Problem 23 - University Physics with Modern Physics

Question

A 2.50 -kg mass is pushed against a horizontal spring of force constant 25.0 \(\mathrm{N} / \mathrm{cm}\) on a frictionless air table. The spring is attached to the tabletop, and the mass is not attached to the spring in any way. When the spring has been compressed enough to store 11.5 J of potential energy in it, the mass is suddenly released from rest. (a) Find the greatest speed the mass reaches. When does this occur? (b) What is the greatest acceleration of the mass, and when does it occur?

Step-by-Step Solution

Answer

The greatest speed is 3.03 m/s when the spring is at equilibrium. The greatest acceleration is 96 m/s² when the spring is maximally compressed.

Step 1: Understand the Potential Energy Stored in the Spring

The potential energy stored in a compressed or stretched spring is given by \[ U = \frac{1}{2} k x^2 \]where \( U \) is the potential energy, \( k \) is the spring constant, and \( x \) is the compression of the spring. We're given that \( U = 11.5 \, \text{J} \), and \( k = 25.0 \, \text{N/cm} \) or \( 2500 \, \text{N/m} \) after converting to SI units.

Step 2: Solve for the Compression of the Spring

Rearrange the equation for potential energy to solve for \( x \):\[ 11.5 = \frac{1}{2} (2500) x^2 \]\[ x^2 = \frac{11.5 \times 2}{2500} = 0.0092 \]\[ x = \sqrt{0.0092} \approx 0.096 \, \text{m} \].The spring is compressed 0.096 meters.

Step 3: Calculate the Maximum Speed of the Mass

The work done by the spring or the potential energy stored in the spring converts entirely into kinetic energy when the spring is fully relaxed. The kinetic energy (KE) is \[ KE = \frac{1}{2} mv^2 \]where \( m = 2.50 \, \text{kg} \) is the mass. Set potential energy equal to kinetic energy:\[ 11.5 = \frac{1}{2} \times 2.50 \times v^2 \]Solving for \( v \) gives:\[ 11.5 = 1.25v^2 \]\[ v^2 = \frac{11.5}{1.25} = 9.2 \]\[ v = \sqrt{9.2} \approx 3.03 \, \text{m/s} \].The greatest speed is 3.03 m/s and this occurs when the spring returns to its equilibrium position.

Step 4: Calculate the Maximum Acceleration of the Mass

The maximum acceleration happens when the force is maximum, which occurs when the spring is compressed the most (at the start). Force \( F \) exerted by the spring is given by Hooke's Law: \[ F = kx \]The acceleration \( a \) is given by \[ a = \frac{F}{m} \]Substitute maximally compressed \( x = 0.096 \, \text{m} \):\[ F = 2500 \times 0.096 = 240 \, \text{N} \]\[ a = \frac{240}{2.50} = 96 \, \text{m/s}^2 \].The greatest acceleration is 96 m/s², and it occurs when the spring is maximally compressed.

Key Concepts

Potential Energy

Potential energy is the stored energy in an object due to its position or configuration. In the context of springs, it refers to the energy stored when a spring is compressed or stretched. The formula for potential energy in a spring is given by \[ U = \frac{1}{2} k x^2 \] where

• \( U \) is the potential energy, measured in joules (J),
• \( k \) is the spring constant, representing the stiffness of the spring, measured in newtons per meter (N/m),
• \( x \) is the displacement from the spring's equilibrium position, measured in meters (m).

The more a spring is compressed or stretched, the more potential energy it stores. In the given problem, the spring has been compressed enough to store 11.5 J of potential energy. This energy is later converted to kinetic energy when the spring returns to its equilibrium position.

Hooke's Law

Hooke's Law describes the behavior of springs. It states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. Mathematically, it is expressed as: \[ F = kx \] where

• \( F \) is the force applied by the spring, measured in newtons (N),
• \( k \) is the spring constant, an indicator of how stiff the spring is,
• \( x \) is the displacement from equilibrium, measured in meters (m).

In the problem, when the spring is initially compressed by 0.096 meters, it exerts a force, calculated using Hooke's Law, that pushes the mass. This force is maximum when the spring is maximally compressed.

Kinetic Energy

Kinetic energy is the energy of motion. When an object is moving, it possesses kinetic energy, which is given by the formula: \[ KE = \frac{1}{2} mv^2 \] where

• \( KE \) is the kinetic energy, measured in joules (J),
• \( m \) is the mass of the object, measured in kilograms (kg),
• \( v \) is the velocity of the object, measured in meters per second (m/s).

In this exercise, the potential energy stored in the spring is converted into kinetic energy, propelling the mass. This happens as the spring releases its stored energy, resulting in the greatest speed (3.03 m/s) of the mass when the spring returns to its equilibrium position.

Spring Constant

The spring constant, denoted as \( k \), is a key property of a spring that defines its stiffness. It is a measure of the force required to compress or stretch the spring by a unit length. The unit for the spring constant is newtons per meter (N/m). In our exercise, the spring constant is given as 25.0 N/cm, which is converted to 2500 N/m for calculation purposes. This conversion is necessary because working with SI units (meters) simplifies mathematical calculations and ensures uniformity in physics equations. The spring constant plays a crucial role in determining the behavior of the spring according to Hooke's Law, as well as in calculating the potential energy stored when the spring is compressed.