As shown in figure, string PQR is stretched by force F=(3−10Kt) N, where K is a constant and t is time in seconds. At time t=0, a small pulse is generated at the fixed end P of the string. Find the value of K (in N/s) if the value of force becomes zero as pulse reaches point Q. (Given mass per unit length of string is 0.03 kg/m, and PQ=1 m.)
Physics · Mixed Chapter
JEE Mains

Question

As shown in figure, string PQR is stretched by force F=(3−10Kt) N, where K is a constant and t is time in seconds. At time t=0, a small pulse is generated at the fixed end P of the string. Find the value of K (in N/s) if the value of force becomes zero as pulse reaches point Q. (Given mass per unit length of string is 0.03 kg/m, and PQ=1 m.)

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: A transverse pulse propagates at instantaneous speed v(t)=√[F(t)/μ]=√[(3−10Kt)/0.03] m/s. Step 2: Force reaches zero at t_f=3/(10K). The total pulse distance by this time is L=∫₀ᵗᶠ√[(3−10Kt)/0.03]dt. Step 3: Substitute u=3−10Kt; L=[1/(10K√0.03)]∫₀³√u du=2×3³ᐟ²/(30K√0.03)=2/K metres. Step 4: As PQ=1 m, 1=2/K and K=2 N/s.