A mass 𝑚, suspended by a spring of stiffness 𝑘, is set into simple harmonic motion. The period of oscillation is given by: 
t=2√(π&m/k)

Using dimensional analysis, demonstrate that the period of oscillation does, indeed, have units of time. 


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.

---

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.

A mass 𝑚, suspended by a spring of stiffness 𝑘, is set into simple harmonic motion. The period of oscillation is given by: t=2√(πm/k). Using dimensional analysis, demonstrate that the period of oscillation does, indeed, have units of time.

Use dimensional analysis to demonstrate that the period of oscillation for a mass suspended by a spring has units of time. This problem involves key physics concepts such as simple harmonic motion, the spring constant, and the relationship between mass and stiffness. Suitable for students in Grades 10-12.

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