Question

A 1.00 -kg block compresses a spring for which \(k=\) 100. \(\mathrm{N} / \mathrm{m}\) by \(20.0 \mathrm{~cm}\) and is then released to move across a horizontal, frictionless table, where it hits and compresses another spring, for which \(k=50.0 \mathrm{~N} / \mathrm{m}\). Determine a) the total mechanical energy of the system, b) the speed of the mass while moving freely between springs, and c) the maximum compression of the second spring.

Short answer

Answer: The total mechanical energy of the system is 2J. 2) What is the speed of the mass while moving freely between the springs? Answer: The speed of the mass while moving freely between the springs is 2 m/s. 3) What is the maximum compression of the second spring? Answer: The maximum compression of the second spring is 1.26 m (or 126 cm).

Step-by-step solution

Calculate the potential energy stored in the first spring

To find the potential energy in the first spring, we can use the formula for potential energy stored in a spring: \(U_{spring} = \frac{1}{2}kx^2\) , where \(k\) is the spring constant, and \(x\) is the compression length. \(k = 100 N/m\), \(x = 20.0 cm = 0.2 m\) \(U_{spring1} = \frac{1}{2}(100)(0.2)^2 = 2J\)

Determine the total mechanical energy of the system

In this system, since there is no external force acting on it, the total mechanical energy is conserved. The mechanical energy consists of the potential energy in the spring and the kinetic energy of the block. Initially, the block is at rest, so it has no kinetic energy. Therefore, the total mechanical energy of the system is equal to the potential energy in the first spring. \(E_{total} = U_{spring1} = 2J\)

Calculate the speed of the mass while moving freely between springs

When the block moves freely between the springs, its velocity is maximum, and the kinetic energy is also maximum. Since the mechanical energy is conserved, the kinetic energy of the block equals the initial potential energy stored in the first spring. \(K_{max} = E_{total} = 2J\) Now, we can use the kinetic energy formula, \(K = \frac{1}{2}mv^2\), to find the velocity of the block: \(2J = \frac{1}{2}(1kg)v^2\) \(v^2 = 4\) \(v=\sqrt{4} = 2m/s\) The maximum speed of the mass while moving freely between springs is 2 m/s.

Find the maximum compression of the second spring

When the block compresses the second spring, it again comes to a momentary stop, so its kinetic energy becomes zero, and all the mechanical energy is converted into potential energy stored in the second spring. \(U_{spring2} = E_{total} = 2J\) Using the potential energy formula for the second spring, \(U_{spring} = \frac{1}{2}kx^2\), we can now solve for the maximum compression distance \(x\): \(2J = \frac{1}{2}(50 N/m)x^2\) \(x^2 = \frac{8}{5} = 1.6\) \(x = \sqrt{1.6} = 1.26 m\) The maximum compression of the second spring is 1.26 m (or 126 cm). In conclusion: a) The total mechanical energy of the system is \(2J\). b) The speed of the mass while moving freely between springs is \(2m/s\). c) The maximum compression of the second spring is \(1.26 m\).

Key concepts

Potential Energy in Springs

The concept of potential energy in springs is essential for understanding the behavior of systems involving elastic forces. A spring stores potential energy when it is compressed or stretched away from its equilibrium position. The amount of energy stored is directly related to the displacement and the spring's resistance to deformation, known as the spring constant.

In our example, when you compress the spring by a certain amount, you're effectively loading it with energy. This energy remains stored as potential energy until the spring is released. At that point, the energy is converted into other forms, typically kinetic energy as the spring returns to its original shape and pushes on any attached mass, in this case, a block. The equation representing this energy is given by: \(U_{spring} = \frac{1}{2}kx^2\), where \(k\) is the spring constant and \(x\) is the amount of compression.

Kinetic Energy of the Block

Once the spring releases its stored energy, the block attached to it begins to move. The kinetic energy of the block represents its energy due to motion and can be determined using the formula \(K = \frac{1}{2}mv^2\). Kinetic energy is always associated with moving objects and is a way of quantifying the energy an object has because of its velocity.

The block accelerates and reaches its maximum speed when all the potential energy in the spring has been transferred into kinetic energy. It's important to discern that in a frictionless environment, the speed of the block becomes maximum when all the stored potential energy in the spring is entirely transformed into motion energy (kinetic energy). This energy conversion showcases the beautiful symmetry between potential and kinetic energy in a system governed by conservation laws.

Spring Constant

The spring constant, represented by the symbol \(k\) in our formulas, is a measure of a spring's stiffness. Essentially, it tells you how much force is needed to extend or compress the spring by a certain distance. A larger spring constant means a stiffer spring, requiring more force to deform it.

In the context of our problem, there are two different spring constants due to the two distinct springs in play. They have unique values reflecting their individual stiffness. The spring constant comes into play when calculating both the potential energy stored in the spring and the force exerted by the spring on the block. It governs how much energy is needed to compress or stretch the spring and how much energy is available to be converted into kinetic energy when the spring is released.

Mechanical Energy Conservation Principle

The mechanical energy conservation principle is a cornerstone of physics, stating that if no external forces are doing work on a system, the total mechanical energy of that system remains constant over time. This principle combines both kinetic and potential energy into a single framework.

For the block and spring system described in the exercise, the mechanical energy conservation principle means that the energy we initially put into compressing the spring (potential energy) is equal to the maximum kinetic energy the block has as it moves between the springs. This principle allows us to solve for quantities like the speed of the block or the compression of the second spring, as seen in the problem solution. It's the reason why, with the initial potential energy known, we're able to determine how the energy of the system transforms and transfers between various components, ensuring the total energy in this closed system remains constant.