Physics · Mechanical Properties Of Solids · NEET
No, they are related but different. Elastic potential energy (U) is the total energy stored in the whole wire, measured in joules (J). Elastic energy density (u) is that energy divided by the volume of the wire, measured in J/m^3. So u = U / (A x L), where A is the area of cross-section and L is the length. For NEET, read the question carefully: 'per unit volume' or 'energy density' means use u = 1/2 x stress x strain; 'energy stored in the wire' means use U = 1/2 x F x l.
The SI unit is J/m^3 (joule per cubic metre). One J/m^3 equals one N/m^2 (pascal), the same unit as stress and pressure. This is because energy density = stress x strain (dimensionless strain), so it carries the same units as stress. In dimensions it is [M L^-1 T^-2]. Do not write it as J/m^2 - a common slip that loses marks.
When you stretch a wire, the internal restoring force is not constant. It starts at zero and grows in proportion to the extension (Hooke's law). So the average force is only half the final force. Work done = average force x extension gives the 1/2. This is the same reason a spring stores 1/2 k x^2, not k x^2. The next page 'Why elastic energy is half stress times strain' derives this fully.
Start from U = 1/2 x F x l (total energy). Write stress = F/A and strain = l/L. Then U = 1/2 x (stress x A) x (strain x L) = 1/2 x stress x strain x (A x L). Since A x L is the volume V, dividing both sides by V gives energy density u = U/V = 1/2 x stress x strain. This one step (dividing total energy by volume A x L) is the key link between the two formulas.
Both are correct and give the same answer, since stress = Y x strain. Use (1/2) Y (strain)^2 when the question gives you the extension or strain (like length and change in length). Use (stress)^2/(2Y) when the question gives you the applied stress, force per area, or pressure. Pick whichever matches the data given so you avoid extra steps.
The amount of elastic potential energy per unit volume (in SI unit) of a steel wire of length 100 cm stretched by 1 mm is (Young's modulus of the wire = 2.0 x 10^11 N/m^2)
Try the real previous-year questions from this chapter — each with the answer and a full solution.
u = 1/2 x stress x strain. Using Young's modulus Y (with stress = Y x strain) it can be written as u = 1/2 x Y x (strain)^2 or u = (stress)^2 / (2Y). All three forms are equal.
J/m^3 (joule per cubic metre), which is the same as N/m^2 or pascal. Its dimensional formula is [M L^-1 T^-2].
Total elastic potential energy U (in joules) is for the whole wire; energy density u (in J/m^3) is energy per unit volume. They are linked by u = U / volume = U / (A x L).
The restoring force grows from zero to its final value as the wire stretches, so the average force is half the maximum. Work = average force x extension gives the factor 1/2, just like 1/2 k x^2 for a spring.
Use (1/2) Y (strain)^2 when strain or extension is given, and (stress)^2/(2Y) when stress or force per area is given. Both give the same result.
