Which of the two have same dimensions

To determine which of the given pairs of physical quantities have the same dimensions, let's analyze each pair step-by-step by finding their dimensional formulas.

1. Force and Strain:
- Force is defined as mass times acceleration ($F=ma$). Its dimensional formula is:
$\left[\text{Force}\right]=\left[M^1{L}^1{T}^{-2}\right]$
- Strain is defined as the ratio of change in dimension to the original dimension (e.g., change in length divided by original length, $\frac{\Delta L}{L}$). Since it is a ratio of two similar physical quantities, it is a dimensionless quantity:
$\left[\text{Strain}\right]=\left[M^0{L}^0{T}^0\right]$
Thus, Force and Strain do not have the same dimensions.

2. Force and Stress:
- Force has the dimensional formula:
$\left[\text{Force}\right]=\left[M^1{L}^1{T}^{-2}\right]$
- Stress is defined as restoring force per unit area ($\frac{\text{Force}}{\text{Area}}$). Its dimensional formula is:
$\left[\text{Stress}\right]=\frac{\left[M^1{L}^1{T}^{-2}\right]}{\left[L^2\right]}=\left[M^1{L}^{-1}{T}^{-2}\right]$
Thus, Force and Stress do not have the same dimensions.

3. Angular Velocity and Frequency:
- Angular velocity ($\omega$) is defined as the rate of change of angular displacement ($\frac{d\theta }{dt}$). Since angle ($\theta$) is dimensionless, the dimensional formula for angular velocity is:
$\left[\text{Angular Velocity}\right]=\left[T^{-1}\right]=\left[M^0{L}^0{T}^{-1}\right]$
- Frequency ($\nu$ or $f$) is defined as the number of oscillations per unit time ($\frac{1}{T}$). Its dimensional formula is:
$\left[\text{Frequency}\right]=\left[T^{-1}\right]=\left[M^0{L}^0{T}^{-1}\right]$
Both Angular velocity and Frequency share the exact same dimensional formula, $\left[M^0{L}^0{T}^{-1}\right]$.

4. Energy and Strain:
- Energy (work done) has the dimensional formula:
$\left[\text{Energy}\right]=\left[M^1{L}^2{T}^{-2}\right]$
- Strain is dimensionless:
$\left[\text{Strain}\right]=\left[M^0{L}^0{T}^0\right]$
Thus, Energy and Strain do not have the same dimensions.

Therefore, Angular velocity and frequency have the same dimensions.