A 3.0 kg pendulum swings from point A of height ya = 0.04 m to point B of height yb = 0.12 m, as seen in the diagram below.
The Correct Answer and Explanation is :
To analyze the pendulum’s motion from point A to point B, we can use the principle of conservation of mechanical energy, which states that the total mechanical energy (potential energy + kinetic energy) in an isolated system remains constant if only conservative forces are acting.
- Determine the Potential Energy at Points A and B:
The potential energy (PE) of an object is given by the formula:
[
PE = mgh
]
where:
- (m) = mass of the pendulum (3.0 kg),
- (g) = acceleration due to gravity (approximately (9.81 \, \text{m/s}^2)),
- (h) = height above a reference point. At Point A:
[
PE_A = mgh_a = 3.0 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 0.04 \, \text{m} = 1.176 \, \text{J}
] At Point B:
[
PE_B = mgh_b = 3.0 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 0.12 \, \text{m} = 3.529 \, \text{J}
]
- Calculate the Change in Potential Energy:
The change in potential energy as the pendulum moves from point A to point B can be calculated as:
[
\Delta PE = PE_B – PE_A = 3.529 \, \text{J} – 1.176 \, \text{J} = 2.353 \, \text{J}
] - Energy Conservation Principle:
According to the conservation of mechanical energy, the total mechanical energy at point A (which is all potential energy since it’s at rest) equals the total mechanical energy at point B (potential plus kinetic energy). If we consider that the pendulum swings from A to B, the kinetic energy (KE) will be maximum at the lowest point of the swing. - Conclusion:
As the pendulum rises from point A to point B, it converts some of its kinetic energy into potential energy. The increase in potential energy (approximately (2.353 \, \text{J})) shows that work is done against gravity, which manifests as a rise in height. The analysis confirms that energy is conserved throughout the motion of the pendulum, illustrating fundamental principles of physics related to energy conservation and the transformation between kinetic and potential energy.