Potential energy of a unit-mass body in a one-dimensional conservative field is U = x² - 4x + 3 (SI units). (i) Find the equilibrium position. (ii) Show that small oscillations about equilibrium are SHM and find the period. (iii) Find the amplitude if speed at equilibrium is 2√6 m s⁻¹.
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Question

Potential energy of a unit-mass body in a one-dimensional conservative field is U = x² - 4x + 3 (SI units). (i) Find the equilibrium position. (ii) Show that small oscillations about equilibrium are SHM and find the period. (iii) Find the amplitude if speed at equilibrium is 2√6 m s⁻¹.

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G1 Physics · Accepted Answer

U=x²−4x+3=(x−2)²−1, so equilibrium is x₀=2 m. Comparing U−Uₘᵢₙ=(x−2)² with (1/2)k(Δx)² gives k=2 N m⁻¹. For m=1 kg, T=2π/√2=√2π s. At equilibrium v=2√6 m s⁻¹, so E=(1/2)v²=12 J=(1/2)kA², giving A=2√3 m. All four statements are correct.

Potential energy of a unit-mass body in a one-dimensional conservative field is U = x² − 4x + 3 (SI units). (i) Find the equilibrium position. (ii) Show that small oscillations about equilibrium are SHM and find the period. (iii) Find the amplitude if speed at equilibrium is 2√6 m s⁻¹.