Question

A spring of negligible mass and force constant $k=$ 400 N/m is hung vertically, and a 0.200-kg pan is suspended from its lower end. A butcher drops a 2.2-kg steak onto the pan from a height of 0.40 m. The steak makes a totally inelastic collision with the pan and sets the system into vertical SHM. What are (a) the speed of the pan and steak immediately after the collision; (b) the amplitude of the subsequent motion; (c) the period of that motion?

Short answer

(a) 2.57 m/s; (b) 0.263 m; (c) 0.487 s.

Step-by-step solution

Determine Potential Energy of the Steak

First, calculate the potential energy of the steak before it is dropped. The potential energy (PE) is calculated using the formula:$$PE=mgh$$where $m=2.2\text{kg}$, $g=9.81{\text{m/s}}^2$, and $h=0.40\text{m}$. Plug these values into the formula to find:$$PE=2.2\times 9.81\times 0.40=8.6\text{Joules}$$.

Calculate Speed After Collision Using Energy Conservation

With the potential energy we calculated, we'll determine the speed just before the collision. Use the conservation of energy principle where the entire potential energy converts to kinetic energy (KE) just before hitting the pan. Let $v$ be the speed; the equation becomes:$$\frac{1}{2}m{v}^2=8.6$$Plug in $m=2.2\text{kg}$ and solve for $v$:$$v=\sqrt{\frac{2\times 8.6}{2.2}}=2.8\text{m/s}$$.

Use Conservation of Momentum for Speed After Collision

Next, apply the conservation of linear momentum to find the speed of the pan and steak immediately after the collision. Let $V$ be the speed post-collision:$$m_1{v}_1+m_2{v}_2=(m_1+m_2)V$$where $m_1=0.200\text{kg}$, $m_2=2.2\text{kg}$, $v_1=0\text{m/s}$, and $v_2=2.8\text{m/s}$. Solve for $V$:$$V=\frac{2.2\times 2.8}{2.4}=2.5667\text{m/s}$$.

Determine Amplitude of Motion

Use energy conservation between kinetic and elastic potential energy to find the amplitude ($A$) of the motion after the collision. The total kinetic energy after collision (now in the form of elastic potential energy) is:$$\frac{1}{2}(m_1+m_2)V^2=\frac{1}{2}k{A}^2$$Insert known values $k=400\text{N/m}$, $V=2.5667\text{m/s}$. Solve for $A$:$$A=\sqrt{\frac{(0.200+2.2)\times {2.5667}^2}{400}}=0.263\text{m}$$.

Calculate Period of Simple Harmonic Motion

Finally, calculate the period using the formula for the period $T$ of a mass-spring system:$$T=2\pi \sqrt{\frac{M}{k}}$$where $M=m_1+m_2=2.4\text{kg}$. Substitute the values:$$T=2\pi \sqrt{\frac{2.4}{400}}=0.487\text{s}$$.

Key concepts

Conservation of Energy

The concept of conservation of energy is essential when analysing systems like mass-spring systems. It states that the total energy in a closed system remains constant, though it may change forms. In this context, potential energy (PE) can convert into kinetic energy (KE), and vice-versa. For example, before the steak hits the pan, it possesses gravitational potential energy due to its height. As it falls, this energy transforms into kinetic energy at the moment of impact. Post-collision, the energy further transitions into elastic potential energy stored in the spring, initiating simple harmonic motion. Understanding these transformations helps predict the system's behaviour, such as the speed of the pan, the amplitude of motion, and overall mechanical energy during the motion.
Key Points:

• Energy transitions within a system can be used to predict motion.
• Gravitational potential energy changes to kinetic energy during free fall.
• Post-collision energy is stored as elastic potential energy in the spring.

Momentum Conservation

Conservation of momentum is crucial in understanding the motion post-collision between the steak and the pan. Momentum, a product of mass and velocity, is conserved in isolated systems, regardless of the interactions occurring internally. During the impact, the momentum of the falling steak transfers to the combined mass of the steak and pan, allowing us to calculate the post-collision velocity. Initial momentum was contained solely by the steak, but after impact, the momentum redistributes between both the pan and the steak. Knowing this principle allows us to solve for quantities like velocity after collision by equating the initial and final momentum.
Key Points:

• Momentum is conserved in isolated systems.
• Calculating post-collision velocities involve initial momentum considerations.
• This principle helps predict velocities and directions in collisions.

Mass-Spring System

The mass-spring system described here involves a spring of known spring constant and a combined mass of the pan and steak. Such systems are classic examples of simple harmonic motion (SHM), where the mass's oscillations around an equilibrium position are influenced by the spring's elasticity. The motion's analysis typically involves understanding how potential and kinetic energy interchangeably manifest in the system. As the pan and steak system oscillates, the energy continues to alternate between kinetic and elastic potential, with the spring's restorative force acting to return the system to equilibrium. This cycle repeats consistently given no external damping forces.
Important Considerations:

• Mass-spring systems are ideal for studying simple harmonic motion.
• Understanding spring constant and mass helps predict oscillation characteristics.
• The system's energy alternates between kinetic and elastic potential.

Kinetic Energy

Kinetic energy is the energy of motion and becomes particularly significant as the steak falls and impacts the pan. Initially, the falling steak converts all its gravitational potential energy into kinetic energy, gaining speed in accordance with energy conservation. After the inelastic collision with the pan, energy conservation principles guide us to consider the kinetic energy of the whole system (pan and steak combined) to predict future motion, like velocity or amplitude of oscillations. The essential aspect of kinetic energy in this scenario involves understanding how potential energy transition results in a velocity that is crucial for determining further dynamic properties of motion.
Key Insights:

• Kinetic energy represents motion within the system.
• The conversion from potential to kinetic energy is a key step in falling bodies.
• Post-collision, kinetic energy influences further motion characteristics.

Elastic Potential Energy

In the mass-spring system setup, elastic potential energy becomes a central component post-collision. Upon impact, the kinetic energy of the steak and pan is transferred into the spring, compressing it, and converting kinetic energy into elastic potential energy. This energy is stored in the spring as it deforms, and is directly related to the spring constant and the displacement (amplitude of motion). Elastic potential energy provides the force needed to bring the system back to equilibrium, perpetuating the simple harmonic motion. Understanding how energy is stored and released by the spring is crucial for analyzing oscillatory systems like this.
Key Elements:

• Elastic potential energy is stored when spring is compressed or stretched.
• The spring constant affects how much energy is stored.
• The energy contributes to restoring force, driving the system's motion.