A massless spring of constant k =84.8 N/m is fixed on the left side of a level track. A block of mass m = 0.5 kg is pressed against the spring and compresses it a distance d, as in the figure below. The block (initially at rest) is then released and travels toward a circular loop-the-loop of radius R = 1.6 m. The entire track and the loop-the-loop are frictionless, except for the section of track between points A and B. Given that the coefficient of kinetic friction between the block and the track along AB is μk=0.3, and that the length of AB is 2.2 m, determine the minimum compression d of the spring that enables the block to just make it through the loop-the-loop at point C. (Hint: The force exerted by the track on the block will be zero if the block barely makes it through the loop-the-loop.)

Answer 1

PE(spring) =PE + W(fr)
k•x²/2 =m•g•(2R) +μ•m•g•s.

x= sqrt{2(m•g• 2R +μ•m•g•s)/k} =
= sqrt{2m•g• (2R +μ•s)/k}=
=sqrt{2•0.5•9.8(3.2 + 0.3•2.2)/84.8}=
=0.668 m