Question

A block of mass \(0.773 \mathrm{~kg}\) on a spring with spring constant \(239.5 \mathrm{~N} / \mathrm{m}\) oscillates vertically with amplitude \(0.551 \mathrm{~m}\). What is the speed of this block at a distance of \(0.331 \mathrm{~m}\) from the equilibrium position?

Short answer

Answer: The speed of the block at a distance of 0.331 meters from the equilibrium position is 7.334 m/s.

Step-by-step solution

Calculate the initial total mechanical energy

Initially, when the spring is fully compressed or stretched, the block is at the amplitude (\(A = 0.551 m\)) and has maximum potential energy and zero kinetic energy. The potential energy of the spring can be found using Hooke's Law: \(PE = \frac{1}{2} kA^2\), where \(k\) is the spring constant and \(A\) is the amplitude. Given: \(k = 239.5 \mathrm{~N}/\mathrm{m}\) \(A = 0.551 \mathrm{~m}\) \(PE = \frac{1}{2} (239.5 \mathrm{~N}/\mathrm{m}) (0.551 \mathrm{~m})^2 = 36.398 \mathrm{~J}\) At the amplitude, the initial total mechanical energy is only potential energy: \(E_{initial} = PE = 36.398 \mathrm{~J}\)

Calculate total mechanical energy at the given distance

Let's call \(x\) the distance from the equilibrium position at which we want to find the speed of the block. Given: \(x = 0.331 \mathrm{~m}\) At this point, the total mechanical energy is the sum of potential energy \(PE_x\) and kinetic energy \(KE_x\): \(E_x = PE_x + KE_x\) Since the energy is conserved, we can say that: \(E_{initial} = E_x\)

Calculate the potential energy at the given distance

The potential energy at distance \(x\) can be found using Hooke's Law: \(PE_x = \frac{1}{2} kx^2\) \(PE_x = \frac{1}{2} (239.5 \mathrm{~N}/\mathrm{m}) (0.331 \mathrm{~m})^2 = 13.078 \mathrm{~J}\)

Find the kinetic energy at the given distance

From the conservation of energy equation (\(E_{initial} = E_x\)), we can find the kinetic energy at distance \(x\): \(KE_x = E_{initial} - PE_x\) \(KE_x = 36.398 \mathrm{~J} - 13.078 \mathrm{~J} = 23.320 \mathrm{~J}\)

Calculate the speed of the block at the given distance

We can use the kinetic energy formula to find the speed of the block: \(KE_x = \frac{1}{2} mv^2\), where \(m\) is the mass of the block and \(v\) is the speed. \(v = \sqrt{\frac{2 KE_x}{m}}\) Given: \(m = 0.773 \mathrm{~kg}\) \(v = \sqrt{\frac{2 (23.320 \mathrm{~J})}{0.773 \mathrm{~kg}}} = 7.334 \mathrm{~m/s}\) The speed of the block at a distance of \(0.331 \mathrm{~m}\) from the equilibrium position is \(7.334 \mathrm{~m/s}\).

Key concepts

Mechanical Energy Conservation

Mechanical energy conservation is a fundamental concept in physics that states that the total mechanical energy of an isolated system remains constant, provided that the only forces doing work are conservative forces. In the context of a spring-mass system, this principle tells us that the total energy remains constant as the block oscillates up and down.

In our exercise, this total energy is initially in the form of potential energy when the spring is either fully compressed or stretched. As the block moves through its motion, some of this potential energy is converted into kinetic energy. Nevertheless, the sum of potential and kinetic energy at any point in time equals the initial total energy at the maximum amplitude.

This principle allows us to calculate various unknowns in a system by equating the total energy at different stages of motion. For instance, knowing the initial energy and the energy configuration at a specific point helps us determine the speed or kinetic energy of the block at that point.

Spring-Mass System

A spring-mass system is a classic example of a harmonic oscillator. It consists of a mass attached to a spring, capable of moving back and forth when displaced from its equilibrium position. This system is commonly used to demonstrate the principles of simple harmonic motion (SHM).

The system undergoes oscillations in a repetitive manner, described by sinusoidal functions for position, velocity, and acceleration with time. The spring-mass system has inherent characteristics, such as amplitude, frequency, and period, that determine how the system behaves.

• **Amplitude (A)** is the maximum displacement from the equilibrium position.
• **Frequency (f)** is the number of oscillations per unit of time.
• **Period (T)** is the time it takes for one complete oscillation.

In the exercise, the spring-mass system consists of a block with mass 0.773 kg and a spring with a spring constant of 239.5 N/m. Understanding the dynamics of this system is essential for solving problems related to oscillations, like finding the speed of the block at a specific displacement from equilibrium.

Hooke's Law

Hooke's Law is a principle of physics that states the force exerted by a spring is proportional to its displacement from its natural length. It is given by the formula:\[ F = -kx \]where:

• \( F \) is the force exerted by the spring (negative indicates a restoring force)
• \( k \) is the spring constant, indicating the stiffness of the spring
• \( x \) is the displacement from the equilibrium position

Hooke's Law is integral to understanding potential energy in spring-mass systems. The potential energy stored in a spring is determined by:\[ PE = \frac{1}{2} kx^2 \]In the given problem, Hooke's Law allows us to calculate the potential energy at different displacements from equilibrium, an essential step in using energy conservation to find the velocity of the block under these conditions. It highlights the relationship between force, energy, and displacement, showing how they are intertwined in the behavior of oscillating systems.