Derive the equation of motion using the principle of conservation of energy for each of the syst

Derive the equation of motion using the principle of...

Key Concepts

Conservation of Energy

This principle states that in an isolated system the total energy remains constant over time, as energy can only be transformed from one form to another and not created or destroyed. In mechanical systems, it often involves the interplay between kinetic energy and potential energy, and is a foundational principle for deriving equations of motion when dissipative forces are negligible.

Kinetic Energy

Kinetic energy is the energy that a body possesses due to its motion. It is commonly expressed as one-half the product of the mass and the square of its velocity. In deriving equations of motion through energy methods, the expression for kinetic energy is differentiated with respect to time to relate the variations in motion to forces acting on the system.

Potential Energy

Potential energy represents stored energy due to the configuration of a system, such as gravitational or elastic potential energy. In many dynamic problems, changes in potential energy are used alongside changes in kinetic energy to formulate the conservation of energy equation, enabling the derivation of the system's governing differential equations.

Derivation of Equations of Motion

Deriving the equation of motion involves setting up an energy balance where the sum of kinetic and potential energy remains constant. By expressing both energies in terms of the system's generalized coordinates and their time derivatives and then differentiating, one can obtain differential equations that describe the system's dynamics over time.

Energy Methods in Dynamics

Energy methods provide an alternative approach to Newton's laws by using energy balances instead of force analysis to derive motion equations. This method simplifies the process, especially in systems with complex constraints, by leveraging the conservative nature of the energy transformations inherent in the system.

Transcript

00:01 All right, so first for figure 285, the mass is on the inner radius r, so no slip.

00:09 So x is equal to r theta, which means theta is x over r.

00:16 And so theta dot would be x dot over r.

00:20 The spring is on the outer radius for r, so the stretch is delta equals 4r times theta, which would be 4r times x over r, which would be 4x.

00:35 Now the energies.

00:36 The kinetic energy, t is one -half -m -x -dott -squared plus one -half j -not -theta -dot squared, which is one -half -m -dot, not m -dot, but m -x dot squared, plus one -half j -k -not over x -over r squared.

00:59 The potential energy from the spring would be v equals one -half -k, delta squared, which is one half k times 4x squared, which is one half k times 16x squared, which is 8kx squared.

01:18 And then we'll use ddt, which is times dt dx dot plus dvdx equals zero.

01:34 And we'll compute dt dx dot would be mx dot plus j.

01:40 R squared times x dot which would be m plus j not over r times x dot so d d t that would be m plus j not over r squared times x double dot and d v d x is d d d x times 8 kx squared which would be 16 kx so m plus j knot over r squared times x double dot plus 16 k x equals 0 and now for figure 286 when the moving pulley goes down by x the two rope segments around at each change by x giving 2x total change also the left spring extension e1 takes up rope in the opposite sense and the right spring extension e5 uses rope length so the small change constrained is negative e1 plus 2x minus e5 equals 0.

02:55 So e5 equals 2x minus e1.

03:00 The same rope tension t everywhere.

03:03 It's a massless rope and frictionless pulle...

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