Question

(a) What is the energy of a pendulum \((L=1.0 \mathrm{m}, m=\) $0.50 \mathrm{kg})\( oscillating with an amplitude of \)5.0 \mathrm{cm} ?$ (b) The pendulum's energy loss (due to damping) is replaced in a clock by allowing a \(2.0-\mathrm{kg}\) mass to drop \(1.0 \mathrm{m}\) in 1 week. What average percentage of the pendulum's energy is lost during one cycle?

Short answer

Answer: The average percentage of energy loss during one cycle for a pendulum with a given mass and amplitude is given by the formula: Percent energy loss = \( \frac{256}{T} \% \) where T is the period of the pendulum's oscillation.

Step-by-step solution

Calculate the maximum potential energy of the pendulum

The formula for potential energy is: \(PE = mgh\) ,where m is the mass of the pendulum (0.5 kg), g is the acceleration due to gravity (9.81 \(m/s^2\)) and h is the maximum height of the pendulum during its oscillation. We can calculate h using amplitude: $$ h = \sqrt{L^2 - (L-A)^2} $$ where A is the amplitude (5.0 cm = 0.05 m) and L is the length of the pendulum (1.0 m). Calculating the maximum height, we have: $$ h = \sqrt{(1.0)^2 - (1.0-0.05)^2} = \sqrt{1-0.95^2} = 0.15605025 m $$ Now, calculate the maximum potential energy: $$ PE_{max} = mgh = (0.50 kg)(9.81 m/s^2)(0.15605025 m) ≈ 0.765 J $$

Calculate the energy loss during one cycle

The pendulum loses some of its energy due to damping and is replaced by allowing a 2.0 kg mass to drop 1.0 m during one week. We can calculate the energy replaced in one cycle. Let T be the period of the pendulum's oscillation, then the number of oscillation cycles per week is \(N = \frac{7 \times 24 \times 60 \times 60}{T}\). Now, we calculate the energy loss during one week: $$ E_{week} = mgh = (2.0 kg)(9.81 m/s^2)(1.0 m) = 19.62 J $$ Now we can find the energy loss during one cycle: $$ E_{cycle} = \frac{E_{week}}{N} = \frac{19.62 J}{\frac{7\times 24\times 60\times 60}{T}} = 19.62 J \times \frac{T}{7\times 24\times 60\times 60} $$

Calculate the average percentage of energy loss during one cycle

To find the percentage of energy loss during one cycle, we can use the following formula: Percent energy loss = \(\frac{E_{cycle}}{PE_{max}} \times 100\) Substituting the values, we have: $$ \text{Percent energy loss} = \frac{19.62 J \times \frac{T}{7\times 24\times 60\times 60}}{0.765 J} \times 100 = \frac{256}{T} \% $$ Since we don't know the period (T), we cannot provide an exact representation, but we provided the percentage of energy loss during one cycle for any given period T.