spring constant

(noun)

a characteristic of a spring which is defined as the ratio of the force affecting the spring to the displacement caused by it

Related Terms

  • force

Examples of spring constant in the following topics:

  • Work

    • If FFF is constant, in addition to being directed along the line, then the integral simplifies further to:
    • This calculation can be generalized for a constant force that is not directed along the line, followed by the particle.
    • Let's consider an object with mass mmm attached to an ideal spring with spring constant kkk.
    • When the object moves from x=x0x=x_0x=x​0​​ to x=0x=0x=0, work done by the spring would be:
    • The spring applies a restoring force (−k⋅x-k \cdot x−k⋅x) on the object located at xxx.
  • Applications of Second-Order Differential Equations

    • In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, FFF, proportional to the displacement, xxx: F⃗=−kx⃗\vec F = -k \vec x \,​F​⃗​​=−k​x​⃗​​, where kkk is a positive constant.
    • The system under consideration could be an object attached to a spring, a pendulum, etc.
    • If FFF is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency.
    • ω0\omega_0ω​0​​ is called angular velocity, and the constants AAA and ϕ\phiϕ are determined from initial conditions of the motion.
  • Indefinite Integrals and the Net Change Theorem

    • An indefinite integral is defined as ∫f(x)dx=F(x)+C\int f(x)dx = F(x)+ C∫f(x)dx=F(x)+C, where FFF satisfies F′(x)=f(x)F'(x) = f(x)F​′​​(x)=f(x) and where CCC is any constant.
    • We can add any constant CCC to FFF without changing the derivative.
    • With this in mind, we define the indefinite integral as follows: ∫f(x)dx=F(x)+C\int f(x)dx = F(x)+ C∫f(x)dx=F(x)+C , where FFF satisfies F′(x)=f(x)F'(x) = f(x)F​′​​(x)=f(x) and CCC is any constant.
    • Therefore, all the antiderivatives of x2x^2x​2​​ can be obtained by changing the value of CCC in F(x)=(x33)+CF(x) = \left ( \frac{x^3}{3} \right ) + CF(x)=(​3​​x​3​​​​)+C, where CCC is an arbitrary constant known as the constant of integration.
    • Apply the basic properties of indefinite integrals, including the constant, sum, and difference rules
  • Solving Differential Equations

    • A complete solution contains the same number of arbitrary constants as the order of the original equation.
    • Since our example above is a first-order equation, it will have just one arbitrary constant in the complete solution.
    • Therefore, the general solution is f(x)=Ce−xf(x) = Ce^{-x}f(x)=Ce​−x​​, where CCC stands for an arbitrary constant.
    • You can see that the differential equation still holds true with this constant.
    • For a specific solution, replace the constants in the general solution with actual numeric values.
  • Differentiation Rules

    • If f(x)f(x)f(x) is a constant, then f′(x)=0f'(x) = 0f​′​​(x)=0, since the rate of change of a constant is always zero.
    • By extension, this means that the derivative of a constant times a function is the constant times the derivative of the function.
    • The known derivatives of the elementary functions x2x^2x​2​​, x4x^4x​4​​, ln(x)\ln(x)ln(x), and exe^xe​x​​, as well as that of the constant 7, were also used.
  • Linear Equations

    • A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.
    • where mmm and bbb designate constants.
    • In this particular equation, the constant mmm determines the slope or gradient of that line, and the constant term bbb determines the point at which the line crosses the yyy-axis, otherwise known as the yyy-intercept.
  • Partial Derivatives

    • A partial derivative of a function of several variables is its derivative with respect to a single variable, with the others held constant.
    • Usually, the lines of most interest are those which are parallel to the xzxzxz-plane and those which are parallel to the yzyzyz-plane (which result from holding either yyy or xxx constant, respectively).
    • To find the slope of the line tangent to the function at P(1,1,3)P(1, 1, 3)P(1,1,3) that is parallel to the xzxzxz-plane, the yyy variable is treated as constant.
    • By finding the derivative of the equation while assuming that yyy is a constant, the slope of fff at the point (x,y,z)(x, y, z)(x,y,z) is found to be:
    • For the partial derivative at (1,1,3)(1, 1, 3)(1,1,3) that leaves yyy constant, the corresponding tangent line is parallel to the xzxzxz-plane.
  • Antiderivatives

    • As the derivative of a constant is zero, x2x^2x​2​​ will have an infinite number of antiderivatives, such as x33+0\frac{x^3}{3} + 0​3​​x​3​​​​+0, x33+7\frac{x^3}{3} + 7​3​​x​3​​​​+7, x33−42\frac{x^3}{3} - 42​3​​x​3​​​​−42, x33+293\frac{x^3}{3} + 293​3​​x​3​​​​+293, etc.
    • Therefore, all the antiderivatives of x2x^2x​2​​ can be obtained by adding the value of CCC in F(x)=x33+CF(x) = \frac{x^3}{3} + CF(x)=​3​​x​3​​​​+C, where CCC is an arbitrary constant known as the constant of integration.
    • If FFF is an antiderivative of fff, and the function fff is defined on some interval, then every other antiderivative GGG of fff differs from FFF by a constant: there exists a number CCC such that G(x)=F(x)+CG(x) = F(x) + CG(x)=F(x)+C for all xxx.
    • CCC is called the arbitrary constant of integration.
    • If the domain of FFF is a disjoint union of two or more intervals, then a different constant of integration may be chosen for each of the intervals.
  • Derivatives of Exponential Functions

    • Functions of the form cexce^xce​x​​ for constant ccc are the only functions with this property.
    • If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth, continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time.
    • Explicitly for any real constant kkk, a function $f: R→R$ satisfies $f′ = kf $ if and only if f(x)=cekxf(x) = ce^{kx}f(x)=ce​kx​​ for some constant ccc.
  • Volumes

    • A volume integral is a triple integral of the constant function 111, which gives the volume of the region DDD.
    • of the constant function 111 calculated on the cuboid itself.
    • Triple integral of a constant function 111 over the shaded region gives the volume.
    • Calculate the volume of a shape by using the triple integral of the constant function 1
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