Problem 23 - University Physics with Modern Physics

Question

Two identical 1.50-kg masses are pressed against opposite ends of a light spring of force constant \(1.75 \mathrm{N} / \mathrm{cm},\) compressing the spring by 20.0 \(\mathrm{cm}\) from its normal length. Find the speed of each mass when it has moved free of the spring on a frictionless horizontal table.

Step-by-Step Solution

Answer

Each mass has a speed of approximately 1.53 m/s after being released from the spring.

Step 1: Identify Hooke's Law

The potential energy stored in a spring can be calculated using Hooke's Law: \[ U = \frac{1}{2} k x^2 \] where \( k \) is the force constant, and \( x \) is the displacement from the equilibrium position. Here, \( k = 1.75 \mathrm{N/cm} \) (which must be converted to \(1.75 \times 10^2 \mathrm{N/m}\)) and \( x = 0.20 \mathrm{m} \).

Step 2: Calculate Potential Energy

Substitute the given values into the potential energy formula: \[ U = \frac{1}{2} \times 1.75 \times 10^2 \mathrm{N/m} \times (0.20 \mathrm{m})^2 \]. This gives \( U = \frac{1}{2} \times 175 \times 0.04 \), resulting in \( U = 3.5 \mathrm{J} \).

Step 3: Convert Potential to Kinetic Energy

The potential energy stored in the spring will convert into the kinetic energy of the two masses once the spring is released. Therefore, total kinetic energy \( K = U = 3.5 \mathrm{J} \). Since the masses are identical and on a frictionless surface, this energy will be equally distributed: \( \frac{K}{2} = \frac{3.5}{2} = 1.75 \mathrm{J} \) per mass.

Step 4: Apply Kinetic Energy Formula

Express the kinetic energy in terms of the mass and velocity: \[ K = \frac{1}{2}mv^2 \]. Rearrange it to solve for velocity (\(v\)): \[ v = \sqrt{\frac{2K}{m}} \].

Step 5: Calculate Each Mass's Speed

Substitute \( K = 1.75 \mathrm{J} \) and \( m = 1.50 \mathrm{kg} \) into the velocity equation: \[ v = \sqrt{\frac{2 \times 1.75}{1.50}} \]. Simplify to find \( v = \sqrt{\frac{3.5}{1.5}} \approx \sqrt{2.33} \). This gives approximately \( v \approx 1.53 \mathrm{m/s} \).

Key Concepts

Hooke's Law

Hooke's Law is a principle that describes the behavior of springs when they are compressed or stretched. It's a straightforward concept with the formula:\[ F = -kx \]Here,

• \( F \) is the force exerted by the spring,
• \( k \) is the spring constant (expressed in Newtons per meter, indicating the stiffness of the spring),
• and \( x \) is the displacement from the spring's original position.

The negative sign in Hooke's Law indicates that the force exerted by the spring opposes the direction of the displacement.

In other words, if you stretch the spring, it pulls back, and if you compress it, it pushes forward.Understanding this law helps us determine how much potential energy can be stored in a spring, which will convert to kinetic energy upon release.

Potential energy

Potential energy, in the context of springs, is the energy stored in the spring when it is compressed or extended from its equilibrium position.

This type of energy can be calculated using the formula:\[ U = \frac{1}{2} k x^2 \]Where:

• \( U \) is the potential energy,
• \( k \) is the spring constant,
• and \( x \) is the displacement of the spring from its normal length.

This formula shows us that the potential energy is dependent on both the spring's stiffness and the degree to which it is displaced.

In real-world applications, these principles allow us to calculate the energy stored in various spring-related systems like car suspensions or even trampolines, where potential energy is stored when they are compressed, and energy is released when they return to their natural length.

Kinetic energy formula

Once the spring releases, the energy stored in it as potential energy is transferred to the masses as kinetic energy.

Kinetic energy is the energy that an object has due to its motion, which can be calculated using:\[ K = \frac{1}{2} mv^2 \]Where:

• \( K \) is the kinetic energy,
• \( m \) is the mass of the object,
• and \( v \) is the velocity of the object.

This equation reflects how mass and velocity directly affect the amount of kinetic energy an object possesses.

In the described system, once the spring is released, each mass gains kinetic energy equally because they share the total potential energy previously stored in the spring. The kinetic energy is then used to determine how fast each mass will be moving. This conversion from stored potential energy to active motion energy is a fundamental concept in mechanics and helps explain how objects interact and move in various systems.