A 2.66-kg rock is attached at the end of a thin, very light rope 1.45 m long and is started swinging by releasing it when the rope makes an 11.9 ∘ angle with the vertical. You record the observation that it rises only to an angle of 4.10 ∘ with the vertical after 10.5 swings.
To find the period (T) of the pendulum, we first need to find the angular frequency (ω), which is given by the formula:
ω = √(g / L)
where g is the acceleration due to gravity (approximately 9.8 m/s²) and L is the length of the pendulum.
In this case, the length of the pendulum is 1.45 m, so we can substitute the values into the formula to find the angular frequency:
ω = √(9.8 / 1.45)
Next, we can find the period (T) from the angular frequency:
T = 2π / ω
Substituting the value of ω into the formula:
T = 2π / √(9.8 / 1.45)
Now, we can calculate the period of the pendulum.
To find the period after 10.5 swings, we need to multiply the period by the number of swings:
T_total = T * 10.5
To find the time it takes for each swing (T_swing), we can divide the total period by the number of swings:
T_swing = T_total / 10.5
Finally, we have all the information required to answer the question.