The Potential Energy of a Spring

The Potential Energy of a Spring

The spring force is an example of a variable force which is conservative. Fig. 6.7 shows a block attached to a spring and resting on a smooth horizontal surface.The other end of the spring is attached to a rigid wall.The spring is light and may be treated as massless. In an ideal spring, the spring force Fs is proportional to xwhere x is the displacement of the block from the equilibrium position. The displacement could be either positive [Fig. 6.7(b)] or negative [Fig. 6.7(c)]. This force law for the spring is called Hooke’s law and is mathematically stated as \(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa % aaleaacaWGZbaabeaakiabg2da9iabgkHiTiaadUgacaWG4baaaa!3BCF! {F_s} = - kx\)

The constant k is called the spring constant. Its unit is N m^-1.The spring is said to be stiff if k is large and soft if k is small.

Suppose that we pull the block outwards as in Fig. 6.7(b). If the extension is x_m, the work done by the spring force is 

\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGxb % WaaSbaaSqaaiaadohaaeqaaOGaeyypa0Zaa8qCaeaacaWGgbWaaSba % aSqaaiaadohaaeqaaOGaamizaiaadIhaaSqaaiaaicdaaeaacaWG4b % WaaSbaaWqaaiaad2gaaeqaaaqdcqGHRiI8aOGaeyypa0JaeyOeI0Ya % a8qCaeaacaWGRbGaamiEaiaadsgacaWG4baaleaacaaIWaaabaGaam % iEamaaBaaameaacaWGTbaabeaaa0Gaey4kIipaaOqaaiabg2da9iab % gkHiTmaalaaabaGaam4AaiaadIhadaqhaaWcbaGaamyBaaqaaiaaik % daaaaakeaacaaIYaaaaaaaaa!53D5! \begin{array}{l} {W_s} = \int\limits_0^{{x_m}} {{F_s}dx} = - \int\limits_0^{{x_m}} {kxdx} \\ = - \frac{{kx_m^2}}{2} \end{array}\)

This expression may also be obtained by considering the area of the triangle as in Fig. 6.7(d). Note that the work done by the external pulling force F is positive since it overcomes the spring force.

\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 % da9iabgUcaRmaalaaabaGaam4AaiaadIhadaqhaaWcbaGaamyBaaqa % aiaaikdaaaaakeaacaaIYaaaaaaa!3D58! W = + \frac{{kx_m^2}}{2}\)

fig.6.7. Illustration of the spring force with a block attached to the free end of the spring.

     (a) The spring force Fs is zero when the displacement x from the equilibrium position is zero. (b) For the stretched spring x > 0 and Fs< 0 (c)              For the compressed spring x < 0 and Fs > 0.(d) The plot of Fs versus x.The area of the shaded triangle represents the work done by the                 spring force. Due to the opposing signs of Fs and x, this work done is negative,\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaBa % aaleaacaWGZbaabeaakiabg2da9iabgkHiTiaadUgacaWG4bWaa0ba % aSqaaiaad2gaaeaacaaIYaaaaOGaai4laiaaikdaaaa!3F34! {W_s} = - kx_m^2/2\)

The same is true when the spring is compressed with a displacement x_c (< 0).The spring force does work \(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaBa % aaleaacaWGZbaabeaakiabg2da9iabgkHiTiaadUgacaWG4bWaa0ba % aSqaaiaadogaaeaacaaIYaaaaOGaai4laiaaikdaaaa!3F2A! {W_s} = - kx_c^2/2\) while the external force F does work\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSIaam % 4AaiaadIhadaqhaaWcbaGaam4yaaqaaiaaikdaaaGccaGGVaGaaGOm % aaaa!3C0F! + kx_c^2/2\).If the block is moved from an initial displacement x_i to a final displacement x_f ,the work done by the spring force W_s is

\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaBa % aaleaacaWGZbaabeaakiabg2da9iabgkHiTmaapehabaGaam4Aaiaa % dIhacaWGKbGaamiEaaWcbaGaamiEamaaBaaameaacaWGPbaabeaaaS % qaaiaadIhadaWgaaadbaGaamOzaaqabaaaniabgUIiYdGccqGH9aqp % daWcaaqaaiaadUgacaWG4bWaa0baaSqaaiaadMgaaeaacaaIYaaaaa % GcbaGaaGOmaaaacqGHsisldaWcaaqaaiaadUgacaWG4bWaa0baaSqa % aiaadAgaaeaacaaIYaaaaaGcbaGaaGOmaaaadaqadaqaaiaaiAdaca % GGUaGaaGymaiaaiEdaaiaawIcacaGLPaaaaaa!540C! {W_s} = - \int\limits_{{x_i}}^{{x_f}} {kxdx} = \frac{{kx_i^2}}{2} - \frac{{kx_f^2}}{2}\left( {6.17} \right)\)

Thus the work done by the spring force depends only on the end points. Specifically,if the block is pulled from x_i and allowed to return to x_i ;

