Problem 28 - College Physics

Question

A \(2.00 \mathrm{~kg}\) frictionless block is attached to an ideal spring with force constant \(315 \mathrm{~N} / \mathrm{m} .\) Initially the spring is neither stretched nor compressed, but the block is moving in the negative direction at \(12.0 \mathrm{~m} / \mathrm{s}\) Find (a) the amplitude of the motion, (b) the maximum acceleration of the block, and (c) the maximum force the spring exerts on the block.

Step-by-Step Solution

Answer

Amplitude is 1.35 m, max acceleration is 212.625 m/s², and max force is 425.25 N.

Step 1: Understanding the Problem

We need to find the amplitude, maximum acceleration, and maximum force on a block attached to a spring. We have a mass \( m = 2.00 \, \text{kg} \), spring constant \( k = 315 \, \text{N/m} \), and initial velocity \( v_i = -12.0 \, \text{m/s} \). The spring is neither stretched nor compressed initially.

Step 2: Calculate Amplitude Using Energy Conservation

The total mechanical energy in a spring-mass system is conserved. Initially, all energy is kinetic: \( E = \frac{1}{2}mv_i^2 \). At maximum amplitude, all energy is potential: \( E = \frac{1}{2}kA^2 \). Therefore, set the kinetic energy equal to the potential energy:\[ \frac{1}{2}mv_i^2 = \frac{1}{2}kA^2 \]Solve for amplitude \( A \):\[ A = \sqrt{\frac{m}{k}} \times v_i \] Substituting the values:\[ A = \sqrt{\frac{2.00}{315}} \times 12.0 \approx 1.35 \, \text{m} \]

Step 3: Calculate Maximum Acceleration

Maximum acceleration occurs at maximum displacement (amplitude). The formula for maximum acceleration in a spring system is \( a_{\text{max}} = \frac{kA}{m} \). Substitute the known values:\[ a_{\text{max}} = \frac{315 \times 1.35}{2.00} = 212.625 \, \text{m/s}^2 \]

Step 4: Calculate Maximum Force Exerted by the Spring

The maximum force exerted by the spring is at maximum displacement using Hooke's Law: \( F = kA \). Substitute the known values:\[ F = 315 \times 1.35 = 425.25 \, \text{N} \]

Key Concepts

Amplitude

In simple harmonic motion, the amplitude is a crucial concept and refers to the maximum extent of the oscillation from its equilibrium position. Basically, it's the furthest point that the block and spring will stretch or compress from their starting place. The amplitude is affected by the overall energy in the system. In this example, the energy begins as kinetic because the block is moving, then it swaps to potential energy at the highest stretch/compression.

• For this exercise, the block carries all its energy as kinetic initially since it's on the move.
• Using energy conservation principles, the energy at the start equals the potential energy at maximum displacement.
• This allows the calculation of amplitude through the relationship: \( A = \sqrt{\frac{m}{k}} \times v_i \).

Substituting the numbers, the amplitude of the motion calculates out to approximately \( 1.35 \, \text{m} \). Every time the block reaches this point, all kinetic energy has been swapped into potential. Understanding amplitude helps you predict how far the block will travel before changing direction.

Maximum Acceleration

In simple harmonic motion, the point of maximum acceleration happens at the amplitude positions. This means when the block is either fully compressed or fully stretched. Acceleration is all about how quickly velocity changes, and at these points, velocity shifts most rapidly, driving the highest acceleration.

• Mathematically, maximum acceleration can be found using: \( a_{\text{max}} = \frac{kA}{m} \).
• Placing in our values, we find that the maximum acceleration is about \( 212.625 \, \text{m/s}^2 \).
• This acceleration is directed towards the equilibrium position, trying to bring the block back from its extreme positions.

It's the balance between stretch/compression and returning force that governs acceleration here. Understanding this helps you see how and why things try to return to balance when disturbed.

Spring Force

In this exercise involving a block on a spring, the spring exerts a force which obeys Hooke's Law. Hooke’s Law says this force is directly related to how much the spring is stretched/compressed, expressed as \( F = kA \), where \( F \) is the force, \( k \) is the spring constant, and \( A \) is the amplitude.

• At maximum displacement, or at the amplitude, the spring force reaches its peak value.
• Inserting our numbers, the maximum spring force ends up being around \( 425.25 \, \text{N} \).
• It's this force that generates the maximum acceleration, pushing or pulling the block back to equilibrium.

Spring force is what gives simple harmonic motion its "yo-yo" like behavior, always trying to restore balance by pulling back from stretched or compressed positions. At every point in the motion, the spring force works hand in hand with the mass to dictate the motion entirely.