Energy and the Simple Harmonic Oscillator
By the end of this section, you will be able to:
To study the energy of a simple harmonic oscillator, we first consider all the forms of energy it can have. We know from Hooke’s Law: Stress and Strain Revisited that the energy stored in the deformation of a simple harmonic oscillator is a form of potential energy given by:
Because a simple harmonic oscillator has no dissipative forces, the other important form of energy is kinetic energy \(\mathrm{KE}\). Conservation of energy for these two forms is:
or
This statement of conservation of energy is valid for all simple harmonic oscillators, including ones where the gravitational force plays a role
Namely, for a simple pendulum we replace the velocity with \(v = L \omega\), the spring constant with \(k = mg/L\), and the displacement term with \(x = L \theta\). Thus
In the case of undamped simple harmonic motion, the energy oscillates back and forth between kinetic and potential, going completely from one to the other as the system oscillates. So for the simple example of an object on a frictionless surface attached to a spring, as shown again in Figure 1, the motion starts with all of the energy stored in the spring. As the object starts to move, the elastic potential energy is converted to kinetic energy, becoming entirely kinetic energy at the equilibrium position. It is then converted back into elastic potential energy by the spring, the velocity becomes zero when the kinetic energy is completely converted, and so on. This concept provides extra insight here and in later applications of simple harmonic motion, such as alternating current circuits.
Figure 1 The transformation of energy in simple harmonic motion is illustrated for an object attached to a spring on a frictionless surface.
The conservation of energy principle can be used to derive an expression for velocity \(v\). If we start our simple harmonic motion with zero velocity and maximum displacement \((x = X)\), then the total energy is
This total energy is constant and is shifted back and forth between kinetic energy and potential energy, at most times being shared by each. The conservation of energy for this system in equation form is thus:
Solving this equation for \(v\) yields:
Manipulating this expression algebraically gives:
where
From this expression, we see that the velocity is a maximum \((v_{\mathrm{max}}) \) at \(x =0\), as stated earlier in \(v(t) = -v_{\mathrm{max}} \sin \frac{2 \pi t}{T}\). Notice that the maximum velocity depends on three factors. Maximum velocity is directly proportional to amplitude. As you might guess, the greater the maximum displacement the greater the maximum velocity. Maximum velocity is also greater for stiffer systems, because they exert greater force for the same displacement. This observation is seen in the expression for vmax;vmax; it is proportional to the square root of the force constant \(k\). Finally, the maximum velocity is smaller for objects that have larger masses, because the maximum velocity is inversely proportional to the square root of \(m\). For a given force, objects that have large masses accelerate more slowly.
A similar calculation for the simple pendulum produces a similar result, namely:
Consider a mass m attached to a spring, with spring constant k, fixed to a wall. When the mass is displaced from its equilibrium position and released, the mass undergoes simple harmonic motion. The spring exerts a force \(F = - kv\) on the mass. The potential energy of the system is stored in the spring. It will be zero when the spring is in the equilibrium position. All the internal energy exists in the form of kinetic energy, given by \(KE = \frac{1}{2} mv^2\). As the system oscillates, which means that the spring compresses and expands, there is a change in the structure of the system and a corresponding change in its internal energy. Its kinetic energy is converted to potential energy and vice versa. This occurs at an equal rate, which means that a loss of kinetic energy yields a gain in potential energy, thus preserving the work-energy theorem and the law of conservation of energy.
Suppose that a car is 900 kg and has a suspension system that has a force constant \(k= 6.53 \times 10^4 \thinspace \mathrm{N/m}\). The car hits a bump and bounces with an amplitude of 0.100 m. What is its maximum vertical velocity if you assume no damping occurs?
Strategy
We can use the expression for \(v_{\mathrm{max}}\) given in \(v_{\mathrm{max}} = \sqrt{\frac{k}{m}}X\) to determine the maximum vertical velocity. The variables \(m\) and \(k\) are given in the problem statement, and the maximum displacement \(X\) is 0.100 m.
Solution
Discussion
This answer seems reasonable for a bouncing car. There are other ways to use conservation of energy to find \(v_{\mathrm{max}}\). We could use it directly, as was done in the example featured in Hooke’s Law: Stress and Strain Revisited.
The small vertical displacement y of an oscillating simple pendulum, starting from its equilibrium position, is given as
where \(a\) is the amplitude, \(\omega\) is the angular velocity and \(t\) is the time taken. Substituting \(\omega = \frac{2 \pi}{T}\), we have
Thus, the displacement of pendulum is a function of time as shown above.
Also the velocity of the pendulum is given by
so the motion of the pendulum is a function of time.