\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGxb % WaaSbaaSqaaiaadohaaeqaaOGaeyypa0JaeyOeI0Yaa8qCaeaacaWG % RbGaamiEaiaadsgacaWG4baaleaacaWG4bWaaSbaaWqaaiaadMgaae % qaaaWcbaGaamiEamaaBaaameaacaWGPbaabeaaa0Gaey4kIipakiab % g2da9maalaaabaGaam4AaiaadIhadaqhaaWcbaGaamyAaaqaaiaaik % daaaaakeaacaaIYaaaaiabgkHiTmaalaaabaGaam4AaiaadIhadaqh % aaWcbaGaamyAaaqaaiaaikdaaaaakeaacaaIYaaaaaqaaiabg2da9i % aaicdadaqadaqaaiaaiAdacaGGUaGaaGymaiaaiIdaaiaawIcacaGL % Paaaaaaa!55DA! \begin{array}{l} {W_s} = - \int\limits_{{x_i}}^{{x_i}} {kxdx} = \frac{{kx_i^2}}{2} - \frac{{kx_i^2}}{2}\\ = 0\left( {6.18} \right) \end{array}\)

The work done by the spring force in a cyclic process is zero. We have explicitly demonstrated that the spring force (i) is position dependent only as first stated by Hooke, (F_s = - kx); (ii) does work which only depends on the initial and final positions, e.g. Eq. (6.17). Thus, the spring force is a conservative force.

We define the potential energy V(x) of the spring to be zero when block and spring system is in the equilibrium position. For an extension (or compression) x the above analysis suggests that

 \(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaabm % aabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaam4Aaiaa % dIhadaahaaWcbeqaaiaaikdaaaaakeaacaaIYaaaaaaa!3E09! V\left( x \right) = \frac{{k{x^2}}}{2}\)       (6.19)

You may easily verify that -dV/dx = -k x, the spring force. If the block of mass m in Fig. 6.7 is extended to xm and released from rest, then its total mechanical energy at any arbitrary point x, where x lies between – x_m  and + x_m, will be given by

\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % aIXaaabaGaaGOmaaaacaWGRbGaamiEamaaDaaaleaacaWGTbaabaGa % aGOmaaaakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaam4Aai % aadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkdaWcaaqaaiaaigda % aeaacaaIYaaaaiaad2gacaWG2bWaaWbaaSqabeaacaaIYaaaaaaa!45FB! \frac{1}{2}kx_m^2 = \frac{1}{2}k{x^2} + \frac{1}{2}m{v^2}\)

where we have invoked the conservation of mechanical energy. This suggests that the speed and the kinetic energy will be maximum at the equilibrium position, x = 0, i.e.,

\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % aIXaaabaGaaGOmaaaacaWGTbGaamODamaaDaaaleaacaWGTbaabaGa % aGOmaaaakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaam4Aai % aadIhadaqhaaWcbaGaamyBaaqaaiaaikdaaaaaaa!41A4! \frac{1}{2}mv_m^2 = \frac{1}{2}kx_m^2\)   where v_m is the maximum speed.

or \(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBa % aaleaacaWGTbaabeaakiabg2da9maakaaabaWaaSaaaeaacaWGRbaa % baGaamyBaaaaaSqabaGccaWG4bWaaSbaaSqaaiaad2gaaeqaaaaa!3D51! {v_m} = \sqrt {\frac{k}{m}} {x_m}\)

​​​​​​We obtain

\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa % aaleaacaWGTbaabeaakiabg2da9iaaikdacaGGUaGaaGimaiaaicda % caWGTbaaaa!3CF5! {x_m} = 2.00m\)

We note that we have idealised the situation. The spring is considered to be massless.The surface has been considered to possess negligible friction.

We conclude this section by making a few remarks on conservative forces.

1. Information on time is absent from the above discussions. In the example considered above, we can calculate the compression, but not the time over which the compression occurs. A solution of Newton’s Second Law for this system is required for temporal information.

2. Not all forces are conservative. Friction, for example, is a non-conservative force. The principle of conservation of energy will have to be modified in this case. This is illustrated in Example 6.9.

3. The zero of the potential energy is arbitrary. It is set according to convenience.For the spring force we took V(x) = 0, at x = 0, i.e. the unstretched spring had zero potential energy. For the constant gravitational force mg, we took V = 0 on the earth’s surface. In a later chapter we shall see that for the force due to the universal law of gravitation, the zero is best defined at an infinite distance from the gravitational source. However, once the zero of the potential energy is fixed in a given discussion, it must be consistently adhered to throughout the discussion. You cannot change horses in midstream !

EXAMPLE 9

Consider Example 6.8 taking the coefficient of friction,\(\mu\), to be 0.5 and calculate the maximum compression of the spring.

ANSWER

In presence of friction, both the spring force and the frictional force act so as to oppose the compression of the spring as shown in Fig. 6.9.

We invoke the work-energy theorem, rather than the conservation of mechanical energy.

The change in kinetic energy is

                             ​​​​​​​

                     Fig.6.9.The forces acting on the car.

