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Work = Force times Distance = Energy

Work is defined as force times distance. Work is a measure of the energy expended in applying a force to move an object.B.8

The work required to compress a spring $ k$ through a displacement of $ x$ meters, starting from rest, is then

$\displaystyle W_k(x) = \int_0^x k\, \xi\, d\xi = \frac{1}{2} k\, x^2. \protect$ (B.6)

Work can also be negative. For example, when uncompressing an ideal spring, the (positive) work done by the spring on its moving end support can be interpreted also as saying that the end support performs negative work on the spring as it allows the spring to uncompress. When negative work is performed, the driving system is always accepting energy from the driven system. This is all simply accounting. Physically, one normally considers the driver as the agent performing the positive work, i.e., the one expending energy to move the driven object. Thus, when allowing a spring to uncompress, we consider the spring as performing (positive) work on whatever is attached to its moving end.

During a sinusoidal mass-spring oscillation, as derived in §B.1.4, each period of the oscillation can be divided into equal sections during which either the mass performs work on the spring, or vice versa.

Gravity, spring forces, and electrostatic forces are examples of conservative forces. Conservative forces have the property that the work required to move an object from point $ a$ to point $ b$, either with or against the force, depends only on the locations of points $ a$ and $ b$ in space, not on the path taken from $ a$ to $ b$.

Potential Energy in a Spring

When compressing an ideal spring, work is performed, and this work is stored in the spring in the form of what we call potential energy. Equation (B.6) above gives the quantitative formula for the potential energy $ W_k(x)$ stored in an ideal spring after it has been compressed $ x$ meters from rest.

Kinetic Energy of a Mass

Kinetic energy is energy associated with motion. For example, when a spring uncompresses and accelerates a mass, as in the configuration of Fig.B.2, work is performed on the mass by the spring, and we say that the potential energy of the spring is converted to kinetic energy of the mass.

Suppose in Fig.B.2 we have an initial spring compression by $ A$ meters at time $ t=0$, and the mass velocity is zero at $ t=0$. Then from the equation of motion Eq.$ \,$(B.5), we can calculate when the spring returns to rest ($ x(t)=0$). This first happens at the first zero of $ \cos(\omega_0t)$, which is time $ t=(\pi/2)/\omega_0=(\pi/2)\sqrt{m/k}$. At this time, the velocity, given by the time-derivative of Eq.$ \,$(B.5),

$\displaystyle v(t) = -A\omega_0\sin(\omega_0 t), $

can be evaluated at $ t=(\pi/2)/\omega_0$ to yield the mass velocity $ v[(\pi/2)/\omega_0] = -A\omega_0 = -A\sqrt{k/m}$, which is when all potential energy from the spring has been converted to kinetic energy in the mass. The square of this value is

$\displaystyle v^2_{\mbox{max}} = A^2\omega_0^2 = A^2\frac{k}{m}, $

and we see that if we multiply $ v^2_{\mbox{max}}$ by $ m/2$, we get

$\displaystyle \frac{1}{2}m\,v^2_{\mbox{max}} = \frac{1}{2}k\,A^2, $

which is the initial potential energy stored in the spring. We require this result. Therefore, the kinetic energy of a mass must be given by

$\displaystyle E_m(v) = \frac{1}{2}m\, v^2 $

in order that the kinetic energy of the mass when spring compression is zero equals the original potential energy in the spring when the kinetic energy of the mass was zero. In the next section we derive this result in a more general way.

Mass Kinetic Energy from Virtual Work

From Newton's second law, $ f=ma=m{\ddot x}$ (introduced in Eq.$ \,$(B.1)), we can use d'Alembert's idea of virtual work to derive the formula for the kinetic energy of a mass given its speed $ v={\dot x}$. Let $ d x$ denote a small (infinitesimal) displacement of the mass in the $ x$ direction. Then we have, using the calculus of differentials,

\begin{eqnarray*} f(t) &=& m\, {\ddot x}(t)\\ \,\,\Rightarrow\,\,\quad d W\isde... ...{1}{2}{\dot x}^2\right)\\ &=& d\left(\frac{1}{2}m\,v^2\right). \end{eqnarray*}

