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A massless inextensible string passes over a pulley which is a fixed distance above the floor. A bunch of bananas of mass m is attached to one end of the string. A monkey of mass M is initially at the other end B. The monkey climbs the string, and her displacement d(t) with respect to the end B is a given function of time. The system is initially at rest, so that the initial conditions are d(0) = 0, d(0) = 0. (a) Introduce suitable generalized coordinates, write the transformation equations, and calculate the lagrangian of the system in terms of these coordinates. Show that the equation governing the height z(t) of the monkey above the floor is (M+m)2-md = (m - M)g. (b) Find the hamiltonian H and Hamilton's equations. Rederive the equation for 2 from Hamilton's equations. (c) Is H = E=T +V? Is H conserved? Is E conserved? Justify briefly. (d) Integrate the equation to find z(t) as a function of t. In the special case that m = M, show that the bananas and the operation between them is constant.

A massless inextensible string passes over a pulley which is a fixed distance above the floor. A bunch of bananas of mass m is attached to one end of the string. A monkey of mass M is initially at the other end B. The monkey climbs the string, and her displacement d(t) with respect to the end B is a given function of time. The system is initially at rest, so that the initial conditions are d(0) = 0, d(0) = 0. (a) Introduce suitable generalized coordinates, write the transformation equations, and calculate the lagrangian of the system in terms of these coordinates. Show that the equation governing the height z(t) of the monkey above the floor is (M+m)2-md = (m - M)g. (b) Find the hamiltonian H and Hamilton's equations. Rederive the equation for 2 from Hamilton's equations. (c) Is H = E=T +V? Is H conserved? Is E conserved? Justify briefly. (d) Integrate the equation to find z(t) as a function of t. In the special case that m = M, show that the bananas and the operation between them is constant.

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I only need help for question 2 part a. Using Lagrangian formalism
Here is the transcription of the image: --- 2. A massless inextensible string passes over a pulley which is fixed a distance above the floor. A bunch of bananas of mass \( m \) is attached to one end of the string. A monkey of mass \( M \) is initially at the other end B. The monkey climbs the string, and her displacement \( d(t) \) with respect to the end B is a given function of time. The system is initially at rest, so that the initial conditions are \( d(0) = 0, \dot{d}(0) = 0 \). (a) Introduce suitable generalized coordinates, write the transformation equations, and calculate the Lagrangian of the system in terms of these coordinates. Show that the equation governing the height \( z(t) \) of the monkey above the floor is \( (M + m) \ddot{z} = (m - M)g \). (b) Find the Hamiltonian \( H \) and Hamilton's equations. Derive the equation for \( \ddot{z} \) from Hamilton's equations. (c) Is \( H = E = T + V \)? Is \( H \) conserved? Is \( E \) conserved? Justify briefly. (d) Integrate the equation to find \( z(t) \) as a function of \( t \). In the special case that \( m = M \), show that the bananas and the monkey rise through equal distances so that the vertical separation between them is constant. 3. Consider a single particle of mass \( m \) in spherical coordinates, with the kinetic energy \[ T = \frac{1}{2m} \left( p_r^2 + \frac{p_\theta^2}{r^2} + \frac{p_\phi^2}{r^2 \sin^2 \theta} \right) \] --- Note: There are no graphs or diagrams in this image.
Transcribed Image Text:Here is the transcription of the image: --- 2. A massless inextensible string passes over a pulley which is fixed a distance above the floor. A bunch of bananas of mass \( m \) is attached to one end of the string. A monkey of mass \( M \) is initially at the other end B. The monkey climbs the string, and her displacement \( d(t) \) with respect to the end B is a given function of time. The system is initially at rest, so that the initial conditions are \( d(0) = 0, \dot{d}(0) = 0 \). (a) Introduce suitable generalized coordinates, write the transformation equations, and calculate the Lagrangian of the system in terms of these coordinates. Show that the equation governing the height \( z(t) \) of the monkey above the floor is \( (M + m) \ddot{z} = (m - M)g \). (b) Find the Hamiltonian \( H \) and Hamilton's equations. Derive the equation for \( \ddot{z} \) from Hamilton's equations. (c) Is \( H = E = T + V \)? Is \( H \) conserved? Is \( E \) conserved? Justify briefly. (d) Integrate the equation to find \( z(t) \) as a function of \( t \). In the special case that \( m = M \), show that the bananas and the monkey rise through equal distances so that the vertical separation between them is constant. 3. Consider a single particle of mass \( m \) in spherical coordinates, with the kinetic energy \[ T = \frac{1}{2m} \left( p_r^2 + \frac{p_\theta^2}{r^2} + \frac{p_\phi^2}{r^2 \sin^2 \theta} \right) \] --- Note: There are no graphs or diagrams in this image.
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