The dimensions of the quantities in one (or more) of the following pairs are the same. Identify the pair(s)

Step 1: Define the dimensions of force.

Force is defined as mass times acceleration.

$$[F] = [M L T^{-2}]$$

Step 2: Compare the dimensions of Torque and Work.

Torque ($\tau$) is the product of force and perpendicular distance.

$$[\tau] = [r \times F] = [L][M L T^{-2}] = [M L^{2} T^{-2}]$$

Work ($W$) is the product of force and displacement.

$$[W] = [F \cdot d] = [M L T^{-2}][L] = [M L^{2} T^{-2}]$$

Both quantities have the same dimensions.

Step 3: Compare the dimensions of Angular momentum and Work.

Angular momentum ($L$) is the product of mass, velocity, and radius.

$$[L] = [mvr] = [M][L T^{-1}][L] = [M L^{2} T^{-1}]$$

Since the dimension of work is $[M L^{2} T^{-2}]$, these two quantities have different dimensions.

Step 4: Compare the dimensions of Energy and Young's modulus.

Energy ($E$) has the same dimensions as work.

$$[E] = [M L^{2} T^{-2}]$$

Young's modulus ($Y$) is the ratio of stress to strain, where strain is dimensionless.

$$[Y] = \frac{[F]}{[A]} = \frac{[M L T^{-2}]}{[L^{2}]} = [M L^{-1} T^{-2}]$$

These two quantities have different dimensions.

Step 5: Compare the dimensions of Light year and Wavelength.

A light year is the distance traveled by light in one year.

$$[\text{Light year}] = [L]$$

Wavelength ($\lambda$) is the physical length of a wave cycle.

$$[\lambda] = [L]$$

Both quantities have the same dimensions.

Answer: The pairs with the same dimensions are (1) Torque and Work and (4) Light year and Wavelength.