A block executes SHM under two springs on a smooth surface as shown. The time period in case (a) is double that in case (b). If k₁ = 10 N m⁻¹, find k₂ (N m⁻¹)
.

Question

A block executes SHM under two springs on a smooth surface as shown. The time period in case (a) is double that in case (b). If k₁ = 10 N m⁻¹, find k₂ (N m⁻¹) .

G1 Physics · Accepted Answer

Case (a) has springs in series: kₐ=k₁k₂/(k₁+k₂). Case (b) has them in parallel: kᵦ=k₁+k₂. Given Tₐ=2Tᵦ, Tₐ/Tᵦ=√(kᵦ/kₐ)=(k₁+k₂)/√(k₁k₂)=2. Equality holds only for k₁=k₂. Since k₁=10 N m⁻¹, k₂=10 N m⁻¹.