A 500 g air-track glider attached to a spring with spring constant 10 N/m is sitting at rest on a frictionless air track. A 250 g glider is pushed toward it from the far end of the track at a speed of 120 cm/s. It collides with and sticks to the 500 g glider. What are the amplitude and period of the subsequent oscillations?

Verified step by step guidance

1. Convert all given quantities to SI units. The mass of the first glider is \( m_1 = 0.500 \; \text{kg} \), the mass of the second glider is \( m_2 = 0.250 \; \text{kg} \), the spring constant is \( k = 10 \; \text{N/m} \), and the velocity of the second glider is \( v_2 = 1.20 \; \text{m/s} \).
2. Determine the total mass of the system after the collision. Since the two gliders stick together, the combined mass is \( M = m_1 + m_2 \).
3. Use the principle of conservation of momentum to find the velocity of the combined gliders immediately after the collision. The total momentum before the collision is \( p_{\text{initial}} = m_2 \cdot v_2 \), and the total momentum after the collision is \( p_{\text{final}} = M \cdot v_{\text{final}} \). Solve for \( v_{\text{final}} \) using \( p_{\text{initial}} = p_{\text{final}} \).
4. Determine the amplitude of oscillation. The kinetic energy of the combined gliders immediately after the collision is converted into the potential energy of the spring at maximum compression. Use the equation \( \frac{1}{2} M v_{\text{final}}^2 = \frac{1}{2} k A^2 \), where \( A \) is the amplitude. Solve for \( A \).
5. Calculate the period of oscillation using the formula for the period of a mass-spring system: \( T = 2 \pi \sqrt{\frac{M}{k}} \), where \( M \) is the total mass of the system and \( k \) is the spring constant.

Key Concepts

Conservation of Momentum

In a closed system, the total momentum before a collision is equal to the total momentum after the collision. This principle is crucial for analyzing the collision between the two gliders, as it allows us to determine their combined velocity immediately after they stick together.

Simple Harmonic Motion (SHM)

SHM describes the oscillatory motion of an object when it is displaced from its equilibrium position and experiences a restoring force proportional to that displacement. The gliders will undergo SHM after the collision due to the restoring force provided by the spring, which is characterized by its spring constant.

Amplitude and Period of Oscillation

The amplitude of oscillation is the maximum displacement from the equilibrium position, while the period is the time taken to complete one full cycle of motion. For the combined mass of the gliders and the spring constant, these parameters can be calculated using the formulas for a mass-spring system, which relate mass, spring constant, and energy conservation.