6.1: Work Done by a Constant Force

• Work is a force applied over a distance.

  • Work = force x displacement

    • $W=Fs$

  • Scalar quantity

  • SI Unit: joule (J)

    • $1N\cdot m=1joule\left( J\right)$

• Definition of work:

  • $W=F\cos \theta s$

    • $F$ = force of the object

    • $s$ = displacement

    • $\theta$ = angle between $\overrightarrow{F}$ and $\overrightarrow{s}$ when places tail to tail.

    • $\cos0$ = 1

    • $\cos90$ = 0

    • $\cos180$ = -1

• When a force is PARALLEL to the displacement, the work by the force is POSITIVE.

• When a force is OPPOSITE to the displacement, the work done by the force is NEGATIVE.

• When the force is PERPENDICULAR to the displacement, the work done by the force is ZERO.

6.2 The Work Energy Theorem and KE

• The kinetic energy (KE) of an object with mass m and speed v is given by:

  • $KE=\dfrac{1}{2}mv^{2}$

  • KE will always be positive because mass can not be negative and v² is always positive.

  • Scalar quantity

  • SI Unit: J

• The work-energy theorem - the net work done by the forces on an object equals the change in its kinetic energy.

  • When a net external force does work on an object, the kinetic energy of the object changes by the amount of work done on it:

    • $W=KE_{f}-KE_{0}$

6.3: Gravitational Potential Energy

• Gravitational potential energy - the energy that an object of mass m has by virtue of its position relative to the surface of the earth.

  • $W_{gravity}=mgh_{0}-mgh_{f}$

• That position is measured by the height $h$ of the object relative to an arbitrary zero level:

  • $PE=mgh$

  • Scalar quantity

  • SI Unit: J

• Potential and Kinetic energy

  • When your PE is at maximum, your KE is zero.

  • When your PE is zero, your KE is at maximum.

6.4: Conservative vs. Nonconservative Forces

Conservative Forces

• The gravitational force is called a conservative force and occurs when:

  • The work it does on a moving object is independent of the path between the object’s initial and final positions.

  • It does no work on an object moving around a closed path, starting and finishing at the same point.

• The total mechanical energy (kinetic energy + potential energy) of the system remains constant if only conservative forces are acting on it.

• Gravity is a conservative force. It does no work on an object, executing a closed path.

• When a path has the same starting and ending point, it is called a closed path.

  • For a closed path, the work done by gravity will equal zero joules.

  • W_gravity = 0 J for a closed path.

Working Together

• When conservative forces and nonconservative forces act simultaneously on an object, we can write work as:

  • $W=W_{c}+W_{nc}$

Conservative vs Nonconservative Table

6.5: The Conservation of Mechanical Energy

• If the net work on an object by nonconservative forces is zero, then its energy does not change.

• Conservation of energy: $E_{f}=E_{0}$

  • This equation can also broken down into: $\dfrac{1}{2}mv_{f}^{2}+mgh_{f}=\dfrac{1}{2}mv_{0}^{2}+mgh_{0}$

• Total mechanical energy: $E=KE+PE$

  • The total mechanical energy of an object remains constant as the object moves, provided that the net work done by external nonconservative forces is zero

6.6: Nonconservative Forces and the Work-Energy Theorem

• Most moving objects experience nonconservative forces, such as friction, air resistance, and propulsive forces, and the work W_nc done by the net external nonconservative force is not zero. In these situations, the difference between the final and initial total mechanical energies is equal W_nc. Can be written in two ways:

  • $W_{nc}=E_{f}-E_{0}$

  • $W_{nc}=\left( mgh_{f}+\dfrac{1}{2}mv_{f}^{2}\right) -\left( mgh_{0}+\dfrac{1}{2}mv_{0}^{2}\right)$

• The concept of potential energy is not defined for a nonconservative force.

6.7: Power

• Average power ($\overline{P}$) is the time rate at which work is done.

  • It is defined as Work divided by time.

  • $\overline{P}=\dfrac{W}{t}$

    • W is also equal Fs

    • s/t is equal to $\overline{v}$

    • Another way to write this equation is $\overline{P}=F\overline{v}$

  • Scalar quantity

  • SI Unit: watt (W) aka joule/s

• The work–energy theorem relates the work done by a net external force to the change in the energy of the object.

  • We can also define average power as the rate at which the energy is changing, or as the change in energy/time during which the change occurs.

• 1 horsepower = $550ft\cdot lbs/\sec$ = 745.7 watts

6.8: Other Forms of Energy and the Conservation of Energy

The Principle of Conservation of Energy

• Energy can neither be created nor destroyed, but can only be converted from one form to another.