PHY.7:5.1 - Represent a constraint, then move its support

A point mass m moves in a vertical plane on an ideal rigid, massless rod of length l. Its frictionless pivot is initially fixed. Gravity is uniform with acceleration g. Let theta be the angle from the downward vertical; relative to the pivot,

x=l*sin(theta); y=-l*cos(theta).

The rod constraint is built into this configuration. For its ideal reaction, virtual motion along the circle has no radial displacement and the reaction does no virtual work. Kinetic energy is T=m*l^2*theta_dot^2/2, and potential energy is V=-m*g*l*cos(theta). Thus

L=m*l^2*theta_dot^2/2+m*g*l*cos(theta).

Varying theta with fixed temporal endpoints gives

m*l^2*theta_ddot+m*g*l*sin(theta)=0.

For l=1 m, g=10 m/s^2 and initial theta=pi/6, the angular acceleration is -5 rad/s^2. The radial reaction need not be solved to obtain that consequence. It can be recovered from the physical acceleration if a later question concerns the rod load.

Changed support. Prescribe a horizontal pivot position X(t). The physical position becomes x=X(t)+l*sin(theta) while y is unchanged. Differentiating the complete position gives

T=m*(X_dot^2+2*X_dot*l*cos(theta)*theta_dot+l^2*theta_dot^2)/2.

Keeping the same gravitational potential and varying theta gives

m*l^2*theta_ddot+m*l*X_ddot*cos(theta)+m*g*l*sin(theta)=0.

With pivot acceleration X_ddot=2 m/s^2 at the same angle, the angular acceleration is about -6.732 rad/s^2. Omitting the pivot term from the velocity would preserve the old answer while losing a real forcing. The pivot’s prescribed motion may do work, so the fixed-pivot mechanical-energy conservation claim does not automatically transfer.

Changed interaction. For the fixed pivot, add an established damping torque Q=-b*theta_dot, with b>=0. The virtual-work equation gives m*l^2*theta_ddot+b*theta_dot+m*g*l*sin(theta)=0. The mechanical energy has derivative -b*theta_dot^2. Appending this dissipative torque to a conservative scalar potential would require a different physical account. PHY.6 supplies the corresponding balance and receiving energy form; a more elaborate dissipative action is useful only when the intended work needs it.

The general move is to construct the allowed configuration and its velocities, derive the consequence, and rebuild only the affected contribution when the support or interaction changes.