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A point mass m is hanging from a massless spring attached at one end at a point O. The mass can swing in the vertical plane, see Figure 1. Assume that the coordinate x is in the horizontal direction while the coordinate z is in the (upwards) vertical direction. The spring potential energy is k(r - R)² where r is the length of the spring and k, R are positive constants. The system is in a uniform gravitational field. m Figure 1: (a) Write down the Lagrangian functions in terms of the generalised coordinates (x, z) shown in Figure 1. (b) Write down the Lagrange equations of motion for the system. (c) Find the equilibria of the system.

A point mass m is hanging from a massless spring attached at one end at a point O. The mass can swing in the vertical plane, see Figure 1. Assume that the coordinate x is in the horizontal direction while the coordinate z is in the (upwards) vertical direction. The spring potential energy is k(r - R)² where r is the length of the spring and k, R are positive constants. The system is in a uniform gravitational field. m Figure 1: (a) Write down the Lagrangian functions in terms of the generalised coordinates (x, z) shown in Figure 1. (b) Write down the Lagrange equations of motion for the system. (c) Find the equilibria of the system.

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A point mass m is hanging from a massless spring attached at one end at a point O. The mass can swing in the vertical plane, see Figure 1. Assume that the coordinate x is in the horizontal direction while the coordinate z is in the (upwards) vertical direction. The spring potential energy is k(r – R)? where r is the length of the spring and k, R are positive constants. The system is in a uniform gravitational field. - Z Figure 1: (a) Write dowm the Lagrangian functions in terms of the generalised coordinates (x, z) shown in Figure 1. (b) Write down the Lagrange equations of motion for the system. (c) Find the equilibria of the system.
Transcribed Image Text:A point mass m is hanging from a massless spring attached at one end at a point O. The mass can swing in the vertical plane, see Figure 1. Assume that the coordinate x is in the horizontal direction while the coordinate z is in the (upwards) vertical direction. The spring potential energy is k(r – R)? where r is the length of the spring and k, R are positive constants. The system is in a uniform gravitational field. - Z Figure 1: (a) Write dowm the Lagrangian functions in terms of the generalised coordinates (x, z) shown in Figure 1. (b) Write down the Lagrange equations of motion for the system. (c) Find the equilibria of the system.
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