Elastic Potential Energy in a Stretched Wire - UNSOLVED PRACTICE SET

Q7. Derive the expression for elastic potential energy stored in a stretched wire. Show that it equals ½ FΔL.

Q8. A wire of length L and cross-sectional area A is stretched by an amount l. Show that the energy stored per unit volume is ½ × Y × (l/L)².

Q9. A spring with force constant 200 N/m is stretched by 5 cm. Calculate the energy stored in the spring.

Q10. In your school, a student stretches two identical springs — one by 2 cm and another by 4 cm. Compare the energy stored in each. What does this tell you about how energy depends on extension?

Q11. A wire of length 2 m and cross-sectional area 10⁻⁶ m² is stretched by 2 mm. Young's modulus is 2 × 10¹¹ Pa. Calculate the energy stored in the wire.

Q12. Why is the factor ½ present in the expression for elastic potential energy? Explain using the force-extension graph.

Q13. Derive the expression for elastic potential energy stored in a stretched wire starting from first principles. Show that:

(a) U = ½ FΔL

(b) U = ½ × stress × strain × volume

(c) U = ½ Y(strain)² × volume

(d) U = ½ (stress)²/Y × volume

Explain why the energy density is given by ½ σ ε and discuss its significance.

Q14. A steel wire of length 3 m and diameter 1 mm is stretched by a load of 10 kg.

(a) Calculate the extension of the wire. (Y = 2 × 10¹¹ Pa)

(b) Calculate the elastic potential energy stored.

(c) Calculate the energy stored per unit volume.

(d) If this energy were converted entirely to heat, calculate the temperature rise. (Specific heat capacity of steel = 460 J/kg·K, density = 7860 kg/m³)

(e) If the load is suddenly removed, what happens to this energy?

Q15. Compare the energy storage in different systems:

(a) A steel spring compressed by 5 cm

(b) A rubber band stretched by 5 cm

(c) A compressed gas in a cylinder

(d) A raised weight

For each system, discuss:

(i) The formula for stored energy

(ii) Which is most efficient for energy storage

(iii) Practical applications

(iv) Advantages and limitations

Q16. In a school experiment, a student investigates the energy stored in stretched rubber bands of different cross-sections:

Table

Rubber Band Length (cm) Cross-section (mm²) Extension (cm) Force at Extension (N)

A 10 2 2 4

B 10 4 2 8

C 20 2 2 2

(a) Calculate the energy stored in each rubber band.

(b) Calculate the energy stored per unit volume for each.

(c) Which rubber band stores the most energy per unit volume?

(d) If rubber band A is used in a catapult and stretched by 4 cm instead of 2 cm, how much more energy is stored?

(e) Discuss why rubber is preferred over steel for applications like catapults and slingshots.

Q17. A 2 kg mass is attached to a steel wire of length 4 m and cross-sectional area 0.5 mm². The mass is pulled sideways until the wire makes an angle of 30° with the vertical and then released.

(a) Calculate the extension of the wire when the mass is at the lowest point.

(b) Calculate the elastic potential energy stored in the wire at this point.

(c) Calculate the speed of the mass as it passes through the lowest point.

(d) Calculate the maximum angle on the other side if energy is conserved.

(e) Discuss why the actual angle will be slightly less than calculated.

Q18. A bow used in archery can be modelled as a spring. When drawn by 50 cm, it exerts an average force of 200 N.

(a) Calculate the effective force constant of the bow.

(b) Calculate the energy stored in the drawn bow.

(c) If this energy is transferred to an arrow of mass 30 g, calculate the arrow's speed.

(d) If the bow is drawn by 75 cm instead, calculate the new energy and speed.

(e) Discuss why real bows do not obey Hooke's law perfectly and how this affects the calculation.