A simple pendulum, 1.00 m in length, is released from rest when the support string is at an angle of \( 35.0^{\circ} \) from the vertical. What is the speed of the suspended mass at the bottom of the swing? ( \( g=9.80 \mathrm{~m} / \mathrm{s}^{2} \) and ignore air resistance) 1) \( 0.67 \mathrm{~m} / \mathrm{s} \) 2) \( 0.94 \mathrm{~m} / \mathrm{s} \) 3) \( 1.33 \mathrm{~m} / \mathrm{s} \) 4) \( 1.88 \mathrm{~m} / \mathrm{s} \) 5) \( 2.45 \mathrm{~m} / \mathrm{s} \)

To find the speed of the suspended mass at the bottom of the swing of a simple pendulum, we can use the principles of energy conservation. At the highest point, the pendulum has potential energy, and at the lowest point, all this energy converts into kinetic energy. First, we need to find the height (h) from which the pendulum is released. The height can be found using the length of the pendulum (L = 1.00 m) and the angle (θ = 35.0°): \[ h = L - L \cos(θ) = 1.00 - 1.00 \cos(35^\circ) \approx 1.00 - 1.00 \times 0.8192 \approx 0.181 \text{ m} \] Now using the potential energy (PE) at the height to find the speed (v) at the bottom, where all this potential energy converts to kinetic energy (KE): \[ PE = mgh \quad \text{and} \quad KE = \frac{1}{2} mv^2 \] Setting PE equal to KE: \[ mgh = \frac{1}{2} mv^2 \] The mass (m) cancels out, giving: \[ gh = \frac{1}{2} v^2 \] Solving for v: \[ v^2 = 2gh \qquad \to \qquad v = \sqrt{2gh} \] Substituting g = 9.80 m/s² and h = 0.181 m: \[ v = \sqrt{2 \times 9.80 \times 0.181} \approx \sqrt{3.55} \approx 1.88 \text{ m/s} \] Thus, the speed of the suspended mass at the bottom of the swing is approximately \( 1.88 \mathrm{~m} / \mathrm{s} \) (Option 4). --- Did you know that the pendulum was crucial in creating accurate clocks in the 17th century? Inventor Christiaan Huygens famously used the pendulum mechanism to construct a clock that was incredibly precise for the time, revolutionizing timekeeping! It helped mariners determine longitude at sea, making voyages far safer and more reliable. In real-world applications, simple pendulum concepts also apply in modern-day technologies. For example, seismographs use pendulum motions to measure and record vibrations of the Earth during earthquakes. Understanding how pendulums work helps improve our engineering practices, ensuring buildings can withstand tremors effectively!