Vertical springs and energy conservation | Work and energy | Physics | Khan Academy

Khan Academy Physics
29 Jul 201614:27
EducationalLearning
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TLDRThe video script explores the dynamics of a mass attached to a spring on both horizontal and vertical planes. It explains that at the spring's natural length, there is no net force on the mass, making it an equilibrium position. The script then delves into the conservation of energy, showing how the potential energy stored in a stretched spring is converted into kinetic energy as the mass moves through the equilibrium position. The concept is extended to a vertical spring, where the script clarifies that despite the presence of gravitational potential energy, it cancels out when using the new equilibrium position as the reference point, allowing for the same energy conservation approach as with a horizontal spring. The explanation is mathematically grounded, providing equations and examples to illustrate the principles discussed.

Takeaways
  • 📍 The equilibrium position of a mass on a horizontal spring is where the spring is at its natural length and the net force on the mass is zero.
  • 🏋️‍♂️ When a mass on a horizontal spring is pulled and released, it gains kinetic energy and moves through the equilibrium position at a speed determined by the conservation of energy principle.
  • 📶 The speed of the mass at the equilibrium position can be calculated using the formula v = √(k/m * d²), where k is the spring constant, m is the mass, and d is the distance the mass is pulled from the equilibrium position.
  • 🔄 For a vertical spring with a mass hanging from it, the spring's natural length is not the equilibrium position due to the influence of gravity.
  • ⚖️ The equilibrium position in a vertical spring system is where the spring force (kx) equals the gravitational force (mg), resulting in a net force of zero.
  • 📈 In a vertical spring system, the initial potential energy includes both spring potential energy and gravitational potential energy, but the latter cancels out when using conservation of energy to solve for the mass's speed at the equilibrium position.
  • 🌐 When calculating the speed of a mass in a vertical spring system, it's sufficient to measure the spring displacement from the new equilibrium position, ignoring gravitational potential energy.
  • 🎯 The conservation of energy principle applies equally to both horizontal and vertical spring systems, allowing for the calculation of the mass's speed at the equilibrium position without considering gravitational potential energy when displacement is measured from the new equilibrium position.
  • 🤔 Understanding the concept of equilibrium and the application of conservation of energy provides insight into the behavior of masses on springs, regardless of orientation.
  • 📚 The script demonstrates the mathematical and physical principles behind spring-mass systems, emphasizing the importance of recognizing how gravitational forces affect vertical systems differently than horizontal ones.
Q & A

  • What is the equilibrium position of a mass on a horizontal spring?

    -The equilibrium position of a mass on a horizontal spring is the point where the spring is at its natural length, where it neither pushes nor pulls, and the net force on the mass is zero.

  • How does the speed of a mass passing through the equilibrium position on a horizontal spring relate to the spring constant and the distance it's stretched?

    -The speed of the mass at the equilibrium position can be calculated using the formula v = sqrt(k/m * d^2), where v is the speed, k is the spring constant, m is the mass, and d is the distance the spring has been stretched.

  • What happens when a mass is pulled to the right and released on a horizontal spring?

    -When a mass is pulled to the right and released on a horizontal spring, the spring's potential energy is converted into kinetic energy as the mass moves back towards the equilibrium position, passing through it with a certain speed determined by the conservation of energy.

  • How does the concept of equilibrium position change for a mass on a vertical spring compared to a horizontal spring?

    -For a mass on a vertical spring, the equilibrium position is not at the spring's natural length but at a point where the spring's upward force equals the downward gravitational force, which is a distance a from the spring's natural length, calculated as a = m*g/k.

  • Why is it possible to ignore gravitational potential energy when calculating the speed of a mass passing through the equilibrium position on a vertical spring?

    -Gravitational potential energy can be ignored in these calculations because it cancels out when using conservation of energy. The mass and spring system can be treated as if it were on a horizontal surface, with only spring displacement from the new equilibrium position being considered.

  • What is the relationship between the spring force and gravitational force at the equilibrium position on a vertical spring?

    -At the equilibrium position on a vertical spring, the spring force (k*x) exactly equals the gravitational force (m*g), resulting in a net force of zero that keeps the mass in equilibrium.

