Problem 39 - University Physics with Modern Physics

Question

A 6.0-kg box moving at 3.0 m/s on a horizontal, frictionless surface runs into a light spring of force constant 75 N/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 0.084 meters.

Step 1: Understand the Problem

We need to find the maximum compression of a spring when a 6.0-kg box moving at 3.0 m/s hits it. We'll use the work-energy theorem, which relates the kinetic energy of the box and the potential energy stored in the spring at maximum compression.

Step 2: Convert Units

The spring constant is given as 75 N/cm. Convert this to N/m for consistency in SI units: \[ k = 75 \times 100 = 7500 \text{ N/m} \]

Step 3: Calculate Initial Kinetic Energy

Calculate the initial kinetic energy of the box using the formula: \[ KE = \frac{1}{2} m v^2 \]Substitute the given values: \[ KE = \frac{1}{2} \times 6.0 \times 3.0^2 = 27 \, \text{J} \]

Step 4: Apply Work-Energy Theorem

According to the work-energy theorem, the work done on the spring (potential energy at maximum compression) equals the initial kinetic energy of the box:\[ \frac{1}{2} k x^2 = KE \]Substituting the known values:\[ \frac{1}{2} \times 7500 \times x^2 = 27 \]

Step 5: Solve for Maximum Compression

Simplify the equation to find \( x \):\[ 7500 x^2 = 54 \]\[ x^2 = \frac{54}{7500} \]\[ x = \sqrt{\frac{54}{7500}} \approx 0.084 \text{ m} \]

Step 6: Conclusion

The maximum compression of the spring is approximately 0.084 meters.

Key Concepts

Kinetic Energy

Kinetic energy is an essential concept in physics, describing the energy of an object due to its motion. Whenever an object moves, it possesses kinetic energy, which depends on two key factors: the object's mass and velocity. The formula to calculate kinetic energy is: \[ KE = \frac{1}{2} mv^2 \]

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

Kinetic energy increases with the square of the velocity. So, a slight increase in speed results in a much larger increase in kinetic energy. In our example, the 6.0-kg box moving at a speed of 3.0 m/s has a kinetic energy of 27 joules. This energy is what gets transferred into the spring, leading to its compression.

Potential Energy

Potential energy is the energy that an object possesses due to its position or configuration. Specifically, in the context of springs, potential energy is stored when the spring is compressed or stretched.For springs, we use Hooke's Law to determine their potential energy. The formula is:\[ PE = \frac{1}{2} k x^2 \]

• \( k \) is the spring constant, indicating the stiffness of the spring.
• \( x \) is the displacement from the spring's natural length, in meters (m).

As the box impacts the spring, its kinetic energy is transformed into potential energy, stored as the spring compresses. The greater the compression, the more potential energy is stored. The potential energy at maximum compression will be equal to the initial kinetic energy according to the work-energy theorem.

Spring Constant

The spring constant, denoted as \( k \), is a measure of a spring's stiffness. It is expressed in units of Newtons per meter (N/m). A higher spring constant means a stiffer spring that is harder to compress or stretch. Conversely, a lower spring constant means a more flexible spring. To find the spring constant in the given problem, we first convert it from N/cm to N/m by multiplying by 100:\[ k = 75 \times 100 = 7500 \, \text{N/m} \]Understanding the spring constant is crucial when analyzing problems involving springs, as it directly affects how much force the spring can exert and how much it compresses for a given force. In this problem, the spring constant helps us calculate the potential energy and thereby find the maximum compression of the spring when the box hits it.

Maximum Compression

Maximum compression refers to the furthest point a spring is compressed when an external force is applied. When dealing with springs, this occurs when the kinetic energy is completely converted into potential energy.To find the maximum compression in our exercise, we set the potential energy equal to the initial kinetic energy using the work-energy theorem:\[ \frac{1}{2} k x^2 = KE \]After substituting the given values and solving for \( x \):

• \( k = 7500 \, \text{N/m} \)
• \( KE = 27 \, \text{J} \)

We solve for \( x \), yielding:\[ x = \sqrt{\frac{54}{7500}} \approx 0.084 \, \text{m} \]This calculation shows that the maximum compression of the spring is approximately 0.084 meters, which represents the point at which all kinetic energy is stored as potential energy in the spring.