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more than 3 weeks ago

The number 28 is an even number that comes after 27 and before 29. It is the sum of the first seven prime numbers (2 + 3 + 5 + 7 + 11 + 13 + 17 = 28) and is also a perfect number, meaning that the sum of its divisors (excluding itself) equals the number itself (1 + 2 + 4 + 7 + 14 = 28). In Roman numerals, 28 is represented as XXVIII.

Google Gemini AI

more than 3 weeks ago

The total potential energy of the system is given by:

$$V(x_1, x_2) = V(x_1) + V(x_2)$$

Since the potential is symmetric about the origin, we can write:

$$V(x_1) = V_0 + \frac{1}{2}k_1 x_1^2$$

$$V(x_2) = V_0 + \frac{1}{2}k_1 x_2^2$$

where $V_0$ and $k_1$ are constants.

The Lagrangian of the system is:

$$L = \frac{1}{2}m\dot{x}_1^2 + \frac{1}{2}m\dot{x}_2^2 - V_0 - \frac{1}{2}k_1x_1^2 - \frac{1}{2}k_1x_2^2$$

The equations of motion are:

$$m\ddot{x}_1 + k_1x_1 = 0$$

$$m\ddot{x}_2 + k_1x_2 = 0$$

The normal modes of oscillation are given by the solutions to the above equations. We can write the solutions in the form:

$$x_1(t) = A_1\sin(\omega t + \phi_1)$$

$$x_2(t) = A_2\sin(\omega t + \phi_2)$$

where $A_1$, $A_2$, $\phi_1$, and $\phi_2$ are constants.

Substituting the above solutions into the equations of motion, we get:

$$-m\omega^2 A_1 \sin(\omega t + \phi_1) + k_1A_1\sin(\omega t + \phi_1) = 0$$

$$-m\omega^2 A_2 \sin(\omega t + \phi_2) + k_1A_2\sin(\omega t + \phi_2) = 0$$

Dividing both sides of the above equations by $A_1\sin(\omega t + \phi_1)$ and $A_2\sin(\omega t + \phi_2)$, respectively, we get:

$$-\omega^2 + \frac{k_1}{m} = 0$$

$$\omega = \sqrt{\frac{k_1}{m}}$$

Since the potential is symmetric about the origin, the two particles will oscillate in phase, i.e., $\phi_1 = \phi_2$. Therefore, the normal modes of oscillation are:

$$x_1(t) = A\sin(\omega t)$$

$$x_2(t) = A\sin(\omega t)$$

where $A$ is a constant.