length needed for a beat of 105 beats per minute is approximately 0.8 meters.