Thus, by Newton's second law, a differential of work $ dW$ applied to a mass $ m$ by force $ f$ through distance $ d x$ boosts the kinetic energy of the mass by $ d(m\,v^2/2)$. The kinetic energy of a mass moving at speed $ v$ is then given by the integral of all such differential boosts from 0 to $ v$:

$\displaystyle E_m(v) = \int_0^v dW = \int_0^v d\left(\frac{1}{2}m \nu^2\right) = \frac{1}{2}m v^2 = \frac{1}{2}m\,{\dot x}^2, $

where $ E_m(v)$ denotes the kinetic energy of mass $ m$ traveling at speed $ v$.

The quantity $ dW=f\,dx$ is classically called the virtual work associated with force $ f$, and $ d x$ a virtual displacement [544].

Energy in the Mass-Spring Oscillator

Summarizing the previous sections, we say that a compressed spring holds a potential energy equal to the work required to compress the spring from rest to its current displacement. If a compressed spring is allowed to expand by pushing a mass, as in the system of Fig.B.2, the potential energy in the spring is converted to kinetic energy in the moving mass.

We can draw some inferences from the oscillatory motion of the mass-spring system written in Eq.$ \,$(B.5):

$\displaystyle x(t) = A\cos(\omega_0 t), \quad t\ge 0 $

  • From a global point of view, we see that energy is conserved, since the oscillation never decays.
  • At the peaks of the displacement $ x(t)$ (when $ \cos(\omega_0t)$ is either $ 1$ or $ -1$), all energy is in the form of potential energy, i.e., the spring is either maximally compressed or stretched, and the mass is momentarily stopped as it is changing direction.
  • At the zero-crossings of $ x(t)$, the spring is momentarily relaxed, thereby holding no potential energy; at these instants, all energy is in the form of kinetic energy, stored in the motion of the mass.
  • Since total energy is conserved (§B.2.5), the kinetic energy of the mass at the displacement zero-crossings is exactly the amount needed to stretch the spring to displacement $ -A$ (or compress it to $ +A$) before the mass stops and changes direction. At all times, the total energy $ E$ is equal to the sum of the potential energy $ E_k(t)$ stored in the spring, and the kinetic energy $ E_m(t)$ stored in the mass:

    $\displaystyle E = E_k(t) + E_m(t) =$   $\displaystyle \mbox{constant [$E_k(0)$]}$

Regarding the last point, the potential energy, $ E_k(t)=k\,x^2(t)/2$ was defined as the work required to displace the spring by $ x$ meters, where work was defined in Eq.$ \,$(B.6). The kinetic energy of a mass $ m$ moving at speed $ v$ was found to be $ E_m(t)=m\,v^2(t)/2=m\,{\dot x}^2/2$. The constance of the potential plus kinetic energy at all times in the mass-spring oscillator is easily obtained from its equation of motion using the trigonometric identity $ \cos^2(\theta)+\sin^2(\theta)=1$ (see Problem 3).

Energy Conservation

It is a remarkable property of our universe that energy is conserved under all circumstances. There are no known exceptions to the conservation of energy, even when relativistic and quantum effects are considered.B.9

Energy may be defined as the ability to do work, where work may be defined as force times distance (§B.2).

Energy Conservation in the Mass-Spring System

Recall that Newton's second law applied to a mass-spring system, as in §B.1.4, yields

$\displaystyle f_m(t) + f_k(t) = 0, \quad \forall t, $

which led to the differential equation obeyed by the mass-spring system:

$\displaystyle m{\ddot x}(t) + k\,x(t) = 0 \quad \forall t $

Multiplying through by $ {\dot x}(t)=v(t)$ gives

\begin{eqnarray*} 0 &=& m{\ddot x}(t){\dot x}(t) + k\,x(t){\dot x}(t)\\ &=& m\... ...{d}{dt} \left[ E_m(t) + E_k(t) \right]\\ &=& \frac{d}{dt} E(t). \end{eqnarray*}

Thus, Newton's second law and Hooke's law imply conservation of energy in the mass-spring system of Fig.B.2.

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