Conservation of Energy - Vertical Springs

Physics Ninja

21 Oct 202322:59

EducationalLearning

32 Likes 10 Comments

TLDRIn this educational video, Physics Ninja tackles a vertical spring problem using the conservation of energy principle. The video introduces a spring with a 300 G mass attached, stretched beyond its equilibrium position by 11 cm, and released with initial kinetic energy. Two methods are presented to find the mass's maximum height: a more algebraically complex first method and a simpler second method that omits gravitational potential energy. The video guides viewers through the calculations, emphasizing the importance of understanding the first method to appreciate the trick in the second, ultimately revealing the mass's maximum height.

Takeaways

😀 The video is a physics tutorial by Physics Ninja, focusing on solving a vertical spring problem using the conservation of energy method.
🔍 The problem involves a spring with a natural length \( l_0 \), a mass of 300g, and an initial stretch to an equilibrium position where the spring constant \( K \) is calculated.
🤔 The video presents two methods to solve the problem: a more algebraically complex first method and a simpler second method that uses a trick.
📚 The first method involves setting a zero reference point before connecting the mass, which helps in defining the energies and solving the problem algebraically.
🔄 The conservation of energy principle is applied by comparing the initial and final configurations of the system, considering kinetic, spring potential, and gravitational potential energy.
📉 The initial kinetic energy is calculated using the formula \( \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the initial velocity.
🌐 The spring potential energy is given by \( \frac{1}{2}K(y_0 + a)^2 \), with \( y_0 \) being the initial stretch and \( a \) the additional stretch beyond equilibrium.
📈 The gravitational potential energy is calculated as \( -mgy \), with \( y \) being the height relative to the zero reference point.
🔢 The final configuration at maximum height involves solving a quadratic equation to find \( y_{final} \), the maximum height the mass reaches.
🎯 The second method simplifies the problem by setting the zero at the equilibrium position and excluding gravitational potential energy from the calculations.
📝 The final result of both methods should yield the maximum height the mass reaches, which is found to be 8.7 cm above the initial reference point.
🔬 The video concludes by explaining the mathematical justification behind the second method, showing that it simplifies to the same result as the first method due to the inclusion of constant terms on both sides of the equation.

Q & A

What is the main topic of the video?
-The main topic of the video is solving a vertical spring problem using the conservation of energy method.
What is the initial mass connected to the spring in the problem?
-The initial mass connected to the spring is 300 grams, which is equivalent to 0.3 kilograms.
What is the natural length of the spring in the problem?
-The natural length of the spring is not explicitly given, but it is referred to as 'l0' in the problem.
What is the equilibrium position of the spring in the problem?
-The equilibrium position of the spring is the stretched position when the mass is connected, which is 25 cm from the natural length.
What is the additional stretch of the spring from the equilibrium position before release?
-The additional stretch of the spring from the equilibrium position before release is 11 cm.
What is the spring constant K calculated in the video?
-The spring constant K is calculated to be 11.76 Newtons per meter.
What are the two methods presented in the video to solve the problem?
-The two methods presented in the video are an algebraic method and a trick method, with the latter being simpler but requiring understanding of the former.
What is the significance of setting a zero point in method one?
-Setting a zero point in method one is crucial for defining the positive and negative values for height and energy calculations, with everything above the zero being positive and below being negative.
How is the initial kinetic energy calculated in the problem?
-The initial kinetic energy is calculated using the formula 1/2 * mass * (initial velocity)^2.
What is the role of Hooke's law in the problem?
-Hooke's law is used to calculate the spring force, which is given by -Kx, where K is the spring constant and x is the amount of stretching from the natural length of the spring.
What is the final step in solving the problem using method one?
-The final step in solving the problem using method one is to solve for the final height (y_final) by setting the total energy at the initial and final configurations equal to each other and solving the resulting quadratic equation.
Why does method two exclude gravitational potential energy?
-Method two excludes gravitational potential energy because it is a trick method that simplifies the calculations by redefining the zero position at the equilibrium and focusing only on the spring's potential and kinetic energy changes.
What is the mathematical trick used in method two to simplify the problem?
-The mathematical trick in method two is to redefine the zero position at the equilibrium, which allows for the exclusion of gravitational potential energy and simplifies the energy conservation equation to a direct calculation of the maximum height.
How does the video explain the two solutions obtained from the quadratic equation?
-The video explains that the two solutions represent the mass's position relative to the zero point, with one being above and the other below. The positive solution is the correct one for the maximum height when the mass is launched upwards.
What is the final maximum height calculated in the video using method two?
-The final maximum height calculated using method two is 8.7 cm above the equilibrium position, which is the same result obtained from method one but with simpler calculations.

Outlines

00:00

🔍 Introduction to the Vertical Spring Problem

Physics Ninja introduces a vertical spring problem involving conservation of energy. A spring with a natural length is connected to a 300 G mass, which stretches the spring to an equilibrium position. The spring is further stretched by 11 cm, and the mass is given an initial velocity. The goal is to find the maximum height the mass will reach. Two methods are presented to solve the problem: a more algebraically complex first method and a second method involving a trick that requires understanding the first. The video encourages viewers to like, subscribe, and support the channel.

05:01

📚 Method One: Algebraic Approach to the Spring Problem

The first method begins with an equilibrium analysis, calculating the spring constant 'K' using the mass, gravitational acceleration, and the initial stretch 'y0'. The energy conservation principle is applied by setting the initial kinetic energy, spring potential energy, and gravitational potential energy equal to the final energy at the maximum height. A quadratic equation is derived to solve for the final position 'y final'. The process involves substituting values and simplifying the equation to find the maximum height the mass can reach.

