Problem 51 - College Physics

Question

\(\bullet\) \(\bullet\) When a piece of wood is pressed against a spring and com- presses the spring by \(5.0 \mathrm{cm},\) the wood gains a maximum kinetic energy \(K\) when it is released. How much kinetic energy (in terms of \(K\) would the piece of wood gain if the spring were compressed 10.0 \(\mathrm{cm}\) instead?

Step-by-Step Solution

Answer

The kinetic energy would be quadruple, or \(4K\).

Step 1: Understanding Spring Compression and Potential Energy

The kinetic energy gained by the piece of wood when released from the spring is due to the potential energy stored in the spring. This potential energy is given by the formula \( U = \frac{1}{2} kx^2 \), where \( k \) is the spring constant and \( x \) is the compression distance.

Step 2: Calculate Potential Energy in Terms of Kinetic Energy K

When the spring is compressed by \(5 \; \text{cm} = 0.05 \; \text{m}\), the kinetic energy \( K \) is equal to the potential energy: \( K = \frac{1}{2} k (0.05)^2 \).

Step 3: Express Kinetic Energy for 10 cm Compression

For a compression of 10 cm, which is \(0.10 \; \text{m}\), the potential energy becomes \( U_{10} = \frac{1}{2} k (0.10)^2 \).

Step 4: Relating New Compression to Original Compression

By comparing the potential energies: \( U_{10} = \frac{1}{2} k (0.10)^2 = \frac{1}{2} k \cdot 4(0.05)^2 \), which simplifies to \( U_{10} = 4 \cdot \frac{1}{2} k (0.05)^2 = 4K \).

Step 5: Calculate The Kinetic Energy for the New Compression

Since all potential energy is converted to kinetic energy when the wood is released (ignoring losses), the new kinetic energy is \( 4K \).

Key Concepts

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. In this exercise, when a piece of wood is released from a compressed spring, it moves and gains kinetic energy. This movement comes from the conversion of potential energy that was initially stored in the spring.
The formula for kinetic energy is given by \[ KE = \frac{1}{2} mv^2 \] where:

• \emph{m} is the mass of the object
• \emph{v} is the velocity of the object

In the context of a spring system, the kinetic energy is equal to the potential energy that is stored in the spring due to compression. Hence, the energy stored when the spring is compressed is released as kinetic energy when the object is allowed to move. Kinetic energy, unlike potential energy, depends on the velocity of the moving object, so it varies as the object speeds up or slows down. In this problem, analyzing the transition of stored energy to kinetic energy helps us understand how compression affects the object's ability to move.

Spring Compression

Spring compression occurs when a spring is pressed together, reducing its length and storing potential energy. This potential energy is calculated using Hooke's Law for springs, which states that the force required to compress or extend a spring by some distance \( x \) is proportional to that distance.
When a spring is compressed, it stores energy in a manner that can be calculated using the equation for potential energy:\[ U = \frac{1}{2} k x^2 \]where:

• \( U \) is the potential energy
• \( k \) is the spring constant
• \( x \) is the displacement from the spring's natural length

This stored potential energy can be fully converted into kinetic energy when the spring is released, propelling the attached object. The greater the compression, the more potential energy is stored, and hence more kinetic energy is available for the object when released.In the exercise, we see how compressing the spring from 5 cm to 10 cm results in four times the potential energy, and thus, four times the kinetic energy when the spring is released. This is because the compression distance is squared in the formula, making the energy change quadratically with the compression length.

Spring Constant

The spring constant, denoted as \( k \), is a measure of a spring's stiffness. It tells us how much force is needed to compress or stretch the spring by a certain amount. In the formula \( F = kx \), \( k \) represents the constant of proportionality. It's expressed in units of force per unit length (e.g., Newtons per meter).
Higher spring constants mean that a spring is harder to compress or stretch. Consequently, springs with higher \( k \) values store more potential energy for a given compression distance than springs with lower \( k \) values.In our exercise, the spring constant plays a crucial role in calculating the potential energy stored during compression:\[ U = \frac{1}{2} k x^2 \]This equation shows that potential energy is directly proportional to the spring constant.
Understanding the spring constant helps in analyzing how different springs react under compression. A higher spring constant results in a steeper increase in potential energy with more compression, therefore affecting the amount of kinetic energy an object can gain once released.