Force and Energy in SHM

Force Law For Simple Harmonic Motion ($ F = -kx $)

As established previously, the defining condition for Simple Harmonic Motion (SHM) is the presence of a linear restoring force. This means that the net force acting on the oscillating object is directly proportional to its displacement from the equilibrium position and is always directed towards the equilibrium.

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The Restoring Force

Consider an object undergoing linear SHM along the x-axis, with the equilibrium position at $x=0$. The restoring force $F$ at any displacement $x$ is given by:

$ F = -kx $

where $k$ is the positive force constant (or stiffness constant). The negative sign is crucial; it signifies that the force is always opposite in direction to the displacement. If the object is displaced to the right ($x > 0$), the force is to the left ($F < 0$). If the object is displaced to the left ($x < 0$), the force is to the right ($F > 0$). This force always tries to pull or push the object back towards the equilibrium position ($x=0$).

This force law is a specific case of a restoring force where the relationship is linear with displacement. Many restoring forces in nature are approximately linear for small displacements from equilibrium (e.g., the force exerted by a spring obeying Hooke's Law, the restoring torque on a simple pendulum at small angles).

Applying Newton's Second Law, $F = ma$, gives the differential equation of SHM:

$ m \frac{d^2x}{dt^2} = -kx $

$ \frac{d^2x}{dt^2} + \frac{k}{m}x = 0 $

The angular frequency of the resulting SHM is $\omega = \sqrt{k/m}$. The force constant $k$ is a measure of the stiffness of the system; a larger $k$ means a stronger restoring force for a given displacement and thus a higher frequency (shorter period) of oscillation.

Energy In Simple Harmonic Motion ($ E = \frac{1}{2}kA^2 $)

An object undergoing Simple Harmonic Motion possesses both kinetic energy (due to its motion) and potential energy (due to its position in the conservative force field associated with the restoring force). Since the restoring force in SHM ($F = -kx$) is a conservative force, the total mechanical energy of the system (sum of kinetic and potential energy) is conserved in the absence of non-conservative forces like friction or air resistance.

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Potential Energy in SHM

The restoring force $F = -kx$ is a conservative force. The potential energy $U(x)$ associated with this force can be found by calculating the work done by the force as the object moves from the equilibrium position ($x=0$) to displacement $x$. The change in potential energy is the negative of the work done by the conservative force:

$ \Delta U = U(x) - U(0) = -\int_0^x F(x') \, dx' $

Let's take the potential energy at the equilibrium position ($x=0$) as zero, $U(0) = 0$.

$ U(x) = -\int_0^x (-kx') \, dx' = \int_0^x kx' \, dx' $

$ U(x) = k \left[ \frac{(x')^2}{2} \right]_0^x = k \left(\frac{x^2}{2} - 0\right) $

$ U(x) = \frac{1}{2} kx^2 $

This is the potential energy stored in the system when the object is at displacement $x$ from equilibrium. This is similar to the potential energy stored in a stretched or compressed spring, which is consistent since $F=-kx$ is Hooke's Law.

The potential energy is maximum at the extreme positions ($x = \pm A$), where $U_{max} = \frac{1}{2} kA^2$, and minimum at the equilibrium position ($x=0$), where $U_{min} = 0$.

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Kinetic Energy in SHM

The kinetic energy $KE(t)$ of the object of mass $m$ at time $t$ with velocity $v(t)$ is:

$ KE(t) = \frac{1}{2} m v(t)^2 $

Using the expression for velocity in terms of displacement, $v = \pm \omega \sqrt{A^2 - x^2}$, and $\omega^2 = k/m$ (so $m = k/\omega^2$):

$ KE(x) = \frac{1}{2} m [\omega \sqrt{A^2 - x^2}]^2 = \frac{1}{2} m \omega^2 (A^2 - x^2) $

$ KE(x) = \frac{1}{2} \left(\frac{k}{\omega^2}\right) \omega^2 (A^2 - x^2) = \frac{1}{2} k (A^2 - x^2) $

The kinetic energy is maximum at the equilibrium position ($x=0$), where $KE_{max} = \frac{1}{2} kA^2$, and minimum (zero) at the extreme positions ($x = \pm A$), where $KE_{min} = 0$.

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Total Mechanical Energy in SHM

The total mechanical energy $E$ is the sum of the kinetic energy and potential energy:

$ E(x) = KE(x) + U(x) $

$ E(x) = \frac{1}{2} k (A^2 - x^2) + \frac{1}{2} kx^2 $

$ E(x) = \frac{1}{2} kA^2 - \frac{1}{2} kx^2 + \frac{1}{2} kx^2 $

$ E(x) = \frac{1}{2} kA^2 $

This shows that the total mechanical energy of a system undergoing SHM is constant and is proportional to the square of the amplitude ($A^2$). It is purely potential energy at the extreme points and purely kinetic energy at the equilibrium point. At any other point, the energy is a mix of kinetic and potential, but their sum is constant.

Using $\omega^2 = k/m$, we can also write $k = m\omega^2$. The total energy is:

$ E = \frac{1}{2} m \omega^2 A^2 $

This formula shows the total energy depends on the mass, angular frequency, and amplitude. Since the energy is conserved in SHM, the amplitude of oscillation remains constant.

(Image Placeholder: A graph with Displacement (x) on the x-axis, ranging from -A to +A. Y-axis is Energy. Show a parabola for Potential Energy U = 1/2 kx^2, with minimum at x=0 and maximum at x=+-A. Show an inverted parabola for Kinetic Energy KE = 1/2 k(A^2-x^2), with maximum at x=0 and minimum (zero) at x=+-A. Show a horizontal line at E = 1/2 kA^2, representing the constant Total Energy, which is the sum of U and KE at any x.)

The conservation of energy in SHM is a powerful tool for analysing the motion and relating velocity and displacement without explicitly using the time-dependent equations.

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