10:03

🔄 Method Two: The Tricky Spring Energy Solution

Method two simplifies the problem by setting the zero position at the equilibrium and excluding gravitational potential energy from the calculations. The initial and final energies are expressed in terms of spring potential energy and initial kinetic energy. The solution involves straightforward algebra to find 'y final', the position of the mass relative to the new zero. Two solutions are obtained, but only the positive one is considered for the maximum height, which is measured from the new equilibrium position.

15:07

📉 Analyzing the Solutions and the Behind-the-Scenes of Method Two

The video script explains the significance of the two solutions obtained from method two, relating them to the initial conditions and the direction of launch. It clarifies that the maximum height is the positive solution, which is 8.7 cm above the new zero. The script then delves into the mathematical justification for method two, showing that method one simplifies to method two by redefining the zero position and recognizing that constant terms do not affect the final result due to their presence on both sides of the energy conservation equation.

20:08

🧩 Conclusion: Understanding the Two Methods for Solving the Spring Problem

The final paragraph wraps up the video by highlighting the importance of understanding method one to appreciate the simplification achieved in method two. It emphasizes that while method two is mathematically simpler, the algebraic groundwork laid by method one is crucial for comprehending the underlying physics. The video concludes by encouraging viewers to solve problems using either method, now that they have all the necessary information and understanding.

Mindmap

Keywords

💡Conservation of Energy

Conservation of Energy is a fundamental principle in physics stating that energy cannot be created or destroyed, only transformed from one form to another. In the video's context, it is used to solve a vertical spring problem by equating the total energy at the initial and final states of the system. The script mentions that since there is no friction, the energy remains constant, allowing the use of this principle to find the maximum height the mass can reach.

💡Spring Constant (K)

The spring constant, denoted as 'K' in the script, is a measure of the stiffness of a spring. It is defined as the force needed to extend or compress a spring by a unit distance. In the video, the spring constant is calculated using Hooke's Law, which relates the spring force to the amount of stretch from its natural length, and is essential for determining the spring's potential energy.

💡Equilibrium Position

An equilibrium position is a state where the net force acting on an object is zero, resulting in no acceleration. In the script, the equilibrium position of the spring is established when the spring force equals the weight of the mass attached to it. This concept is crucial for understanding the initial setup of the problem and calculating the spring constant.

💡Hooke's Law

Hooke's Law describes the linear relationship between the force exerted by a spring and the distance it is stretched or compressed. It is represented as F = -Kx, where 'F' is the force, 'K' is the spring constant, and 'x' is the displacement. The video uses Hooke's Law to establish the force balance at the equilibrium position and to calculate the spring constant.

💡Potential Energy

Potential energy is the stored energy of an object due to its position in a force field, such as gravitational or elastic. In the script, two types of potential energy are discussed: spring potential energy and gravitational potential energy. The spring potential energy is calculated using the formula 1/2 Kx^2, where 'x' is the displacement from the spring's equilibrium position, which is essential for solving the problem using conservation of energy.

💡Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, defined as 1/2 mv^2, where 'm' is the mass and 'v' is the velocity. In the video, the initial kinetic energy of the mass is considered when applying the conservation of energy method to find the maximum height it reaches after being released.

💡Free Body Diagram

A free body diagram is a simplified representation used in physics to visualize all the forces acting on an object. In the script, a free body diagram is mentioned for the mass at equilibrium, showing the weight of the mass and the spring force acting on it, which helps in understanding the force balance and calculating the spring constant.

💡Quadratic Equation

A quadratic equation is a polynomial equation of the second degree, typically in the form ax^2 + bx + c = 0. In the video, the script derives a quadratic equation from the conservation of energy principle to find the final position of the mass. Solving this equation provides the possible heights the mass can reach.

💡Zero Position

The zero position, or reference point, is a point chosen to measure displacements and potential energy in a problem. The script discusses two methods with different zero positions: one at the natural length of the spring before any forces are applied and another at the equilibrium position. Choosing the zero position is crucial for correctly setting up and solving the energy equations.

💡Method 1 and Method 2

These refer to two different approaches to solving the spring problem presented in the video. Method 1 involves a more algebraically complex solution with a specific zero position and includes gravitational potential energy in the calculations. Method 2 is simpler, uses a different zero position at the equilibrium, and intriguingly omits gravitational potential energy in its calculations, yet arrives at the same result due to the underlying physics.

Highlights

Introduction of a physics problem involving a spring, mass, and conservation of energy.

Description of the initial setup with a spring of natural length \( l_0 \), a mass of 300g, and the equilibrium position.

Explanation of Hooke's Law and its application in calculating the spring force.

Derivation of the spring constant \( K \) using the equilibrium condition and the mass's weight.

Introduction of two methods to solve the problem, with a teaser of a trick in the second method.

Setting the zero position for potential energy calculations in method one.

Calculation of initial kinetic, spring potential, and gravitational potential energy for the starting configuration.

Application of energy conservation to equate initial and final total energies of the system.

Derivation of a quadratic equation to find the maximum height of the mass.

Solving the quadratic equation to obtain two potential solutions for the mass's height.

Interpretation of the two solutions in the context of the problem's setup and the zero position.

Introduction of method two with a different approach to setting the zero position and excluding gravitational potential energy.

Simplification of the problem using method two, leading to a direct calculation of the maximum height.

Comparison of the results from both methods, highlighting the mathematical simplicity of method two.

Detailed algebraic explanation of why method two works, despite its apparent exclusion of gravitational potential energy.

Conclusion emphasizing the ability to solve the problem using either method, with an invitation to understand the algebraic background.

Transcripts

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