7 Work and Kinetic Energy
7 Chapter Review
Key Terms
- average power
- work done in a time interval divided by the time interval
- kinetic energy
- energy of motion, one-half an object’s mass times the square of its speed
- net work
- work done by all the forces acting on an object
- power
- (or instantaneous power) rate of doing work
- work
- done when a force acts on something that undergoes a displacement from one position to another
- work done by a force
- integral, from the initial position to the final position, of the dot product of the force and the infinitesimal displacement along the path over which the force acts
- work-energy theorem
- net work done on a particle is equal to the change in its kinetic energy
Key Equations
| Work done by a force over an infinitesimal displacement | [latex]dW=\mathbf{\overset{\to }{F}}\cdot d\mathbf{\overset{\to }{r}}=|\mathbf{\overset{\to }{F}}||d\mathbf{\overset{\to }{r}}|\text{cos}\,\theta[/latex] |
| Work done by a force acting along a path from A to B | [latex]{W}_{AB}=\underset{\text{path}AB}{\int }\mathbf{\overset{\to }{F}}\cdot d\mathbf{\overset{\to }{r}}[/latex] |
| Work done by a constant force of kinetic friction | [latex]{W}_{\text{fr}}=\text{−}{f}_{k}|{l}_{AB}|[/latex] |
| Work done going from A to B by Earth’s gravity, near its surface | [latex]{W}_{\text{grav,}AB}=\text{−}mg({y}_{B}-{y}_{A})[/latex] |
| Work done going from A to B by one-dimensional spring force | [latex]{W}_{\text{spring,}AB}=\text{−}(\frac{1}{2}k)({x}_{B}^{2}-{x}_{A}^{2})[/latex] |
| Kinetic energy of a non-relativistic particle | [latex]K=\frac{1}{2}m{v}^{2}=\frac{{p}^{2}}{2m}[/latex] |
| Work-energy theorem | [latex]{W}_{\text{net}}={K}_{B}-{K}_{A}[/latex] |
| Power as rate of doing work | [latex]P=\frac{dW}{dt}[/latex] |
| Power as the dot product of force and velocity | [latex]P=\mathbf{\overset{\to }{F}}\cdot \mathbf{\overset{\to }{v}}[/latex] |
Summary
7.1 Work
- The infinitesimal increment of work done by a force, acting over an infinitesimal displacement, is the dot product of the force and the displacement.
- The work done by a force, acting over a finite path, is the integral of the infinitesimal increments of work done along the path.
- The work done against a force is the negative of the work done by the force.
- The work done by a normal or frictional contact force must be determined in each particular case.
- The work done by the force of gravity, on an object near the surface of Earth, depends only on the weight of the object and the difference in height through which it moved.
- The work done by a spring force, acting from an initial position to a final position, depends only on the spring constant and the squares of those positions.
7.2 Kinetic Energy
- The kinetic energy of a particle is the product of one-half its mass and the square of its speed, for non-relativistic speeds.
- The kinetic energy of a system is the sum of the kinetic energies of all the particles in the system.
- Kinetic energy is rel