The equation in the image is:

$T = 2\pi \sqrt{\frac{m}{k}}$

Where:

• $T$ is the period of the oscillation,
• $m$ is the mass of the object,
• $k$ is the spring constant (or stiffness),
• $2\pi$ is a constant,
• and the term $\sqrt{\frac{m}{k}}$ describes the relationship between mass and the spring constant in harmonic motion.

This equation describes the period $T$ of a mass $m$ attached to a spring undergoing simple harmonic motion, such as in a mass-spring system.

Proof:

This formula comes from Hooke's Law and Newton's second law of motion.

1. Hooke's Law: The restoring force $F$ in a spring is proportional to the displacement $x$ from the equilibrium position: $F = -kx$ where $k$ is the spring constant.

2. Newton's Second Law: $F = ma$, where $a$ is the acceleration, and $m$ is the mass. For a mass on a spring, acceleration $a$ is the second derivative of displacement $x$ with respect to time $t$: $F = m \frac{d^2x}{dt^2}$

   Equating the two expressions for the force: $m \frac{d^2x}{dt^2} = -kx$

3. Rearranging the equation: $\frac{d^2x}{dt^2} + \frac{k}{m}x = 0$ This is the standard form of a simple harmonic motion equation, where the solution is: $x(t) = A \cos(\omega t + \phi)$ where $A$ is the amplitude, $\phi$ is the phase angle, and $\omega$ is the angular frequency.

4. The angular frequency $\omega$ is related to the spring constant and mass by: $\omega = \sqrt{\frac{k}{m}}$

5. The period $T$ of oscillation is related to $\omega$ by the formula: $T = \frac{2\pi}{\omega}$

6. Substituting the expression for $\omega$: $T = \frac{2\pi}{\sqrt{\frac{k}{m}}} = 2\pi \sqrt{\frac{m}{k}}$

Thus, we arrive at the equation:

$T = 2\pi \sqrt{\frac{m}{k}}$

This proves the formula for the period of a mass-spring system in simple harmonic motion.

Let me know if you have any questions or need further clarification!

Related questions:

1. How does increasing the mass $m$ affect the period of oscillation $T$?
2. What happens to the period if the spring constant $k$ is doubled?
3. Can you derive the equation for the frequency $f$ from the period $T$?
4. How does this formula apply to a pendulum system?
5. What assumptions are made in deriving this equation for simple harmonic motion?

Tip:

In any harmonic motion, the period is independent of the amplitude of oscillation, meaning even large oscillations follow the same time period if the system remains in simple harmonic motion.