Energy conservation with conservative forces: A very small 100-g object is attached to one end of a massless 10-cm rod that is pivoted without friction about the opposite end. The rod is held vertical, with the object at the top, and released, allowing the rod to swing. What is the speed of the object at the instant that the rod is horizontal?

Question

Quizplus · Accepted Answer

The speed of the object can be found using the principle of conservation of mechanical energy. Initially, when the rod is vertical, all the energy is in the form of potential energy (PE), and there is no kinetic energy (KE). When the rod is horizontal, the potential energy has been converted into kinetic energy. The initial potential energy can be calculated using PE = mgh, where m is the mass (0.1 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (0.1 m). This gives PE = 0.1 kg * 9.8 m/s^2 * 0.1 m = 0.098 J. Since energy is conserved, this potential energy equals the kinetic energy at the horizontal position, which is given by KE = 1/2 mv^2. Solving for v gives v = sqrt(2*KE/m) = sqrt(2*0.098 J / 0.1 kg) = sqrt(1.96) = 1.4 m/s. However, the correct calculation should account for the fact that the height (h) is the full length of the rod, so h = 0.1 m, and the calculation error in the explanation led to the wrong answer choice being initially selected. Correcting this, the potential energy at the start is PE = mgh = 0.1 * 9.8 * 0.1 = 0.98 J, and solving for v using the correct PE gives v = sqrt(2*0.98/0.1) = sqrt(19.6) = 4.43 m/s, which is not among the provided options, indicating a mistake in the calculation or interpretation of the options. The correct approach involves using the height difference correctly, but none of the calculations provided exactly match an option, suggesting a mistake in the explanation process. The correct approach is to equate the potential energy lost to the kinetic energy gained without the calculation error.

Energy Conservation with Conservative Forces: a Very Small 100-G Object