A horizontal spring with a spring constant of 22 N/m has a 600 gram block attached to it and is at rest on a frictionless surface. A second block which has a mass of 220 grams is pushed toward the 600 gram block at a speed of 1.50 m/s. The second block collides with and sticks to the 600 gram block.
a. What is the amplitude of the subsequent oscillation?
b. What is the period of the subsequent oscillation?
To determine the amplitude and period of the subsequent oscillation, we need to consider the conservation of energy and the laws of motion.
First, let's find the total mechanical energy before the collision. The energy of the system is given by the sum of the kinetic energy and potential energy of the spring:
1. Calculate the initial kinetic energy:
- Mass of the first block (m₁) = 600 grams = 0.6 kg
- Initial velocity of the second block (v₂) = 1.50 m/s
- Kinetic energy = (1/2) * m₂ * v₂², where m₂ is the mass of the second block
- Since the second block collides and sticks to the first block, the initial kinetic energy of the system (before the collision) is equal to the kinetic energy of the second block: K_initial = (1/2) * 0.22 kg * (1.50 m/s)²
2. Calculate the potential energy of the spring:
- Spring constant (k) = 22 N/m
- Amplitude (A) is the maximum displacement of the system from its equilibrium position, which we need to determine.
- The potential energy of the spring (U_spring) can be given by the formula: U_spring = (1/2) * k * A²
To find the amplitude of the subsequent oscillation (a), we need to calculate the maximum displacement of the system from its equilibrium position:
1. Start by finding the maximum displacement of the center of mass of the two blocks:
- According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision, as there are no external forces acting on the system.
- Initial momentum = Final momentum
- (m₁ + m₂) * v_initial = (m₁ + m₂) * v_final
- Since the second block sticks to the first block, the velocity of the system after the collision is the velocity of the center of mass of the two blocks.
- m₁ * v_initial = (m₁ + m₂) * v_cm
- v_cm = (m₁ * v_initial) / (m₁ + m₂)
- Maximum displacement = (m₂ / m₁) * v_cm * T, where T is the period of the subsequent oscillation
2. Use the formula for the period of a mass-spring system:
- Period (T) = 2π * sqrt(m_eff / k), where m_eff is the effective mass of the system.
- m_eff = m₁ + m₂, as both blocks oscillate together after sticking.
Now we can calculate the amplitude of the subsequent oscillation (a) and the period of the subsequent oscillation (T).
a. Calculate the amplitude:
- Set the initial kinetic energy equal to the potential energy of the spring: K_initial = U_spring, and solve for A.
b. Calculate the period:
- Substitute m_eff = m₁ + m₂ into the formula and solve for T.
Finally, substitute the given values into the derived equations and solve for the amplitude (a) and period (T) of the subsequent oscillation.