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Problem 3 A conservative mechanical system consists of a mass m that is constrained to move along a circle of radius R. The centre of the circle is at the origin O of the coordinate system. The mass is connected to a point A along the â-axis at a distance 2R from the centre of circle with a spring of elastic constant k, so that the corresponding elastic potential has the form Vspring = (k/2)d², where d is the (varying) distance between the mass and point A. Gravity acts, as usual, along the vertical direction. See the figure for a depiction of the system. m A X Figure 1: The spring is shown in blue. The y axis points upward. (i) How many degrees of freedom does the system have? (ii) Choose appropriate generalised coordinates and write down the Lagrangian of the system in terms of them. (iii) Write down the Euler-Lagrange equations of the system. (iv) Find the equilibrium positions of the system and discuss their stability.

Problem 3 A conservative mechanical system consists of a mass m that is constrained to move along a circle of radius R. The centre of the circle is at the origin O of the coordinate system. The mass is connected to a point A along the â-axis at a distance 2R from the centre of circle with a spring of elastic constant k, so that the corresponding elastic potential has the form Vspring = (k/2)d², where d is the (varying) distance between the mass and point A. Gravity acts, as usual, along the vertical direction. See the figure for a depiction of the system. m A X Figure 1: The spring is shown in blue. The y axis points upward. (i) How many degrees of freedom does the system have? (ii) Choose appropriate generalised coordinates and write down the Lagrangian of the system in terms of them. (iii) Write down the Euler-Lagrange equations of the system. (iv) Find the equilibrium positions of the system and discuss their stability.

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Problem 3 A conservative mechanical system consists of a mass m that is constrained to move along a circle of radius R. The centre of the circle is at the origin O of the coordinate system. The mass is connected to a point A along the â-axis at a distance 2R from the centre of circle with a spring of elastic constant k, so that the corresponding elastic potential has the form Vspring = (k/2)d², where d is the (varying) distance between the mass and point A. Gravity acts, as usual, along the vertical direction. See the figure for a depiction of the system. m A X Figure 1: The spring is shown in blue. The ŷ axis points upward. (i) How many degrees of freedom does the system have? (ii) Choose appropriate generalised coordinates and write down the Lagrangian of the system in terms of them. (iii) Write down the Euler-Lagrange equations of the system. (iv) Find the equilibrium positions of the system and discuss their stability.
Transcribed Image Text:Problem 3 A conservative mechanical system consists of a mass m that is constrained to move along a circle of radius R. The centre of the circle is at the origin O of the coordinate system. The mass is connected to a point A along the â-axis at a distance 2R from the centre of circle with a spring of elastic constant k, so that the corresponding elastic potential has the form Vspring = (k/2)d², where d is the (varying) distance between the mass and point A. Gravity acts, as usual, along the vertical direction. See the figure for a depiction of the system. m A X Figure 1: The spring is shown in blue. The ŷ axis points upward. (i) How many degrees of freedom does the system have? (ii) Choose appropriate generalised coordinates and write down the Lagrangian of the system in terms of them. (iii) Write down the Euler-Lagrange equations of the system. (iv) Find the equilibrium positions of the system and discuss their stability.
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