Speed of Transverse Wave on a String - UNSOLVED PRACTICE SET

Q7. Derive the expression for the speed of a transverse wave on a stretched string: v = √(T/μ). State the assumptions made.

Q8. A string has mass 0.05 kg and length 2 m. It is stretched with a tension of 10 N. Calculate the speed of transverse waves on this string.

Q9. Explain why the speed of a transverse wave on a string increases when the tension is increased.

Q10. Two strings A and B are made of the same material. String A has twice the radius of string B. Compare their linear mass densities and wave speeds if both are under the same tension.

Q11. A pulse takes 0.1 s to travel the length of a string. If the tension is doubled, how long will the pulse take to travel the same distance?

Q12. Why does a thick string produce a lower note than a thin string of the same length and tension when plucked?

Q13. Derive the expression v = √(T/μ) for the speed of transverse waves on a stretched string using dimensional analysis. Then, using a more rigorous approach, consider a small element of the string and show how the restoring force due to tension leads to wave propagation. Discuss the factors on which the wave speed depends and does not depend.

Q14. A string of length 2 m and mass 10 g is stretched between two fixed points with a tension of 40 N.

(i) Calculate the linear mass density.

(ii) Calculate the speed of transverse waves.

(iii) Calculate the time for a pulse to travel from one end to the other and back.

(iv) If the string is replaced by another of the same length but half the radius (same material), how does the wave speed change?

Q15. Two strings are joined together and stretched with the same tension. String 1 has linear mass density μ₁ and length L₁; String 2 has linear mass density μ₂ and length L₂.

(i) A pulse is sent from String 1 to String 2. What happens at the junction?

(ii) Compare the speeds of the pulse in the two strings if μ₁ = 4μ₂.

(iii) If the pulse takes time t₁ to travel through String 1 and t₂ through String 2, find the ratio t₁/t₂.

(iv) Explain why a pulse partially reflects and partially transmits at the junction.

Q16. A steel wire of length 3 m and diameter 1 mm is stretched with a tension of 100 N. The density of steel is 7800 kg/m³.

(i) Calculate the mass of the wire.

(ii) Calculate the linear mass density.

(iii) Calculate the speed of transverse waves on the wire.

(iv) Calculate the time for a pulse to travel the entire length of the wire.

(v) If the tension is increased to 400 N, calculate the new wave speed and the new travel time.

Q17. A uniform rope of length 10 m and mass 2 kg hangs vertically from a ceiling. A transverse pulse is produced at the bottom end.

(i) Calculate the tension in the rope at a point 2 m from the bottom.

(ii) Calculate the wave speed at this point.

(iii) Calculate the wave speed at the top of the rope.

(iv) Show that the wave speed varies with height and derive the expression v(y) = √(gy), where y is the distance from the bottom.

(v) Calculate the time for the pulse to travel from the bottom to the top.

(Given: g = 9.8 m/s²)

Q18. In a school music room, a guitar has six strings of different thicknesses but approximately the same length (65 cm). The thinnest string (E string) has diameter 0.3 mm and is under 80 N tension. The thickest string (low E) has diameter 0.9 mm and is under 100 N tension. The density of the string material is 8000 kg/m³.

(i) Calculate the linear mass density of each string.

(ii) Calculate the speed of transverse waves on each string.

(iii) If the fundamental frequency is given by f = v/2L, calculate the fundamental frequency of each string.

(iv) Explain why the thicker string produces a lower note even though it has higher tension.

(v) A student wants all strings to have the same fundamental frequency. What adjustment should she make to the tension of the thicker string?