Step 1: Determine the dimensions of Young's modulus
Young's modulus is stress divided by strain. Stress = force/area, dimensions: $[F]/[A] = [M L T^{-2}]/[L^2] = [M L^{-1} T^{-2}]$. Strain is dimensionless. Thus, Young's modulus has dimensions $[M L^{-1} T^{-2}]$.
Step 2: Compare with the options
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(1) Strain: dimensionless, $[M^0 L^0 T^0]$.
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(2) Gravitational potential: energy/mass, $[E]/[M] = [M L^2 T^{-2}]/[M] = [L^2 T^{-2}]$.
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(3) Surface energy: energy/area, $[E]/[A] = [M L^2 T^{-2}]/[L^2] = [M T^{-2}]$.
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(4) Energy density: energy/volume, $[E]/[V] = [M L^2 T^{-2}]/[L^3] = [M L^{-1} T^{-2}]$, matches Young's modulus.
Step 3: Select the correct option
Energy density has the same dimensions as Young's modulus, so the answer is (4).