https://file.aiquickdraw.com/mathbot/file/1727410990083l08jc1j7.jpg

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.

Buktikan persamaan berikut: T = 2π√(m/k)

Understand and prove the equation for the period of a mass-spring system in simple harmonic motion: T = 2π√(m/k). This proof involves key concepts such as Hooke's Law, Newton's Second Law, and harmonic motion. Suitable for students in Grades 10-12.

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