A mass of 3.4 kg is released down a frictionless slope from...

A mass of $3.4 \mathrm{kg}$ is released down a frictionless slope from a height of $h=2.0 \mathrm{m} .$ When it reaches the bottom of the slope, it undergoes a completely inelastic collision with a mass of 1.1 kg attached to a spring of spring constant $k=5.0 \mathrm{kN} / \mathrm{m}$ as illustrated in Eigure $\mathrm{P} 11.66 .$ The combined mass then undergoes periodic motion. Calculate a. the maximum velocity, b. the frequency, c. the maximum amplitude, and d. the period. e. Determine expressions for position as a function of time and velocity as a function of time for the combined mass of this oscillating system.

A mass of $3.4 \mathrm{kg}$ is released down a frictionless slope from a height of $h=2.0 \mathrm{m} .$ When it reaches the bottom of the slope, it undergoes a completely inelastic collision with a mass of 1.1 kg attached
to a spring of spring constant $k=5.0 \mathrm{kN} / \mathrm{m}$ as illustrated in Eigure $\mathrm{P} 11.66 .$ The combined mass then undergoes periodic motion. Calculate
a. the maximum velocity,
b. the frequency,
c. the maximum amplitude, and
d. the period.
e. Determine expressions for position as a function of time and velocity as a function of time for the combined mass of this oscillating system.

Key Concepts

Oscillatory Motion Parameters

The parameters of oscillatory motion include angular frequency, frequency, amplitude, period, and maximum velocity. For a mass-spring system, the angular frequency is determined by the spring constant and the mass involved, the period is the inverse of the frequency, and the maximum velocity occurs at the equilibrium position. These parameters are fundamental for characterizing the motion of oscillatory systems.

Equations of Motion for SHM

The differential equations governing SHM provide expressions for displacement as a function of time and velocity as a function of time, usually given by sinusoidal functions. These equations encapsulate the system’s dynamics, where the displacement is typically described by a cosine (or sine) function and the velocity by its derivative. These expressions allow for a complete description of the motion over time.

Simple Harmonic Motion (SHM)

SHM describes the periodic motion of a mass attached to a spring, where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction. The oscillating system in this scenario undergoes SHM, which allows for the determination of parameters such as maximum velocity, amplitude, angular frequency, period, and frequency.

Conservation of Energy

This concept refers to the idea that the total mechanical energy of a system remains constant when only conservative forces are acting. In problems where an object moves down a frictionless incline, gravitational potential energy is completely converted into kinetic energy. This conversion is used to calculate the velocity at the bottom of the incline before any collisions or further energy transformations take place.

Momentum Conservation in Inelastic Collisions

In a completely inelastic collision, two colliding bodies stick together after the collision. While mechanical energy is not conserved in such collisions, the total linear momentum is conserved. This principle is applied to determine the velocity of the combined mass immediately after the collision between the moving body and the stationary mass attached to the spring.

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