Problem 31 - Physics

Question

A \(1.00 \times 10^{-2} \mathrm{~kg}\) block is resting on a horizontal frictionless surface and is attached to a horizontal spring whose spring constant is \(124 \mathrm{~N} / \mathrm{m}\). The block is shoved parallel to the spring axis and is given an initial speed of \(8.00 \mathrm{~m} / \mathrm{s}\), while the spring is initially unstrained. What is the amplitude of the resulting simple harmonic motion?

Step-by-Step Solution

Answer

The amplitude of the simple harmonic motion is approximately 0.0718 meters.

Step 1: Understand the Equations of Motion for Simple Harmonic Motion

A block attached to a spring exhibits simple harmonic motion (SHM). The key parameters for SHM are mass \(m\), spring constant \(k\), and amplitude \(A\). The amplitude \(A\) represents the maximum displacement from the equilibrium position during oscillation.

Step 2: Identify Relevant Energy Principles

In the absence of friction, the total mechanical energy is conserved. Initially, the block only has kinetic energy since the spring is unstrained. At maximum amplitude, the speed is zero, and all the energy is potential energy of the spring.

Step 3: Write the Energy Conservation Equation

Set the initial kinetic energy equal to the potential energy at maximum amplitude. The kinetic energy \(E_k\) is given by \(\frac{1}{2} m v^2\) and the spring potential energy \(E_p\) is \(\frac{1}{2} k A^2\). Thus: \[\frac{1}{2} m v^2 = \frac{1}{2} k A^2\]

Step 4: Solve for Amplitude \(A\)

Cancel the \(\frac{1}{2}\) on both sides, and substitute the given values: \(m = 1.00 \times 10^{-2} \text{ kg}\), \(v = 8.00 \text{ m/s}\), and \(k = 124 \text{ N/m}\). The equation becomes: \[m v^2 = k A^2\] which simplifies to: \[A = \sqrt{\frac{m v^2}{k}}\].

Step 5: Calculate the Amplitude

Substitute the values into the formula: \[A = \sqrt{\frac{1.00 \times 10^{-2} \times (8.00)^2}{124}}\]. Calculate the result to find \(A\).

Step 6: Final Calculation

Compute the expression: \(A = \sqrt{\frac{1.00 \times 10^{-2} \times 64}{124}}\). This simplifies to \(A = \sqrt{0.00516}\), resulting in \(A \approx 0.0718 \text{ m}\).

Key Concepts

Mechanical Energy Conservation

In any physical system where only conservative forces are acting, the total mechanical energy remains constant over time. This is an essential principle to understand the dynamics of simple harmonic motion (SHM) in a spring-block system.
Mechanical energy in such systems is the sum of kinetic energy and potential energy. Since the surface is frictionless, energy transformation occurs between kinetic and potential energy without any loss.
- When the spring is pushed or pulled, it stores energy as potential energy. - As the block moves, this potential energy converts to kinetic energy. The principle of mechanical energy conservation helps us determine how energy shifts between these different forms during SHM. The total energy here stays constant because no external forces like friction are acting to remove energy from the system. This concept is crucial for determining the maximum displacement or amplitude of motion.

Spring Constant

The spring constant, denoted by the symbol \(k\), is a measure of a spring's stiffness. It tells us how much force is needed to stretch or compress the spring by a certain amount.
In the given exercise, the spring constant is stated as 124 N/m.

• The larger the spring constant, the stiffer the spring, and the more force is needed for a given displacement.
• Conversely, a smaller constant means a more flexible spring.

The spring constant plays a direct role in determining the potential energy stored in the spring when it is compressed or stretched. This is expressed in the formula for potential energy: \[E_p = \frac{1}{2} k A^2\] where \(A\) is the amplitude. Thus, a high spring constant results in greater potential energy stored for the same amplitude.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. In simple harmonic motion, it is a key component of the system's total mechanical energy.
When the block is initially given a speed of 8.00 m/s, it possesses kinetic energy given by the formula:\[E_k = \frac{1}{2} m v^2\] where \(m\) is the mass of the block and \(v\) is its velocity.

• Initially, all mechanical energy is kinetic since the spring is not yet compressed or stretched.
• As the block moves, this kinetic energy will transform into potential energy as it compresses or stretches the spring, ultimately halting momentarily at the maximum displacement or amplitude.

Understanding kinetic energy is vital, as it is the part of energy that transitions to potential energy, helping to define the system's dynamic motion.

Potential Energy

In simple harmonic motion, potential energy is associated with the configuration of the system. For a spring-mass system, this potential energy (\(E_p\)) is stored in the spring. It is crucial at specific points during the oscillation, particularly at the maximum amplitude when all energy is potential.
Potential energy at these points is given by:\[E_p = \frac{1}{2} k A^2\] This equation demonstrates that potential energy, and thus the energy storage capacity of the spring, increases with the square of the amplitude (\(A\)) and the spring constant (\(k\)).
- When the spring is fully compressed or extended, the velocity of the block is zero, meaning kinetic energy is zero, and all energy is potential.- As the block reaches these points, it temporarily stops moving, converting all its kinetic energy into potential energy.This oscillation of energy backward and forward is what creates the continuous motion characteristic of simple harmonic motion.