Potential energy of a spring derivation class 11

https://phys.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fphys.libretexts.org%2FBookshelves%2FUniversity_Physics%2FBook%253A_University_Physics_(OpenStax)%2FBook%253A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)%2F15%253A_Oscillations%2F15.03%253A_Energy_in_Simple_Harmonic_Motion \( \newcommand\) • • • • • • Learning Objectives • Describe the energy conservation of the system of a mass and a spring • Explain the concepts of stable and unstable equilibrium points To produce a deformation in an object, we must do work. That is, whether you pluck a guitar string or compress a car’s shock absorber, a force must be exerted through a distance. If the only result is deformation, and no work goes into thermal, sound, or kinetic energy, then all the work is initially stored in the deformed object as some form of potential energy. Consider the example of a block attached to a spring on a frictionless table, oscillating in SHM. The force of the spring is a conservative force (which you studied in the chapter on potential energy and conservation of energy), and we can define a potential energy for it. This potential energy is the energy stored in the spring when the spring is extended or compressed. In this case, the block oscillates in one dimension with the force of the spring acting parallel to the motion: \[W = \int_ \ldotp\] Energy and the Simple Harmonic Oscillator To study the energy of a simple harmonic oscillator, we need to consider all...

Equation Symbol breakdown Meaning in words U s = 1 2 k x 2 U_s = \dfrac mv^2 K = 2 1 ​ m v 2 K, equals, start fraction, 1, divided by, 2, end fraction, m, v, squared K K K K is translational kinetic energy, m m m m is mass, and v v v v is the speed. Translational kinetic energy is directly proportional to mass and the square of the speed. Elastic potential energy depends upon the position of our system, so a position vs. time graph can be used to find the elastic potential energy U s U_s U s ​ U, start subscript, s, end subscript over time for a simple harmonic oscillator. There are a few important points to note when comparing the position and energy graphs: • U s, max U_\text U s, max ​ U, start subscript, start text, s, comma, space, m, a, x, end text, end subscript occurs when the system is at the maximum displacement of A A A A and − A -A − A minus, A . • U s = 0 U_s=0 U s ​ = 0 U, start subscript, s, end subscript, equals, 0 occurs when the system is at x = 0 x=0 x = 0 x, equals, 0 . • K max K_\text | ∣ − v max ​ ∣ vertical bar, minus, v, start subscript, start text, m, a, x, end text, end subscript, vertical bar . • K = 0 K=0 K = 0 K, equals, 0 occurs when v = 0 v=0 v = 0 v, equals, 0 . Figure 3. A graph of energy vs. time for a simple harmonic oscillator. This graph shows total energy E tot E_\text E tot ​ E, start subscript, start text, t, o, t, end text, end subscript (purple), kinetic energy K K K K (red), and elastic potential energy U s U_s U s ​ U, start su...

https://phys.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fphys.libretexts.org%2FBookshelves%2FUniversity_Physics%2FBook%253A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)%2F13%253A_Simple_Harmonic_Motion%2F13.01%253A_The_motion_of_a_spring-mass_system \( \newcommand\] where \(x\) is the position of the mass. The only other forces exerted on the mass are its weight and the normal force from the horizontal surface, which are equal in magnitude and opposite in direction. Therefore, the net force on the mass is the force from the spring. As we saw in Section 8.4, if the spring is compressed (or extended) by a distance \(A\) relative to the rest position, and the mass is then released, the mass will oscillate back and forth between \(x=\pm A\) \(A\) the “amplitude of the motion”. When the mass is at \(x=\pm A\), its speed is zero, as these points correspond to the location where the mass “turns around”. Description using energy We can describe the motion of the mass using energy, since the mechanical energy of the mass is conserved. At any position, \(x\), the mechanical energy, \(E\), of the mass will have a term from the potential energy, \(U\), associated with the spring force, and kinetic energy, \(K\): \[\begin\] Kinematics of simple harmonic motion We can use Newton’s Second Law to obtain the position, \(x(t)\), velocity, \(v(t)\), and acceleration, \(a(t)\), of the mass as a function of time. The \(x\) componen...

