Question

An inverted pendulum consists of a mass m at the end of a massless stick of length l. The other end of the stick is made to oscillate vertically with a position given by y(t) = A cos(?t), where A ? l. It turns out that if ? is large enough, and if the pendulum is initially nearly upside down, then surprisingly it will not fall over as time goes by. Instead it will (sort of) oscillate back and forth around the vertical position. (a) Find the position of the mass as a function of time. The position can be described by the displacement of the support y(t) and the angle of the mass with respect to the vertical direction. (b) Take the time derivative of the position to obtain the velocity and the kinetic energy from the velocity. (c) Find the Lagrangian of the system. (d) Obtain the equation of motion l?? + sin ?(A?² cos(?t) ? g) = 0 from the Lagrangian.

          An inverted pendulum consists of a mass m at the end of a massless stick of length l. The other end of the stick is made to oscillate vertically with a position given by y(t) = A cos(?t), where A ? l. It turns out that if ? is large enough, and if the pendulum is initially nearly upside down, then surprisingly it will not fall over as time goes by. Instead it will (sort of) oscillate back and forth around the vertical position.

(a) Find the position of the mass as a function of time. The position can be described by the displacement of the support y(t) and the angle of the mass with respect to the vertical direction.

(b) Take the time derivative of the position to obtain the velocity and the kinetic energy from the velocity.

(c) Find the Lagrangian of the system.

(d) Obtain the equation of motion l?? + sin ?(A?² cos(?t) ? g) = 0 from the Lagrangian.

An inverted pendulum consists of a mass m at the end of a massless stick of length l. The other end of the stick is made to oscillate vertically with a position given by y(t) = A cos(?t), where A ? l. It turns out that if ? is large enough, and if the pendulum is initially nearly upside down, then surprisingly it will not fall over as time goes by. Instead it will (sort of) oscillate back and forth around the vertical position.

(a) Find the position of the mass as a function of time. The position can be described by the displacement of the support y(t) and the angle of the mass with respect to the vertical direction.

(b) Take the time derivative of the position to obtain the velocity and the kinetic energy from the velocity.

(c) Find the Lagrangian of the system.

(d) Obtain the equation of motion l?? + sin ?(A?² cos(?t) ? g) = 0 from the Lagrangian.

Added by Mireia C.

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