A pendulum of length \(75 \mathrm{cm}\) and mass \(2.5 \mathrm{kg}\) swings with a mechanical energy of 0.015 J. What is the amplitude?

We are given the following information: - Length of the pendulum, L = 75 cm = 0.75 m - Mass of the pendulum, m = 2.5 kg - Mechanical energy, E = 0.015 J At the highest point, the mechanical energy is equal to the potential energy, which can be calculated as: \(E = mgh\) Where: - E is the mechanical energy - m is the mass of the pendulum - g is the acceleration due to gravity, approximately \(9.8 \mathrm{m/s^2}\) - h is the height of the pendulum at the highest point

To find the height, h, we can rearrange the potential energy formula: \(h = \frac{E}{mg}\) Plugging in the known values: \(h = \frac{0.015\ \mathrm{J}}{{2.5\ \mathrm{kg}}({9.8\ \mathrm{m/s^2}})}\) \(h ≈ 0.0006122\ \mathrm{m}\)

Recall that the amplitude is the maximum displacement from the equilibrium position. When the pendulum swings to its maximum height (h), it forms a triangle with the vertical and horizontal distance making the amplitude. Let A be the amplitude, and θ be the angle between the vertical and the amplitude. We can use the cosine relation: \(cos(\theta) = \frac{A}{L}\) We also know that at the highest point, \(L - h = \frac{Amplitude}{cos(\theta)}\) Rearranging for A: \(A = (L - h)cos(\theta)\) \(h = L - \frac{A}{cos(\theta)}\) \(\frac{A}{cos(\theta)} = L - h\) \(A = (L - h)cos(\theta)\) We know that at the highest point, the pendulum is momentarily at rest, so the potential energy is at its maximum and the kinetic energy is at its minimum (zero). So, at the highest point, the mechanical energy is equal to the potential energy. Thus, the cosine of the angle θ can be determined by: \(cos(\theta) = \frac{h}{L - h}\) Substituting the values of h and L: \(cos(\theta) ≈ \frac{0.0006122\ \mathrm{m}}{{0.75\ \mathrm{m}} - {0.0006122\ \mathrm{m}}}\) \(cos(\theta) ≈ 0.0008177\) Now, we can find the amplitude using the formula: \(A = (L - h)cos(\theta)\) \(≈ (0.75 - 0.0006122) × 0.0008177\) \(A ≈ 0.000113\ \mathrm{m}\) or \(1.13 \times 10^{-4}\ \mathrm{m}\) The amplitude of the pendulum is approximately \(1.13 \times 10^{-4}\ \mathrm{m}\).