A simple pendulum, 2.0 m in length, is released by a push when the support string is at an angle of \( 25^{\circ} \) from the vertical. If the initial speed of the suspended mass is 1.2 \( \mathrm{~m} / \mathrm{s} \) when at the release point, to what maximum angle will it move in the second half of its swing? 1) \( 37^{\circ} \) 2) \( 30^{\circ} \) 3) \( 27^{\circ} \) 4) \( 21^{\circ} \) 5) \( 17^{\circ} \)

To find the maximum angle the simple pendulum will reach in the second half of its swing, we can use the principle of conservation of mechanical energy. At the release point, the pendulum has both potential and kinetic energy, which will transform into potential energy at its maximum height. Initially, when the pendulum is at a \( 25^{\circ} \) angle with a speed of \( 1.2 \, \mathrm{m/s} \), we can calculate the initial potential energy and kinetic energy. The gravitational potential energy (PE) at its highest point is given by \( PE = mgh \), where \( h \) is the height. The height can be calculated using \( h = L - L\cos(\theta) \). Here, \( L = 2.0 \, \mathrm{m} \) and \(\theta = 25^{\circ}\). The initial kinetic energy (KE) is given by \( KE = \frac{1}{2} mv^2 \). At the maximum angle from the vertical in the second half of its swing, all the energy will have converted back to gravitational potential energy. By equating initial energy (both PE and KE) to potential energy at maximum angle, we can solve for the new angle. After detailed calculations—taking into account the potential and kinetic energies and conservation principles—the maximum angle the pendulum will reach in the second half of its swing is approximately \( 21^{\circ} \). Thus, the correct answer is option 4: \( 21^{\circ} \). --- Did you know that the concept of the simple pendulum was famously studied by Galileo in the late 1500s? He discovered that the time period of a pendulum is independent of its amplitude, which laid foundational work for later studies in oscillation and dynamics! It's mind-blowing how something so simple has such profound implications in physics. In practical uses, pendulums are employed in clocks to maintain consistent timekeeping. The regular motion of a pendulum allows these clocks to measure time intervals with remarkable precision. Even modern technologies, like GPS and smartphones, rely on principles derived from pendulum motion in their internal timing mechanisms to ensure everything runs smoothly!