Derivation of Potential Energy The derivation of potential energy is provided here. Potential energy is defined as the energy that is held by an object because of its position with respect to other objects. The SI unit of potential energy is joule whose symbol is J. The gravitational potential energy of an object, elastic potential energy of an extended spring, and electric potential energy of an electric charge are the most common type of potential energies observed. Potential Energy Derivation Potential energy is the work done to take a body to a certain height. For a body with mass m, h is the distance to which it is raised and g is the gravitational force acting on the body, then work done W is given as: W = force * displacement W = mg * h ∴ W = mgh As the work done is equal to mgh which is gained by the object, potential energy Ep is given as: Ep = mgh Therefore, the above is the derivation of potential energy. Stay tuned with BYJU’S for more such interesting articles. Also, register to “BYJU’S – The Learning App” for loads of interactive, engaging Physics-related videos and an unlimited academic assist. Related Physics articles:

Term (symbol) Meaning Spring Object that can extend or contract and return to the original shape. Spring constant ( k k k k ) Measure of a spring’s stiffness, where a more stiff spring has a larger k k k k . SI units of N m \dfrac m N ​ start fraction, start text, N, end text, divided by, start text, m, end text, end fraction . Spring force ( F ⃗ s \vec F_s F s ​ F, with, vector, on top, start subscript, s, end subscript ) Force applied by a spring given by Hooke’s law. SI units of N \text N N start text, N, end text . Elastic potential energy ( U s U_s U s ​ U, start subscript, s, end subscript ) Potential energy stored as a result of applying a force to deform a spring-like object. SI units of J \text J J start text, J, end text . Equation Symbols Meaning in words ∣ F ⃗ s ∣ = k ∣ x ⃗ ∣ \lvert \vec F_s \rvert = k \lvert \vec x \rvert ∣ F s ​ ∣ = k ∣ x ∣ open vertical bar, F, with, vector, on top, start subscript, s, end subscript, close vertical bar, equals, k, open vertical bar, x, with, vector, on top, close vertical bar F ⃗ s \vec F_s F s ​ F, with, vector, on top, start subscript, s, end subscript is spring force, x ⃗ \vec x x x, with, vector, on top is length of extension or compression relative to the unstretched length, and k k k k is spring constant The magnitude of the force required to change the length of a spring-like object is directly proportional to the spring constant and the displacement of the spring. U s = 1 2 k x 2 U_s = \dfrack x ^2 U s ​ = 2 1 ​ k x ...

An object designed to store elastic potential energy will typically have a high elastic limit, however all elastic objects have a limit to the load they can sustain. When deformed beyond the elastic limit, the object will no longer return to its original shape. In earlier generations, wind-up mechanical watches powered by coil springs were popular accessories. Nowadays, we don't tend to use wind-up smartphones because no materials exist with high enough Exercise 1: A truck spring has a spring constant of 5 ⋅ 1 0 4 N / m 5\cdot 10^4~\mathrm 5 ⋅ 1 0 4 N / m 5, dot, 10, start superscript, 4, end superscript, space, N, slash, m . When unloaded, the truck sits 0.8 m above the road. When loaded with goods, it lowers to 0.7 m above the ground. How much potential energy is stored in the four springs? E n e r g y / v o l u m e = 1 2 ( S t r e s s ⋅ S t r a i n ) \mathrm E n e r g y / v o l u m e = 2 1 ​ ( S t r e s s ⋅ S t r a i n ) E, n, e, r, g, y, slash, v, o, l, u, m, e, equals, start fraction, 1, divided by, 2, end fraction, left parenthesis, S, t, r, e, s, s, dot, S, t, r, a, i, n, right parenthesis Exercise 3: Figure 3 shows a stress vs strain plot for a rubber band. As it is stretched (loaded), the curve takes the upper path. Because the rubber band is not ideal, it delivers less force for a given extension when relaxing back (unloaded). The purple shaded area represents the elastic potential energy at maximum extension. The difference in area between the loaded and unloade...

Inside Story • • • • • • • The potential energy of Spring Meaning of potential energy of spring We can see that if we compress or elongate a Actually, this energy stored in any stretched/compressed spring or in stretching of any rubber band, etc, is elastic potential energy as the bodies are resisting change in its shape, size, length, etc. Derivation of the potential energy of spring When compressing a spring from the natural length [Latexpage] \begin Posts • Isobars and Isotopes: definition, examples, and differences, class 11 June 15, 2023 • Atomic number and atomic mass class 11 June 13, 2023 • Rutherford gold foil experiment class 11 June 2, 2023 • Rutherford atomic model: postulates, observations, and limitations, class 11 June 1, 2023 • Thomson model of atom: postulates, drawbacks, & significance, class 11 May 28, 2023