\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam % 4saiabg2da9iaadUeadaWgaaWcbaGaamOzaaqabaGccqGHsislcaWG % lbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaaGimaiabgkHiTmaala % aabaGaaGymaaqaaiaaikdaaaGaamyBaiaadAhadaahaaWcbeqaaiaa % ikdaaaaaaa!450F! \Delta K = {K_f} - {K_i} = 0 - \frac{1}{2}m{v^2}\)

The work done by the net force is

\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 % da9iabgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaGaam4AaiaadIha % daqhaaWcbaGaamyBaaqaaiaaikdaaaGccqGHsislcqaH8oqBcaWGTb % Gaam4zaiaadIhadaWgaaWcbaGaamyBaaqabaaaaa!44BA! W = - \frac{1}{2}kx_m^2 - \mu mg{x_m}\)

Equating we have

\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % aIXaaabaGaaGOmaaaacaWGTbGaamODamaaCaaaleqabaGaaGOmaaaa % kiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaam4AaiaadIhada % qhaaWcbaGaamyBaaqaaiaaikdaaaGccqGHRaWkcqaH8oqBcaWGTbGa % am4zaiaadIhadaWgaaWcbaGaamyBaaqabaaaaa!474D! \frac{1}{2}m{v^2} = \frac{1}{2}kx_m^2 + \mu mg{x_m}\)

Now \(\mu\)mg = 0.5 \(\times\) 10³ \(\times\) 10 \(\times\) 5 \(\times\) 10 ³ N (taking g =10.0 m s^-2).After rearranging the above equation we obtain the following quadratic equation in the unknown x_m.

\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGRb % GaamiEamaaDaaaleaacaWGTbaabaGaaGOmaaaakiabgUcaRiaaikda % cqaH8oqBcaWGTbGaam4zaiaadIhadaWgaaWcbaGaamyBaaqabaGccq % GHsislcaWGTbGaamODamaaCaaaleqabaGaaGOmaaaakiabg2da9iaa % icdaaeaacaWG4bWaaSbaaSqaaiaad2gaaeqaaOGaeyypa0ZaaSaaae % aacqGHsislcqaH8oqBcaWGTbGaam4zaiabgUcaRmaadmaabaGaeqiV % d02aaWbaaSqabeaacaaIYaaaaOGaamyBamaaCaaaleqabaGaaGOmaa % aakiaadEgadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGTbGaam4A % aiaadAhadaahaaWcbeqaaiaaikdaaaaakiaawUfacaGLDbaadaahaa % WcbeqaaiaaigdacaGGVaGaaGOmaaaaaOqaaiaadUgaaaaaaaa!5FB3! \begin{array}{l} kx_m^2 + 2\mu mg{x_m} - m{v^2} = 0\\ {x_m} = \frac{{ - \mu mg + {{\left[ {{\mu ^2}{m^2}{g^2} + mk{v^2}} \right]}^{1/2}}}}{k} \end{array}\)

where we take the positive square root since x_m is positive.Putting in numerical values we obtain x_m=1.35m.which, as expected, is less than the result in Example 6.8.If the two forces on the body consist of a conservative force F_c and a non-conservative force F_nc ,the conservation of mechanical energy formula will have to be modified. By the WE theorem

\(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaqada % qaaiaadAeadaWgaaWcbaGaam4yaaqabaGccqGHRaWkcaWGgbWaaSba % aSqaaiaad6gacaWGJbaabeaaaOGaayjkaiaawMcaaiabgs5aejaadI % hacqGH9aqpcqGHuoarcaWGlbaabaGaamOqaiaadwhacaWG0bGaamOr % amaaBaaaleaacaWGJbaabeaakiabgs5aejaadIhacqGH9aqpcqGHsi % slcqGHuoarcaWGwbaabaGaamisaiaadwgacaWGUbGaam4yaiaadwga % caGGSaGaeyiLdq0aaeWaaeaacaWGlbGaey4kaSIaamOvaaGaayjkai % aawMcaaiabg2da9iaadAeadaWgaaWcbaGaamOBaiaadogaaeqaaOGa % eyiLdqKaamiEaaqaaiabgs5aejaadweacqGH9aqpcaWGgbWaaSbaaS % qaaiaad6gacaWGJbaabeaakiabgs5aejaadIhaaaaa!6772! \begin{array}{l} \left( {{F_c} + {F_{nc}}} \right)\Delta x = \Delta K\\ But{F_c}\Delta x = - \Delta V\\ Hence,\Delta \left( {K + V} \right) = {F_{nc}}\Delta x\\ \Delta E = {F_{nc}}\Delta x \end{array}\)

where E is the total mechanical energy. Over the path this assumes the form \(% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa % aaleaacaWGMbaabeaakiabgkHiTiaadweadaWgaaWcbaGaamyAaaqa % baGccqGH9aqpcaWGxbWaaSbaaSqaaiaad6gacaWGJbaabeaaaaa!3EA5! {E_f} - {E_i} = {W_{nc}}\)