  • How can the initial and final energies of a mass-spring system be described?

    -The initial energy of the system is the sum of the spring potential energy and the gravitational potential energy (if any), while the final energy is the sum of the spring potential energy at the new equilibrium position and the kinetic energy of the mass as it passes through the equilibrium position.

  • What is the formula for the total spring potential energy when a mass is stretched beyond the new equilibrium position on a vertical spring?

    -The total spring potential energy is given by 1/2 * k * (a + b)^2, where k is the spring constant, a is the distance the spring has been stretched from its natural length to the new equilibrium position, and b is the additional stretch beyond the new equilibrium position.

  • How does the choice of the reference line for gravitational potential energy affect the calculation?

    -The choice of the reference line for gravitational potential energy is arbitrary and can be chosen for convenience. It's the difference in gravitational potential energy that matters, so as long as the choice is consistent, it won't affect the outcome of the calculation.

  • What is the final speed of a 3 kg mass hanging from a vertical spring with a spring constant of 50 N/m, when it's pulled down 0.3 m from its equilibrium position and released?

    -The final speed of the 3 kg mass when it reaches the equilibrium position is 1.2 m/s. This is calculated by setting the initial spring potential energy equal to the kinetic energy at the equilibrium position, considering the spring displacement from the new equilibrium position.

  • What is the significance of the result that shows gravitational potential energy can be ignored in vertical spring problems?

    -The significance is that it simplifies the problem-solving process. By measuring the spring displacement from the new equilibrium position and ignoring gravitational potential energy, one can still accurately calculate the mass's speed at the equilibrium position using the same principles as for a horizontal spring.

Outlines
00:00
🌟 Introduction to Mass-Spring System

The paragraph introduces a mass connected to a spring on a frictionless surface. It explains the concept of the spring's natural length, where the spring exerts no force on the mass, thus representing the equilibrium position. The instructor uses humor to relate the concept to everyday life. It then discusses the potential energy stored in the spring when it is stretched and how, upon release, this energy is converted into kinetic energy as the mass moves through the equilibrium position. The paragraph sets up the scenario for further exploration of the mass-spring system, including a comparison between horizontal and vertical springs.

05:02
📈 Conservation of Energy in Vertical Spring

This paragraph delves into the specifics of a vertical spring with a mass hanging from it. It explains how the equilibrium position is determined by the balance between the spring force and gravitational force, resulting in a distance 'a' from the spring's natural length. The paragraph then uses the conservation of energy principle to calculate the speed of the mass as it passes through the equilibrium position in a vertical setup. It highlights the importance of understanding why gravitational potential energy does not need to be considered in the energy conservation equation when calculating the speed, offering deeper insight into the concepts.

10:03
🔄 Simplifying the Vertical Spring Analysis

The final paragraph clarifies that when analyzing a vertical spring system, one can ignore gravitational potential energy by measuring the spring's displacement from the new equilibrium position rather than its natural length. It demonstrates that the gravitational potential energy terms cancel out when using this approach, leading to the same relationship as in a horizontal spring scenario. The paragraph emphasizes that gravity effectively shifts the equilibrium position, allowing the analysis to proceed as if on a horizontal plane when considering displacement from the new equilibrium point. An example problem is provided to illustrate the simplified calculation, showing that the speed of the mass at equilibrium can be determined without considering gravitational potential energy.

