16) Two masses (2.3 Kg and 4.2 Kg) connected by a massless rope hang over opposite ends of a frictionless pulley. Each mass is initially 1.50m above the ground, and the top of the pulley is 6.5 m above the ground. What maximum height does the lighter object reach after the system is released? (3.44 m
To determine the maximum height reached by the lighter object in this physics problem, we can use the principle of conservation of energy.
Let's break down the problem and the steps to solve it:
1. First, calculate the potential energy of both masses at their initial positions. The potential energy (PE) is given by the equation PE = mgh, where m is the mass, g is the acceleration due to gravity (approximated as 9.8 m/s²), and h is the height.
- For the first mass (2.3 kg), the initial potential energy is PE1 = 2.3 kg * 9.8 m/s² * 1.50 m.
- For the second mass (4.2 kg), the initial potential energy is PE2 = 4.2 kg * 9.8 m/s² * 1.50 m.
2. Next, consider the principle of conservation of energy. As the system is released, potential energy is converted into kinetic energy. At the maximum height, the kinetic energy will be zero, and all the initial potential energy will be converted back into potential energy.
- Let's assume the maximum height reached by the lighter object is H (in meters). At this point, the potential energy of the first mass will be PE1 = 2.3 kg * 9.8 m/s² * H.
- Similarly, the potential energy of the second mass will be PE2 = 4.2 kg * 9.8 m/s² * (H + 6.5 m). Since the pulley is 6.5 m above the ground, the second mass will start from a higher height.
3. Equate the initial potential energy (PE1 + PE2) with the final potential energy of the system to solve for H:
PE1 + PE2 = 2.3 kg * 9.8 m/s² * H + 4.2 kg * 9.8 m/s² * (H + 6.5 m).
4. Solve the equation for H:
2.3 kg * 9.8 m/s² * 1.50 m + 4.2 kg * 9.8 m/s² * (1.50 m + 6.5 m) = 2.3 kg * 9.8 m/s² * H + 4.2 kg * 9.8 m/s² * (H + 6.5 m)
Simplify and solve for H:
33.57 + 324.36 = 22.54 H + 277.2 + 42 H
357.93 = 64.54 H
Divide both sides by 64.54 to isolate H:
H = 5.54 m
Therefore, the maximum height reached by the lighter object is approximately 5.54 meters.