Problem 42 - University Physics with Modern Physics

Question

A 4.00-kg block of ice is placed against a horizontal spring that has force constant \(k\) = 200 N/m and is compressed 0.025 m. The spring is released and accelerates the block along a horizontal surface. Ignore friction and the mass of the spring. (a) Calculate the work done on the block by the spring during the motion of the block from its initial position to where the spring has returned to its uncompressed length. (b) What is the speed of the block after it leaves the spring?

Step-by-Step Solution

Answer

The work done is 0.0625 J and the block's speed is 0.177 m/s.

Step 1: Determine the Potential Energy Stored in the Spring

The work done by the spring on the block is equal to the potential energy stored in the spring when it is compressed. The potential energy (or elastic potential energy) can be calculated using the formula:\[ PE = \frac{1}{2} k x^2 \]where \(k\) is the spring constant (200 N/m), and \(x\) is the compression distance (0.025 m). Substitute the values to find the energy:\[ PE = \frac{1}{2} \times 200 \times (0.025)^2 = 0.0625 \, \text{J} \]

Step 2: Calculate the Work Done by the Spring

The work done on the block by the spring is equivalent to the potential energy initially stored in the spring because energy is conserved in this system (ignoring any friction). Therefore, the work done \(W\) on the block by the spring is:\[ W = 0.0625 \, \text{J} \]

Step 3: Use Work-Energy Principle for Speed

The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. The block starts from rest, so its initial kinetic energy is zero. The final kinetic energy \(KE\) is given by the work done, so:\[ KE = \frac{1}{2} mv^2 = W \]We can solve for the speed \(v\) of the block:\[ \frac{1}{2} \times 4.00 \times v^2 = 0.0625 \]

Step 4: Solve for the Speed of the Block

Solve the equation from Step 3 for \(v\):\[ 2 \times 0.0625 = 4.00 \times v^2 \]\[ 0.125 = 4.00 \times v^2 \]\[ v^2 = \frac{0.125}{4.00} \]\[ v^2 = 0.03125 \]\[ v = \sqrt{0.03125} \]\[ v \approx 0.177 \, \text{m/s} \]

Step 5: Conclusion: Results Summary

The work done on the block by the spring is 0.0625 Joules. After being released by the spring, the block achieves a speed of approximately 0.177 m/s.

Key Concepts

Potential Energy

Potential energy is a concept that helps us understand how energy can be stored in an object, ready to be released. It doesn't do any work while in its stored form, but it's all about potential for future action. In our example of a spring and a block, potential energy comes into play when the spring is compressed. Imagine you are squishing the spring. It's like you're filling an invisible energy tank inside it. This stored energy depends on how much you compress the spring, as well as the spring's stiffness, or force constant, denoted by the symbol \( k \). Potential energy (or more specifically, elastic potential energy in the case of springs) can be calculated using the formula:

• \( PE = \frac{1}{2} k x^2 \)

where \( k \) is the spring constant (200 N/m in our exercise), and \( x \) is the compression distance (0.025 m). Substituting these values confirms that the spring stores 0.0625 Joules of potential energy. This energy, initially dormant when the spring is compressed, is what will soon propel our block.

Work-Energy Principle

The Work-Energy Principle is a key concept in physics. It links the work done on an object to the changes in that object's kinetic energy. In simple terms, work is the energy transferred to or from an object via the application of force along a displacement. Now, if we consider the spring and the block, the work done by the spring is actually the stored energy being transferred to the block. The principle tells us that:

• The work done \( W \) equals the change in kinetic energy \( \Delta KE \) of the block.

Since the block begins its journey at rest, its initial kinetic energy is 0 Joules. When released, the spring's stored potential energy (0.0625 Joules) transforms into kinetic energy of the block. This conversion is why the block moves, highlighting the beauty of energy conservation. So, the work done by the spring is exactly 0.0625 Joules, matching the energy initially stored.

Kinetic Energy

Kinetic energy is the energy of motion. When the spring releases, the block moves, and the stored potential energy becomes kinetic energy. Kinetic energy \( KE \) is calculated using the formula:

• \( KE = \frac{1}{2} mv^2 \)

where \( m \) is the mass of the block (4.00 kg) and \( v \) is the velocity of the block. Because the work done by the spring is equivalent to the change in kinetic energy, we set our equation \( \frac{1}{2} mv^2 = 0.0625 \). Solving this, we find that the velocity \( v \) of the block is approximately 0.177 meters per second. This transition demonstrates how the block gains speed as it travels away from the spring, illustrating kinetic energy in action. It's amazing how the stored energy in a compressed spring can be calculated and then seen as motion energy in the block, bringing theory to life.