ans a and b 2. Consider the system depicted in the figure below. Here two beads of mass m slide withoutfriction on a massless circular hoop of radius R. The masses are connected by a springwith spring constant k and equilibrium length lo with lo < 2R. This means at the equatorthe spring is stretched out, at any pole it is pushed together, and somewhere in between thespring is at its minimum of energy. We assume there is no gravity. To add dynamics anexternal motor rotates the hoop about the z axis with constant angular frequency w. Weassume that both masses are always at the same height above the equator on opposite sitesof the hoop as shown in the figure.R.(a) Show that the energy E is given bykE = mR [ö° + w² sin (0)] +5 (2R sin(0) – lo)° ,with the angle 0, radius R, and angular frequency w as shown in the figure. You mayuse that the velocity in spherical coordinates is v = rf + rôn + ro sin(0)ộ, whereî, ô, , are the standard unit vectors that are all orthogonal to each other.(b) Due to the constraints the only remaining degree of freedom is the angle 0, and wewant to write down a Lagrangian L(0, 0) describing its dynamics. Explain why thisLagrangian is not just L = T – V, with T the kinetic energy and V the potentialenergy from the spring, but is instead given byL(0, 0) = T, – V5",with To = mR0? and the effective potentialkve = mRw? sin°(0) +(2R sin(0) – lo)² .You do not have to do a calculation to answer this question.(c) Provide a conserved quantity for the system described by L(0,0). Is this quantity equalto the energy?(d) Determine the three equilibrium positions where 0 is stationary.(e) Explain qualitatively in a few words which equilibrium position is stable at wx 0 andwhich at very large w, and what physical reasoning you applied to find this answer.

Given that: The initial length of the spring is lo

2. Consider the system depicted in the figure below. Here two beads of mass m slide without friction on a massless circular hoop of radius R. The masses are connected by a spring with spring constant k and equilibrium length lo with lo < 2R. This means at the equator the spring is stretched out, at any pole it is pushed together, and somewhere in between the spring is at its minimum of energy. We assume there is no gravity. To add dynamics an external motor rotates the hoop about the z axis with constant angular frequency w. We assume that both masses are always at the same height above the equator on opposite sites of the hoop as shown in the figure. (a) Show that the energy E is given by k E = mR[&° + w° sin*(6)] +(2Rsin(@) – lo)° , with the angle 6, radius R, and angular frequency w as shown in the figure. You may use that the velocity in spherical coordinates is v = řf + rôên + ro sin(0)ộ, where î, 6, 6 are the standard unit vectors that are all orthogonal to each other. (b) Due to the constraints the only remaining degree of freedom is the angle 6, and we want to write down a Lagrangian L(0, é) describing its dynamics. Explain why this Lagrangian is not just L = T – V, with T the kinetic energy and V the potential energy from the spring, but is instead given by L(0,ð) = T, – V, with Te = mR°ở° and the effective potential v = mi*u* sin*(0) +(2Rsin(0) – lo)° . You do not have to do a calculation to answer this question.

2. Consider the system depicted in the figure below. Here two beads of mass m slide without friction on a massless circular hoop of radius R. The masses are connected by a spring with spring constant k and equilibrium length lo with lo < 2R. This means at the equator the spring is stretched out, at any pole it is pushed together, and somewhere in between the spring is at its minimum of energy. We assume there is no gravity. To add dynamics an external motor rotates the hoop about the z axis with constant angular frequency w. We assume that both masses are always at the same height above the equator on opposite sites of the hoop as shown in the figure. (a) Show that the energy E is given by k E = mR[&° + w° sin(6)] +(2Rsin(@) – lo)° , with the angle 6, radius R, and angular frequency w as shown in the figure. You may use that the velocity in spherical coordinates is v = řf + rôên + ro sin(0)ộ, where î, 6, 6 are the standard unit vectors that are all orthogonal to each other. (b) Due to the constraints the only remaining degree of freedom is the angle 6, and we want to write down a Lagrangian L(0, é) describing its dynamics. Explain why this Lagrangian is not just L = T – V, with T the kinetic energy and V the potential energy from the spring, but is instead given by L(0,ð) = T, – V, with Te = mR°ở° and the effective potential v = miu* sin*(0) +(2Rsin(0) – lo)° . You do not have to do a calculation to answer this question.