Problem 32 - University Physics with Modern Physics

Question

On a horizontal, frictionless table, an open-topped \(5.20-\mathrm{kg}\) box is attached to an ideal horizontal spring having force constant 375 \(\mathrm{N} / \mathrm{m}\) . Inside the box is a 3.44 -kg stone. The system is oscillating with an amplitude of 7.50 \(\mathrm{cm} .\) When the box has reached its maximum speed, the stone is suddenly plucked vertically out of the box without touching the box. Find (a) the period and (b) the amplitude of the resulting motion of the box. (c) Without doing any calculations, is the new period greater or smaller than the original period? How do you know?

Step-by-Step Solution

Answer

(a) Original period: 0.953 s. (b) New amplitude: 9.0 cm. (c) New period is smaller.

Step 1: Determine the Original Period

The original period of oscillation for a mass-spring system is given by the formula \( T = 2\pi \sqrt{\frac{m}{k}} \) where \( m \) is the total mass and \( k \) is the spring constant. Here, the total mass is the sum of the box and the stone, \( (5.20 + 3.44) \text{ kg} = 8.64 \text{ kg} \) and \( k = 375 \text{ N/m} \). Plug these values into the formula to find the period:\[ T = 2\pi \sqrt{\frac{8.64}{375}} \approx 0.953 \text{ s} \].

Step 2: Calculate the Maximum Speed Before the Stone is Plucked

The maximum speed \( v_{max} \) of an object in simple harmonic motion is given by \( v_{max} = A \omega \), where \( A \) is the amplitude and \( \omega = \sqrt{\frac{k}{m}} \). First calculate \( \omega \):\[ \omega = \sqrt{\frac{375}{8.64}} \approx 6.58 \text{ rad/s} \].Then, find the maximum speed:\[ v_{max} = 0.075 \text{ m} \times 6.58 \text{ rad/s} \approx 0.494 \text{ m/s} \].

Step 3: Determine the New Mass and New Period

Once the stone is removed, the new mass \( m' \) is just the mass of the box, \( 5.20 \text{ kg} \). The new period \( T' \) is:\[ T' = 2\pi \sqrt{\frac{5.20}{375}} \approx 0.746 \text{ s} \].

Step 4: Calculate the Amplitude of Motion After the Stone is Plucked

Upon removing the stone at maximum speed, the energy of the system should be conserved. This energy is initially stored as kinetic energy: \( KE = \frac{1}{2} m v_{max}^2 \). Use this to find the new amplitude, \( A' \).Given: \( KE = \frac{1}{2} \times 8.64 \times 0.494^2 \approx 1.054 \text{ J} \).When the stone is removed, this energy becomes:\[ KE = \frac{1}{2} imes 5.20 \times v'^2 \]and let \( v' = A' \omega' \), where \( \omega' = \sqrt{\frac{375}{5.20}} \approx 8.49 \text{ rad/s} \).Thus, solve for \( A' \):\[ 1.054 = \frac{1}{2} \times 5.20 \times (A' \cdot 8.49)^2 \]Solving for \( A' \), we find that:\[ A' \approx 0.090 \text{ m} \text{ or } 9.0 \text{ cm} \].

Step 5: Analyze the Change in Period

By comparing the original and new periods, note that the new period \( T' \approx 0.746 \text{ s} \) is less than the original period \( T \approx 0.953 \text{ s} \). This occurs because the system's total mass after removing the stone is less, reducing the period according to the formula \( T = 2\pi \sqrt{\frac{m}{k}} \).

Key Concepts

Mass-Spring System

A mass-spring system is a fundamental model in physics used to understand simple harmonic motion. Imagine a scenario where you attach a mass, like a box, to a spring. The mass can move back and forth as the spring pushes and pulls it. In our initial setup, the box with the stone inside it represents the mass. The spring connected to this setup has a known force constant, which measures the stiffness of the spring.

When you pull the box and let it go, the spring stores potential energy. The system starts to oscillate, or move back and forth, due to the force exerted by the spring. This movement is characteristic of simple harmonic motion. The force constant of the spring, combined with the mass, determines how quickly the system oscillates. The more stiff the spring or the heavier the mass, the slower the oscillation will be.

• System returns to equilibrium due to spring forces.
• Mass oscillates with a specific period and amplitude.
• System's behavior is predictable and periodic.

Oscillation Period

The oscillation period, represented as \( T \), is the time it takes for one complete cycle of motion in a mass-spring system. Understanding the period helps us know how quickly a mass attached to a spring will go from its starting point and come back.

For a mass-spring system, the formula to find the period is \( T = 2\pi \sqrt{\frac{m}{k}} \). Here, \( m \) is the total mass, and \( k \) is the spring constant. In the given problem, the initial period with the box and the stone is approximately 0.953 seconds. This value changes when the stone is removed, becoming older at 0.746 seconds because the system's total mass decreases.

• Period depends on both mass and spring stiffness.
• Decreasing mass results in a shorter period.
• Spring constant remains unchanged.

Amplitude of Motion

Amplitude of motion refers to the maximum distance the mass moves from its equilibrium position during oscillation. For our mass-spring system, this distance was initially 7.50 cm when the box with the stone is considered.

When the stone is removed, the system still conserves energy, impacting the new amplitude. Energy conservation requires that the kinetic energy when the stone is pulled out remains in the mass left behind. This results in a new amplitude of approximately 9.0 cm, slightly larger due to less mass, allowing the box to move further with the same energy.

• Larger amplitude means greater distance from rest position.
• Energy distribution changes as mass changes.
• Conserved energy ensures consistent motion.

Energy Conservation in Physics

Energy conservation in physics is a principle that the total energy in a closed system remains constant. For a mass-spring system, this principle plays a crucial role, especially when examining changes like the one in this problem.

Initially, the energy of the system includes both potential energy in the stretched spring and kinetic energy as the mass moves. When the stone is removed, the total energy doesn't immediately change, but how it's partitioned goes primarily to kinetic energy, influencing the new amplitude.

• Total energy transforms but never disappears.
• Kinetic and potential energies trade off during oscillation.
• Mass removal redistributes energy, impacting motion behavior.