A pendulum of length l and mass m is mounted on a block of mass M. The block can move freely without friction on a horizontal surface as shown in Fig 1 PHYS 2402 1. Consider a particle of mass m moving in a plane under the attractive force μm/r2 directed to the origin of polar coordinates r, θ. Determine the equations of motion. 2. Write down the Lagrangian for a simple pendulum constrained to move in a single vertical plane. Find from it the equation of motion and show that for small displacements from equilibrium the pendulum performs simple harmonic motion. 3. A pendulum of length l and mass m is mounted on a block of mass M. The block can move freely without friction on a horizontal surface as shown in Fig 1. Figure 1 Show that the Lagrangian for the system is L = ( M + m 2 ) ( ̇x)2 + ml ̇x ̇θ + m 2 l2( ̇θ)2 + mgl ( 1 − θ2 2 )

A pendulum of length l and mass m is mounted on a block of mass M. The block can move freely without friction on a horizontal surface as shown in Fig 1 PHYS 2402 1. Consider a particle of mass m moving in a plane under the attractive force μm/r2 directed to the origin of polar coordinates r, θ. Determine the equations of motion. 2. Write down the Lagrangian for a simple pendulum constrained to move in a single vertical plane. Find from it the equation of motion and show that for small displacements from equilibrium the pendulum performs simple harmonic motion. 3. A pendulum of length l and mass m is mounted on a block of mass M. The block can move freely without friction on a horizontal surface as shown in Fig 1. Figure 1 Show that the Lagrangian for the system is L = ( M + m 2 ) ( ̇x)2 + ml ̇x ̇θ + m 2 l2( ̇θ)2 + mgl ( 1 − θ2 2 )

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