Two 500 g air-track gliders are each connected by identical springs with spring constant 25 N/m to the ends of the air track. The gliders are connected to each other by a spring with spring constant 2.0 N/m. One glider is pulled 8.0 cm to the side and released while the other is at rest at its equilibrium position. How long will it take until the glider that was initially at rest has all the motion while the first glider is at rest?

Verified step by step guidance

1. Step 1: Identify the system as two coupled oscillators. Each glider is connected to springs, and the motion of one glider affects the other. The spring constants and masses are given, so we can calculate the angular frequencies of the normal modes of oscillation.
2. Step 2: Write the equations of motion for the two gliders. Let the displacements of the two gliders from their equilibrium positions be x₁ and x₂. Using Hooke's law and Newton's second law, the forces on each glider can be expressed as: mx₁ = -k₁x₁ + k₂(x₂ - x₁) and mx₂ = -k₃x₂ + k₂(x₁ - x₂), where k₁, k₂, and k₃ are the spring constants.
3. Step 3: Solve for the normal modes of the system. Assume solutions of the form x₁(t) = A₁cos(ωt) and x₂(t) = A₂cos(ωt). Substitute these into the equations of motion to derive a system of equations for ω (angular frequency) and the amplitude ratios A₁/A₂. Solve the determinant of the coefficient matrix to find the normal mode frequencies.
4. Step 4: Determine the time period of the beat frequency. The beat frequency arises from the superposition of the two normal modes. The beat period is given by T_beat = 2π/|ω₁ - ω₂|, where ω₁ and ω₂ are the angular frequencies of the two normal modes.
5. Step 5: Calculate the time it takes for energy transfer. The time it takes for the initially stationary glider to have all the motion corresponds to half the beat period, T_transfer = T_beat / 2. Use the values of ω₁ and ω₂ derived earlier to compute this time.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. In this scenario, the gliders connected by springs will exhibit SHM due to the restoring force provided by the springs. The motion is characterized by a constant frequency and amplitude, which can be determined by the mass of the gliders and the spring constants.

Spring Constant and Hooke's Law

The spring constant, denoted as 'k', quantifies the stiffness of a spring, with higher values indicating stiffer springs. According to Hooke's Law, the force exerted by a spring is proportional to its displacement from the equilibrium position, expressed as F = -kx. In this problem, the different spring constants affect the dynamics of the gliders' motion and the time it takes for them to reach their respective positions.

Energy Conservation in Oscillatory Systems

In oscillatory systems like the one described, mechanical energy is conserved, oscillating between potential energy stored in the springs and kinetic energy of the gliders. When one glider is displaced and released, it converts potential energy into kinetic energy as it moves, while the other glider experiences a change in energy as it begins to oscillate. Understanding this energy transfer is crucial for determining the timing of the gliders' motion.