Question

a spring hangs vertically from a bracket at its unweighted equilibrium length, as shown in the left-most image. an object with mass m is attached to the lower end of the spring, and it is gently lowered until the spring reaches its new equilibrium length, as shown in the center figure. referring to the right-most figure, the mass is raised until the spring returns to its original length, and then it is released from rest resulting in vertical oscillations.

Answer 1

**Question:**

Explain how the vertical oscillations of the mass-spring system, as shown in the right-most figure, relate to the initial setup and the release of the mass.

Final Answer:

The vertical oscillations observed in the right-most figure result from the release of the mass after it was initially lowered and the spring reached its new equilibrium length.

Step-by-step explanation:

The oscillations can be understood through the principles of elastic potential energy and the law of conservation of mechanical energy. Initially, when the mass is lowered, it gains potential energy due to its elevation. This potential energy is stored in the spring as elastic potential energy when it is stretched. As the mass is raised back to its original height, this potential energy is converted back into gravitational potential energy. When the mass is released, the potential energy is transformed into kinetic energy, causing the mass to oscillate between kinetic and potential energy, leading to vertical oscillations.

The frequency of these oscillations can be determined by the spring constant (k) and the mass (m) using the formula f = (1/2π) * √(k/m). The amplitude of the oscillations depends on the initial displacement of the mass from the equilibrium position. These oscillations continue until damping forces, such as air resistance, absorb the energy and bring the system to a stop. The phenomenon illustrates the conversion of potential energy into kinetic energy and vice versa, showcasing the interconnected nature of energy transfer in a mass-spring system.

Answer 2

Adding mass to a spring and then displacing it from its new equilibrium position before releasing it will result in vertical oscillations of the mass, with the spring providing the restoring force. These oscillations are described by simple harmonic motion.

To understand and describe the process you've mentioned, let's break it down into step-by-step stages:

1. Initial Equilibrium (Left-Most Figure):

- The spring is hanging vertically from a bracket at its unweighted equilibrium length.

- At this stage, there is no external force acting on the spring other than its own weight, so it's in equilibrium.

2. Adding Mass (Center Figure):

- An object with mass \(m\) is attached to the lower end of the spring.

- The mass is gently lowered until the spring reaches its new equilibrium length.

- At this point, the spring has been stretched or compressed from its original equilibrium position due to the added mass. The spring is now in a new equilibrium position, and it has experienced an elongation or compression.

3. Raising the Mass (Not Shown in Figures):

- Referring to the right-most figure (not explicitly shown), you mentioned that the mass is raised until the spring returns to its original length. This means the spring is being stretched beyond its new equilibrium position.

4. Release and Oscillations (Right-Most Figure):

- After the mass is raised beyond the new equilibrium position and then released from rest, vertical oscillations occur.

- These oscillations are a result of the spring's attempt to return to its equilibrium position. The mass will move up and down around the new equilibrium position, creating a simple harmonic motion.

- The mass will pass through the equilibrium position multiple times, with each oscillation being approximately symmetrical.

Key points to note:

- The spring is acting as a restoring force, trying to bring the mass back to its equilibrium position.

- The mass experiences periodic motion (oscillation) around this new equilibrium position.

- The frequency and amplitude of these oscillations depend on factors like the mass of the object, the spring constant (stiffness) of the spring, and any initial conditions (initial displacement or velocity) when released.