One end of a string is attached to an object of mass; When the object is raised to a position X that is a height H above its lowest point and released from rest, the object undergoes simple harmonic motion. When the object passes through the equilibrium position Y it has a speed of v.  Which of the following methods could a student use to determine the total mechanical energy at Position Y & why. Select 2 answers. A.ME = KE – PE because energy changes from KE to PE when the pendulum swings from X to Y B.ME = PE – KE because energy changes from PE to KE when the pendulum swings from Y to X C.ME = KE because this represents the maximum KE at Y as the system has 0 PE at Y D.ME = PE because this equation represents the maximum PE at X if the system has zero PE at Y
One end of a string is attached to an object of mass M; When the object is raised to a position X that is a height H above its lowest point and released from rest, the object undergoes simple harmonic motion. When the object passes through the equilibrium position Y it has a speed of v.  Which of the following methods could a student use to determine the total mechanical energy at Position Y & why. Select 2 answers.
A.ME = KE – PE because energy changes from KE to PE when the pendulum swings from X to Y
B.ME = PE – KE because energy changes from PE to KE when the pendulum swings from Y to X
C.ME = KE because this represents the maximum KE at Y as the system has 0 PE at Y
D.ME = PE because this equation represents the maximum PE at X if the system has zero PE at Y
The Correct Answer and Explanation is :
To determine the total mechanical energy at Position Y in the context of an object undergoing simple harmonic motion on a pendulum, it’s essential to understand the interplay between kinetic energy (KE) and potential energy (PE). Let’s analyze the provided answers and select the correct ones based on the principles of conservation of mechanical energy.
Answer Choices Analysis:
- A. ME = KE – PE: This choice suggests that mechanical energy (ME) is the difference between KE and PE as the object moves from X to Y. However, this representation is incorrect because the total mechanical energy in a conservative system (like a pendulum) is not the difference but the sum of KE and PE. This choice is incorrect.
- B. ME = PE – KE: This choice similarly suggests that ME is calculated as PE minus KE as the pendulum swings from Y back to X. Again, this is incorrect because mechanical energy is the sum of KE and PE, not the difference.
- C. ME = KE: This choice states that at the equilibrium position Y, ME is represented solely by KE because PE is zero at this point. This is a correct statement. At the equilibrium position, the pendulum has its maximum speed and consequently its maximum kinetic energy, while potential energy is zero (assuming the lowest point as the reference where PE = 0).
- D. ME = PE: This choice asserts that ME is represented solely by PE at position X (the highest point of the swing), assuming zero PE at position Y. However, while it’s true that the maximum PE is at X, this does not provide a method for calculating the total mechanical energy at position Y.
Correct Choices:
- C. ME = KE: This is correct because at the equilibrium position Y, all the mechanical energy of the system is converted into kinetic energy. The potential energy is zero at this midpoint (assuming the zero potential energy level is at the lowest point of the swing).
Explanation:
In simple harmonic motion like that of a pendulum, mechanical energy is conserved, meaning that the total energy (sum of KE and PE) remains constant throughout the motion. When the object is released from position X, it has maximum potential energy (PE = mgh, where h is the height above the lowest point) and zero kinetic energy. As it moves towards Y, potential energy is converted into kinetic energy. At Y, the height (and thus potential energy) is zero, and all the energy of the system is kinetic energy (KE = 0.5mv²).
The correct method to determine the total mechanical energy at Y would be by using KE alone at that position, as represented in choice C. This value remains constant throughout the pendulum’s swing due to the conservation of energy principle. The total mechanical energy at any point in the swing is the same as the total mechanical energy at Y, which can be calculated using the kinetic energy at Y.
I’ll generate the image now based on the described scenario of a pendulum’s simple harmonic motion at equilibrium position Y.
Here is an educational illustration showing a pendulum in simple harmonic motion at the equilibrium position Y. This image depicts the transition of energy states, highlighting zero potential energy and maximum kinetic energy at this point. You can see the clear annotations and labels for a better understanding of the concept.
