Problem 24 - University Physics with Modern Physics

Question

Block \(A\) in Fig. E8.24 has mass \(1.00 \mathrm{kg},\) and block \(B\) has mass 3.00 \(\mathrm{kg}\) . The blocks are forced together, compressing a spring \(S\) between them; then the system is released from rest on a level, frictionless surface. The spring, which has negligible mass, is not fastened to either block and drops to the surface after it has expanded. Block \(B\) acquires a speed of 1.20 \(\mathrm{m} / \mathrm{s}\) . (a) What is the final speed of block \(A\) ? (b) How much potential energy was stored in the compressed spring?

Step-by-Step Solution

Answer

(a) 3.6 m/s, (b) 6.48 J.

Step 1: Analyze the problem

We have two masses, Block A and Block B, with a spring between them. The spring is released, propelling the blocks in opposite directions. We'll use conservation of momentum and energy principles to solve the questions.

Step 2: Apply conservation of momentum

According to the conservation of momentum, the total momentum before and after the release must be equal. Before release, the system is at rest, so initial momentum is zero. Thus, \( m_A \, v_A + m_B \, v_B = 0 \). Given \( m_A = 1.00 \, \text{kg} \), \( m_B = 3.00 \, \text{kg} \), and \( v_B = 1.20 \, \text{m/s} \), we find \( v_A \).

Step 3: Solve for velocity of Block A

Using the equation \( m_A \, v_A + m_B \, v_B = 0 \), substitute the known values: \( 1.00 \, \text{kg} \cdot v_A + 3.00 \, \text{kg} \cdot 1.20 \, \text{m/s} = 0 \). Solving for \( v_A \) yields: \( v_A = -3.6 \, \text{m/s} \). The negative sign indicates Block A moves in the opposite direction to Block B.

Step 4: Calculate the potential energy in the spring

Since the system's kinetic energy came from the potential energy stored in the spring, use energy conservation: \( \frac{1}{2} m_A v_A^2 + \frac{1}{2} m_B v_B^2 = E_{\text{spring}} \). Substitute \( v_A = -3.6 \, \text{m/s} \) and \( v_B = 1.20 \, \text{m/s} \) to find the spring energy.

Step 5: Compute spring potential energy

Substituting values, \( E_{\text{spring}} = \frac{1}{2} \cdot 1.00 \, \text{kg} \cdot (-3.6 \, \text{m/s})^2 + \frac{1}{2} \cdot 3.00 \, \text{kg} \cdot (1.20 \, \text{m/s})^2 \). This results in \( E_{\text{spring}} = 6.48 \, \text{J} \).

Key Concepts

Conservation of Energy

The principle of Conservation of Energy states that energy cannot be created or destroyed, only transformed from one form to another. In the context of our exercise, the system initially holds potential energy in the compressed spring. When released, this energy converts into kinetic energy, propelling the blocks in opposite directions. Understanding this transformation is crucial for solving problems involving mechanical systems.

• Potential energy in the spring is converted into kinetic energy of the blocks.
• The total energy in the system remains constant, assuming no energy loss due to external factors like friction.

In our example, the total potential energy stored in the spring equals the total kinetic energy of both blocks post-release. Ensuring you account for all energy types at different stages is key to accurately analyzing such systems.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It is calculated using the formula: \[ KE = \frac{1}{2}mv^2 \]where \( m \) is the mass of the object and \( v \) is its velocity. In the exercise, both Block A and Block B acquired kinetic energy once the spring was released. The change from being at rest (zero kinetic energy) to moving is directly related to the energy released by the spring.

• Block B, with a mass of 3 kg and a velocity of 1.20 m/s, has a significant kinetic energy component.
• Block A, despite having less mass (1 kg), moves faster (3.6 m/s, found from calculations) and also contributes to the system's total kinetic energy.

Each block's kinetic energy can be calculated individually and added together to verify it matches the potential energy initially stored in the spring. Paying attention to direction and mass distribution is fundamental when assessing kinetic energy distribution.

Potential Energy

Potential energy represents the stored energy in a system due to its position or arrangement. In our exercise, the spring stores potential energy while compressed. Once the system is released, this energy becomes kinetic energy that moves the blocks. The formula for elastic potential energy (like that in a spring) is:\[ PE_{spring} = \frac{1}{2}kx^2 \]where \( k \) is the spring constant, and \( x \) is the displacement from its natural length. Although not directly given in this problem, understanding this relationship helps comprehend how potential energy works in similar contexts.

• The energy stored depends on how much the spring is compressed.
• Upon release, all stored energy transitions into kinetic energy.

This transition highlights how fields and forces, like that of a spring, can store and later release energy, driving motion and change in mechanical systems.