To verify that the period of oscillation has units of time, we can apply dimensional analysis. The equation provided is:

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

Step 1: Identify the dimensions of the parameters

We need to break down the dimensional units for each parameter involved in the equation:

1. $t$ - Time period (we need to verify that this results in units of time).
2. $m$ - Mass, which has the dimension of [M] (mass).
3. $k$ - Spring constant (stiffness), which is related to force per unit displacement. From Hooke's law, $F = kx$, where $F$ is force and $x$ is displacement.
   • The dimension of force is [M][L][T]^{-2}, where:
     • [M] = mass,
     • [L] = length,
     • [T] = time.
   • Displacement $x$ has the dimension of length [L].
   • Therefore, the spring constant $k$ has the dimension of force per length:
     [ [k] = \frac{[M][L][T]^{-2}}{[L]} = [M][T]^{-2} ]
4. $\pi$ - Dimensionless constant, so it has no dimensions.

Step 2: Analyze the right-hand side of the equation

We need to focus on the argument under the square root, $\frac{m}{k}$.

[ \frac{m}{k} = \frac{[M]}{[M][T]^{-2}} = [T]^2 ]

Step 3: Apply the square root

Taking the square root of the result, we get:

[ \sqrt{\frac{m}{k}} = \sqrt{[T]^2} = [T] ]

Step 4: Check the overall dimensions of the equation

Since the expression $\sqrt{\frac{m}{k}}$ has the dimension of time [T], the period of oscillation $t$ must also have units of time:

$t = 2 \sqrt{\frac{m}{k}} \quad \implies \quad t \ \text{has units of} \ [T]$

Thus, by dimensional analysis, we have confirmed that the period of oscillation does indeed have units of time.

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Would you like more details or have any questions?

Here are 5 related questions:

1. What factors affect the period of oscillation in simple harmonic motion?
2. How does the spring constant $k$ influence the period?
3. Can dimensional analysis be applied to other physics problems? How?
4. What is the significance of $\pi$ in the formula?
5. How would the period change if the mass $m$ is doubled?

Tip: Dimensional analysis is a powerful tool for checking the consistency of equations, especially when you're unsure of a derived formula.