Springs & Elastic Potential Energy: Videos & Practice Problems

Elastic potential energy, denoted as $U^{\mathrm{elastic}}=\frac{1}{2}k{x}^2$, is stored in springs when they are compressed or stretched. This energy is similar to gravitational potential energy, which is calculated as $U{g}_0=mgy$. In energy conservation equations, both types of potential energy can be combined, allowing for the analysis of systems involving springs and blocks using Hooke's law and energy principles.

Energy in Horizontal Springs

In the study of energy, we encounter two primary types of potential energy: gravitational potential energy and elastic potential energy. Gravitational potential energy, denoted as \( U_g \), is the energy stored when an object is lifted to a height \( y \) above a reference point, calculated using the formula \( U_g = mgy \), where \( m \) is mass and \( g \) is the acceleration due to gravity. Similarly, elastic potential energy, represented as \( U_{\text{elastic}} \), is the energy stored in a spring when it is either compressed or stretched from its equilibrium position. The formula for elastic potential energy is given by \( U_{\text{elastic}} = \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position.

When analyzing systems involving both gravitational and elastic potential energies, we can combine them in our energy conservation equations. The total mechanical energy can be expressed as the sum of kinetic energy \( K \), gravitational potential energy \( U_g \), and elastic potential energy \( U_{\text{elastic}} \). The general form of the energy conservation equation is \( K_i + U_{g,i} + U_{\text{elastic},i} + W_{\text{non-conservative}} = K_f + U_{g,f} + U_{\text{elastic},f} \), where \( W_{\text{non-conservative}} \) represents work done by non-conservative forces.

To solve problems involving springs, we often apply Hooke's Law, which states that the force exerted by a spring is proportional to its displacement: \( F_{\text{spring}} = kx \). For example, if a block is pushed against a spring with a known applied force, we can determine the compression distance \( x \) by rearranging Hooke's Law to \( x = \frac{F_{\text{applied}}}{k} \). In a scenario where a block is released from a compressed spring, we can calculate its launch speed using energy conservation principles. The initial elastic potential energy stored in the spring converts entirely into kinetic energy as the block is released, leading to the equation \( \frac{1}{2} k x^2 = \frac{1}{2} mv_f^2 \), where \( v_f \) is the final velocity of the block.

By manipulating this equation, we can derive the final velocity as \( v_f = \sqrt{\frac{k}{m}} x \). This relationship highlights how the spring constant and the mass of the block influence the speed at which the block is launched. For instance, substituting specific values for \( k \), \( m \), and \( x \) allows us to calculate the launch speed accurately.

A 4-kg block moving on a flat surface strikes a massless, horizontal spring of force constant 600 N/m with a 20 m/s. The block-surface coefficient of friction is 0.5. Calculate the maximum compression that the spring will experience.

• A. Δx = 0.63 m
• B. Δx = 1.60 m
• C. Δx = 1.63 m
• D. Δx = 1.67 m

Answer: (B)

Here’s what students ask on this topic:

What is elastic potential energy and how is it calculated?

Elastic potential energy is the energy stored in a spring when it is compressed or stretched. It is similar to gravitational potential energy, which is stored when an object is lifted to a height. The formula for elastic potential energy is given by:

$U_{\mathrm{elastic}}=\frac{1}{2}k{x}^2$

where $k$ is the spring constant, and $x$ is the displacement from the equilibrium position. This energy is stored as long as the spring is deformed and can be released when the spring returns to its equilibrium position.

How does elastic potential energy relate to energy conservation?

In energy conservation equations, elastic potential energy can be combined with other forms of potential energy, such as gravitational potential energy. The total mechanical energy of a system is conserved, meaning the sum of kinetic energy and potential energy (both gravitational and elastic) remains constant if no non-conservative forces (like friction) are doing work. The energy conservation equation can be written as:

$K+U_g+U_{\mathrm{elastic}}=K+U_g+U_{\mathrm{elastic}}$

This allows for the analysis of systems involving springs and blocks using Hooke's law and energy principles.

What is Hooke's Law and how is it used in spring problems?

Hooke's Law describes the relationship between the force exerted by a spring and its displacement from the equilibrium position. It is given by:

$F=kx$

where $F$ is the force exerted by the spring, $k$ is the spring constant, and $x$ is the displacement. In spring problems, Hooke's Law is used to determine the force exerted by the spring when it is compressed or stretched. This force can then be used in energy conservation equations to analyze the motion and energy of objects attached to the spring.

How do you calculate the launch speed of a block released from a compressed spring?

To calculate the launch speed of a block released from a compressed spring, you use the principle of energy conservation. Initially, the block has elastic potential energy stored in the compressed spring and no kinetic energy. When the spring is released, this potential energy is converted into kinetic energy. The energy conservation equation is:

$\frac{1}{2}k{x}^2=\frac{1}{2}m{v}^2$

where $k$ is the spring constant, $x$ is the initial compression distance, $m$ is the mass of the block, and $v$ is the final velocity. Solving for $v$ gives:

$v=\sqrt{\frac{k}{m}{x}^2}$

Plug in the values for $k$, $x$, and $m$ to find the launch speed.

When should you use energy principles instead of force principles in spring problems?

Energy principles should be used in spring problems when the force exerted by the spring is not constant, such as when an object is moving between two points on a spring. In these cases, the force changes as the spring compresses or stretches, making it difficult to use force principles and motion equations. Instead, energy conservation equations can be used to analyze the system. For example, when calculating the launch speed of a block released from a compressed spring, energy principles are used because the force exerted by the spring changes as the block moves. By using energy conservation, you can account for the conversion of elastic potential energy into kinetic energy.