Problem 77 - Physics

Question

Refer to Interactive Solution \(\underline{10.77}\) at to review a method by which this problem can be solved. An \(11.2-\mathrm{kg}\) block and a \(21.7-\mathrm{kg}\) block are resting on a horizontal frictionless surface. Between the two is squeezed a spring (spring constant \(=1330 \mathrm{~N} / \mathrm{m}\) ). The spring is compressed by \(0.141 \mathrm{~m}\) from its unstrained length and is not attached permanently to either block. With what speed does each block move away after the mechanism keeping the spring squeezed is released and the spring falls away?

Step-by-Step Solution

Answer

The 11.2 kg block moves at 0.722 m/s, and the 21.7 kg block moves at 0.372 m/s in opposite directions.

Step 1: Analyze the System

This problem involves an isolated system consisting of two blocks and a compressed spring. Since the surface is frictionless, no external horizontal forces act on the blocks. According to the conservation of momentum, the momentum of the system before and after the spring is released must be equal.

Step 2: Apply the Conservation of Momentum

Before the spring is released, both blocks are stationary, meaning the initial momentum of the system is zero. After the spring is released, the two blocks move in opposite directions. The equation for conservation of momentum is: \( m_1 v_1 + m_2 v_2 = 0 \), where \( m_1 = 11.2\,\text{kg} \) and \( m_2 = 21.7 \, \text{kg} \), while \( v_1 \) and \( v_2 \) are their respective velocities.

Step 3: Apply Conservation of Energy

The initial potential energy stored in the compressed spring is transformed into the kinetic energy of the two blocks. The potential energy in a compressed spring is given by \( \frac{1}{2} k x^2 \), where \( k = 1330 \, \text{N/m} \) and \( x = 0.141 \, \text{m}\). The kinetic energy of the blocks is given by \( \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \). Set these two expressions equal: \( \frac{1}{2} k x^2 = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \).

Step 4: Derive the Velocities

Using \( v_2 = -\frac{m_1}{m_2} v_1 \) from momentum conservation in step 2, substitute \( v_2 \) into the energy conservation equation. Solving \( \frac{1}{2} k x^2 = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 \left(-\frac{m_1}{m_2} v_1\right)^2 \), we find \( v_1 \), and subsequently \( v_2 \).

Step 5: Calculate Numerical Values

Plug the known values into the equation derived in Step 4 to find the velocity of each block. Using the data provided: \( m_1 = 11.2 \, \text{kg}, m_2 = 21.7 \, \text{kg}, k = 1330 \, \text{N/m}, x = 0.141 \, \text{m} \). Calculate \( v_1 \) and \( v_2 \).

Step 6: Solve for Each Velocity

After simplifying and solving the equations, we find:\[ v_1 \approx 0.722 \, \text{m/s} \]\[ v_2 \approx -0.372 \, \text{m/s} \]The negative sign indicates that the blocks move in opposite directions.

Key Concepts

Conservation of Energy

In physics, the conservation of energy principle is a fundamental concept that states that energy cannot be created or destroyed, it can only change forms. In the scenario with the two blocks and the spring, this principle allows us to understand how the potential energy stored in the spring is transformed into kinetic energy. When the spring is compressed, it stores potential energy, which is mathematically described by the formula:

• \[U = \frac{1}{2} k x^2\]

where \( U \) is the potential energy, \( k \) is the spring constant, and \( x \) is the compression distance.
Upon release, the spring's potential energy is converted into the kinetic energy of the blocks, which is described by:

• \[K = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2\]

where \( m_1 \) and \( m_2 \) are the masses of the blocks, and \( v_1 \) and \( v_2 \) are their velocities.
Conservation of energy implies that the initial potential energy in the spring equals the total kinetic energy of the blocks, allowing us to write:

• \[\frac{1}{2} k x^2 = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2\]

This equation is key to solving for the velocities of the blocks after the spring is released.

Spring Mechanics

Spring mechanics involves understanding how springs store and release energy. A spring's behavior is typically described by Hooke's Law, which states that the force exerted by a spring is proportional to its displacement from the equilibrium position. The formula for Hooke's Law is:

• \[F = -kx\]

where \( F \) is the restoring force of the spring, \( k \) is the spring constant, and \( x \) is the deformation distance.
In our problem, the spring is compressed between two blocks. When the compression mechanism is released, the spring returns to its original length, transferring its stored potential energy into kinetic energy in the blocks.

• The spring constant \( k \) determines how stiff the spring is; a higher spring constant means a stiffer spring.
• The amount of energy stored in the spring depends on both its stiffness and the amount to which it is compressed.
• This stored energy is calculated using the potential energy formula \( U = \frac{1}{2} k x^2 \).

Understanding these basic mechanics helps us comprehend how energy transitions work in spring systems.

Kinetic Energy

Kinetic energy is the energy of motion. In physics, it describes how much energy an object possesses due to its movement, which is computed by the equation:

• \[K = \frac{1}{2} mv^2\]

where \( K \) is the kinetic energy, \( m \) is the object's mass, and \( v \) is its velocity.
Within the context of our exercise, after the spring releases its stored energy and the blocks begin to move, they gain kinetic energy. The amount of kinetic energy an object has depends on both its velocity and its mass.

• The formula indicates that kinetic energy is proportional to the mass of the object and the square of its velocity.
• For example, doubling the velocity of an object will increase its kinetic energy by four times.
• In the problem, the kinetic energy of each block is determined after the spring's potential energy is transferred to them, using the conservation equations.

By understanding how kinetic energy works, we can analyze the motion of objects resulting from the release of energy, such as when a spring pushes two blocks apart.