Chapter 14: Problem 35
A 2.00-kg frictionless block attached to an ideal spring with force constant 315 N/m is undergoing simple harmonic motion. When the block has displacement \(+\)0.200 m, it is moving in the negative \(x\)-direction with a speed of 4.00 m/s. Find (a) the amplitude of the motion; (b) the block's maximum acceleration; and (c) the maximum force the spring exerts on the block.
Short Answer
Step by step solution
Understand the Problem
Use Energy Conservation to find Amplitude
Calculate Maximum Acceleration
Determine Maximum Force Exerted by the Spring
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Spring Constant
In the case of simple harmonic motion, as we have with the block attached to the spring, the spring constant plays a crucial role. The block's interaction with the spring is what allows it to oscillate, creating the phenomena of simple harmonic motion. The spring constant here is given as 315 N/m, meaning for every meter the spring is stretched, a force of 315 newtons is exerted.
- A higher spring constant means a stiffer spring.
- A lower spring constant indicates a more flexible spring.
Amplitude
Finding amplitude involves using the conservation of energy principle. Energy in the system is conserved and oscillates between kinetic and potential energy.
- Kinetic Energy (KE): \( \frac{1}{2}mv^2 \)
- Potential Energy (PE): \( \frac{1}{2}kx^2 \)
When the block is at its maximum distance from the center, the speed is zero, and all energy is potential: \( \frac{1}{2}kA^2 \). Using known parameters, we solve for \( A \) to find how far the block moves from its resting point in either direction.
Maximum Acceleration
The formula used to find maximum acceleration, \( a_{max} \), is: \[ a_{max} = \frac{k}{m} \times A \] where \( k \) is the spring constant, \( m \) is the mass of the block, and \( A \) is the amplitude.
This formula tells us that the greater the spring constant and the displacement (amplitude combined), the higher the acceleration:
- \( a_{max} \propto k \) means stiffness increases acceleration.
- \( a_{max} \propto A \) means greater displacement increases acceleration.
Conservation of Energy
Even though kinetic and potential energies fluctuate, their sum does not change, as shown in the equation:\[ E = \frac{1}{2}kx^2 + \frac{1}{2}mv^2 \] This essential formula helps determine various aspects of motion, such as amplitude.
- At maximum speed, energy is purely kinetic.
- At maximum displacement, energy is purely potential.