7.4: Solving problems with work and energy
In this chapter two methods to solve dynamics problems using the concepts of work and energy have been described: the principle of work and energy and the concept of energy conservation. Both methods provide one scalar equation, Equation 7.11 and Equation 7.34, that can be used to determine one unknown scalar variable. For example, this unknown scalar variable can be:
- The magnitude of a final or initial velocity
- A final or initial position coordinate
- The work of a force, which can sometimes also be used to determine the magnitude of the force
The advantage of energy methods is that they allow solving a dynamics problems, without requiring one to solve differential equations. Moreover, under certain conditions (energy conservation) they provide solutions even if
the path between the initial and final state is unknown, making the solutions also more generally valid.
Solving the problem then proceeds along the following steps:
- Sketch the point masses, massless mechanisms, force vectors and constraints.
- Choose and draw a suitable coordinate system (CS).
- Determine the constraint equations.
- Determine the kinetic energy function of all point masses in terms of the velocities.
- Determine the work or potential energy function for all forces.
- Determine the change in kinetic energy, change in potential energy, and/or the work done on masses by forces between the initial and final condition.
- Write down the equation for the principle of work and energy, or the conservation of energy equation.
- Determine the unknown scalar variable by solving the energy equation.
Energy methods only give the scalar values of the positions and velocity and not their directions. For instance, a mass with velocity \(v\) has the same kinetic energy as a mass with velocity \(-v\), such that the direction of the velocity cannot be determined from the kinetic energy. Similarly, if the potential energy of a spring is calculated by energy methods, it cannot be determined if it is compressed or extended. Careful evaluation of the problem is needed, based on considerations other than energy, to determine those directions and to select the right solution.