Problem 37 - University Physics with Modern Physics

Question

A 6.0-kg box moving at 3.0 \(\mathrm{m} / \mathrm{s}\) on a horizontal, frictionless surface runs into a light spring of force constant 75 \(\mathrm{N} / \mathrm{cm}\) . Use the work-energy theorem to find the maximum compression of the spring.

Step-by-Step Solution

Answer

The maximum compression of the spring is approximately 0.085 meters.

Step 1: Convert Units for Spring Constant

First, convert the spring constant from newtons per centimeter to newtons per meter. The given spring constant is 75 N/cm, which is equivalent to 7500 N/m (since 1 m = 100 cm).

Step 2: Calculate Initial Kinetic Energy

Find the initial kinetic energy of the box using the formula: \( KE = \frac{1}{2}mv^2 \). With mass \( m = 6.0 \) kg and velocity \( v = 3.0 \) m/s, the kinetic energy is \( KE = \frac{1}{2} \times 6.0 \times (3.0)^2 = 27 \) J.

Step 3: Set Up Work-Energy Equation

According to the work-energy theorem, the initial kinetic energy will be equal to the work done in compressing the spring: \( KE = \frac{1}{2}kx^2 \). Here, \( k = 7500 \) N/m and \( x \) is the compression in meters.

Step 4: Solve for Maximum Compression

Rearrange the equation \( \frac{1}{2}kx^2 = 27 \) to solve for \( x \):\[27 = \frac{1}{2} \times 7500 \times x^2\]\[x^2 = \frac{27 \times 2}{7500} = \frac{54}{7500}\]\[x^2 = 0.0072\]\[x = \sqrt{0.0072} \approx 0.085 \text{ m}\]Thus, the maximum compression of the spring is approximately 0.085 meters.

Key Concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. Imagine being on a swing—when you're moving, you have kinetic energy! The mathematical formula for kinetic energy is given by \[ KE = \frac{1}{2}mv^2 \]where:

• \( m \) is the mass of the object (in kilograms),
• \( v \) is the velocity (speed) of the object (in meters per second).

Kinetic energy is measured in joules (J). In our example, the kinetic energy of the box before it hits the spring is determined by its mass (6.0 kg) and its velocity (3.0 m/s). Plugging these numbers into the formula, we find that the kinetic energy is 27 J. This energy is crucial because it will be transformed by the work done on the spring when the box comes to a stop.

Spring Constant

The spring constant, often represented by \( k \), measures how stiff or strong a spring is. Think of it as the spring's resistance against being compressed or stretched. A larger spring constant indicates a stiffer spring.The units of the spring constant are newtons per meter (N/m). In the exercise, the given spring constant was 75 N/cm, which needed conversion to the standard unit N/m. Since 1 meter equals 100 centimeters, multiplying 75 by 100 gives us 7500 N/m. The spring constant plays an essential role in determining how much a spring will compress under a given force.When dealing with scenarios involving springs, such as the box running into our spring, it is crucial to have the spring constant in appropriate units. This way, you can accurately calculate other factors, such as the compression of the spring.

Maximum Compression of a Spring

The maximum compression of a spring occurs when all the kinetic energy of the moving object has been transferred to the spring, halting the motion. This is where the work-energy theorem comes into play.The work-energy theorem states that the initial kinetic energy of the object is transformed into potential energy in the compressed spring. The relationship can be captured by the formula:\[ KE = \frac{1}{2}kx^2 \]where:

• \( KE \) is the initial kinetic energy,
• \( k \) is the spring constant,
• \( x \) is the compression distance in meters.

In this case, the initial kinetic energy of the box is 27 J, and it comes to a stop by compressing the spring. So, we set up our equation:\[ 27 = \frac{1}{2} \times 7500 \times x^2 \]Solving for \( x \) leads us to a compression of approximately 0.085 meters. This simple calculation shows how the energy from the moving box is used to compress the spring to its maximum potential.