Mindmap
Keywords
💡spring's natural length
The spring's natural length is the position where the spring is neither compressed nor stretched, and it is at its content state. In the video, this is the equilibrium position for a mass on a horizontal spring where the net force on the mass is zero, and the spring has no potential energy. The concept is crucial for understanding the dynamics of the mass-spring system as it forms the basis for potential and kinetic energy transformations.
💡equilibrium position
The equilibrium position is the point where the net force acting on the mass is zero. In the context of the video, for a mass on a horizontal spring, this is where the spring is at its natural length. For a vertical spring, the equilibrium position is slightly different as it is influenced by gravity, causing the spring to stretch a distance (denoted as 'a') from its natural length to balance the gravitational force.
💡spring potential energy
Spring potential energy is the stored energy in a spring when it is either compressed or stretched from its natural length. This energy is converted into kinetic energy as the spring oscillates. In the video, the instructor discusses how stretching the spring gives it potential energy, which is then converted into kinetic energy as the mass moves towards the equilibrium position.
💡conservation of energy
The principle of conservation of energy states that the total amount of energy in a closed system remains constant, though it may change forms. In the video, this principle is used to calculate the speed of the mass as it passes through the equilibrium position by equating the initial potential energy (from the spring and, in the case of a vertical spring, gravity) to the final kinetic energy.
💡kinetic energy
Kinetic energy is the energy of motion, which a body possesses due to its movement. In the context of the video, as the mass moves from a stretched position back towards the equilibrium position, the potential energy stored in the spring is converted into kinetic energy of the mass.
💡spring constant
The spring constant, denoted as 'k', is a measure of a spring's stiffness. It is the ratio of the force exerted by the spring to the distance it is stretched or compressed. In the video, the spring constant is used in the formulas to calculate the potential and kinetic energies of the mass-spring system.
💡gravitational force
Gravitational force is the attractive force that Earth exerts on objects, which is directly proportional to their mass and inversely proportional to the square of the distance between their centers. In the video, the gravitational force acts on the mass hanging from a vertical spring, influencing the equilibrium position and the motion of the mass.
💡gravitational potential energy
Gravitational potential energy is the energy an object possesses due to its position in a gravitational field, typically related to its height above a reference point. In the video, it is discussed in the context of a mass hanging from a vertical spring, where the mass has gravitational potential energy that can convert into other forms of energy.
💡mass
In the context of the video, mass refers to the amount of matter in an object, which determines the strength of its gravitational interactions and its resistance to acceleration. The mass is the object attached to the spring in the mass-spring system, and its movement is central to the energy transformations discussed.
💡frictionless surface
A frictionless surface is an idealized concept where an object can move without any resistance from friction. In the video, the mass is assumed to be on a frictionless surface to simplify the analysis, allowing the focus to be on the interaction between the mass and the spring without additional forces affecting the motion.
💡speed
Speed is a scalar quantity that describes the rate of change of an object's position with respect to time. In the video, the calculation of the mass's speed as it passes through the equilibrium position is a key focus, using principles of energy conservation to derive the formula for the speed in both horizontal and vertical spring scenarios.
Highlights

The concept of a spring's natural length and equilibrium position, where the spring exerts no force and the mass remains at rest.

The transformation of spring potential energy into kinetic energy as the mass moves through the equilibrium position.

The equation for the speed of the mass at the equilibrium position in a horizontal spring system: v = sqrt(k/m * d^2).

The surprising fact that the same speed equation applies to both horizontal and vertical spring systems, despite the presence of gravity.

The explanation that gravitational potential energy does not need to be considered in vertical spring calculations when using conservation of energy.

The demonstration that the spring force and gravitational force cancel each other out in vertical spring systems when calculating the mass's speed at the equilibrium position.

The method for finding the new equilibrium position in a vertical spring system, which is different from the spring's natural length due to the force of gravity.

The equation for the distance the mass hangs down from the spring's natural length at the new equilibrium position: a = m*g/k.

The approach to calculate the mass's speed at the equilibrium position in a vertical spring system using conservation of energy, without including gravitational potential energy.

The practical application of the concepts in solving problems involving vertical springs, where the mass's speed at the equilibrium position can be found without considering gravitational potential energy.

The insight that the presence of gravity shifts the equilibrium position but does not affect the conservation of energy calculation when measuring spring displacement from the new equilibrium position.

The example problem provided to illustrate the calculation of the mass's speed at the equilibrium position in a vertical spring system, with a 3 kg mass and a spring constant of 50 N/m.

The final result of the example problem, showing that the speed of the mass at the equilibrium position is 1.2 m/s when pulled down 0.3 meters and released.

The reiteration of the key point that vertical spring problems can be simplified by measuring spring displacement from the new equilibrium position, negating the need to account for gravitational potential energy.

Transcripts

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