Chapter 6: Work and Energy Flashcards

Concept of Mechanical Work

• Physical Meaning of Work: Mechanical work is performed when a force acts upon an object to cause a displacement.

• Factors Determining Work:

  • Force ($\vec{F}$): Work depends directly on the magnitude and direction of the applied force.

  • Displacement ($\vec{s}$): Work depends directly on the distance over which the force is applied.

Work Done by a Constant Force

• Definition: The work $W$ done on an object by a constant force \table{\vec{F}} is defined as the product of the magnitude of the force, the magnitude of the displacement $\vec{s}$, and the cosine of the angle $\theta$ between the force vector and the displacement vector:

$W = (F \times \cos(\theta)) \times s$

• Units of Work:

  • Standard SI unit: Joule ($\text{J}$), where $1\,\text{J} = 1\,\text{N} \times \text{m}$.

• Variables:

  • $F$: Magnitude of the applied force

  • $s$: Magnitude of the displacement vector

  • $\theta$: Angle between the direction of force and displacement

• Special Cases of Angle ($\theta$):

  • Collinear / Parallel Forces ($\theta = 0^\circ$): Force and displacement point in the exact same direction. Work simplifies to $W = F \times s$ (Maximum positive work).

  • Perpendicular Forces ($\theta = 90^\circ$): Force is applied perpendicular to displacement vector. Since $\cos(90^\circ) = 0$, no work is done ($W = 0$).

  • Opposing Forces ($\theta = 180^\circ$): Force acts in the direction opposite to displacement. Since $\cos(180^\circ) = -1$, work done is negative ($W = -F \times s$).

The Work-Energy Theorem

• Definition of Kinetic Energy: Kinetic energy ($KE$) is the energy of motion possessed by an object of mass $m$ moving at speed $v$:

$KE = \frac{1}{2} \times m \times v^2$

• Work-Energy Theorem: When a net external force does work $W$ on an object, the kinetic energy changes from initial $KE_0$ to final $KE_f$. The net work done equals the total change in kinetic energy:

$W = \Delta KE = KE_f - KE_0$

• Sign Conventions and Speed Changes:

  • Positive Net Work ($W > 0$): Kinetic energy increases ($KE_f > KE_0$), causing speed to increase.

  • Negative Net Work ($W < 0$): Kinetic energy decreases ($KE_f < KE_0$), causing speed to decrease.

  • Zero Net Work ($W = 0$): Kinetic energy remains constant ($KE_f = KE_0$), maintaining constant speed.

Gravitational Potential Energy

• Gravitational Force: Every object near Earth's surface experiences a downward gravitational force $F_{\text{grav}} = m \times g$, where $g = 9.8\,\text{m/s}^2$

• Work Done by Gravity: Vertical displacement $s = h_0 - h_f$ results in gravitational work:

$W_{\text{grav}} = m \times g \times (h_0 - h_f)$

• Path Independence: Work done by gravity depends solely on the vertical height difference $(h_0 - h_f)$ and is entirely independent of the path taken.

• Definition of Gravitational Potential Energy: Gravitational Potential Energy ($PE$) is the position-based energy an object possesses due to its vertical height $h$ relative to a reference level:

$PE = m \times g \times h$

• Comparison of Mechanical Energy Types:

  • Kinetic Energy (Motion): $KE = \frac{1}{2} \times m \times v^2$

  • Potential Energy (Position): $PE = m \times g \times h$

Conservative Versus Non-Conservative Forces

• Conservative Forces:

  • Work done on an object moving between two points is independent of the path taken.

  • Net work done along any closed path is zero ($W_{\text{closed}} = 0$).

  • Examples: Gravitational force, electrostatic force, elastic spring force.

  • Conservative forces conserve total mechanical energy.

• Non-Conservative Forces:

  • Work done depends directly on the path taken.

  • Work along a closed path is non-zero, dissipating mechanical energy into other energy forms.

  • Examples: Friction force, air resistance, propulsion forces.

  • Non-conservative forces change total mechanical energy.

The Conservation of Mechanical Energy

• Total Mechanical Energy ($E$): The sum of kinetic energy ($KE$) and potential energy ($PE$):

$E = KE + PE$

• Principle of Conservation of Mechanical Energy: When net work done by external non-conservative forces is zero ($W_{\text{nc}} = 0$), the total mechanical energy remains constant throughout motion:

$E_{\text{final}} = E_{\text{initial}}$

$KE_f + PE_f = KE_0 + PE_0$

Non-Conservative Forces and The Work-Energy Theorem

• Effect of Non-Conservative Forces: Mechanical energy conservation does not apply when non-conservative forces act.

• General Relationship: Work done by non-conservative forces equals the change in total mechanical energy:

$W_{\text{nc}} = E_f - E_0 = (KE_f + PE_f) - (KE_0 + PE_0)$

Power

• Definition of Average Power ($\bar{P}$): Average rate at which work is performed or energy is transferred:

$\bar{P} = \frac{W}{t}$

• SI Unit: Watt ($\text{W}$), where $1\,\text{W} = 1\,\text{J/s}$.

• Power in Terms of Speed: For a force acting along displacement, average power can be written as:

$\bar{P} = F \times \bar{v}$

where $\bar{v}$ is average speed.

Other Forms of Energy and the Conservation of Energy

• Categories of Energy:

  • Mechanical Forms: Kinetic energy ($KE$) and Gravitational potential energy ($PE$).

  • Non-Mechanical Forms: Electrical energy, chemical energy, thermal energy (heat), and nuclear energy.

• Universal Principle of Conservation of Energy: Energy can neither be created nor destroyed; it can only be transformed from one form to another. Total energy in an isolated system remains constant.

Work Done by a Variable Force

• Variable Forces: When force magnitude or direction varies as a function of position, work cannot be calculated using simple product multiplication.

• Graphical Interpretation: On a graph of force component ($F \times \cos(\theta)$) versus displacement ($s$), total work done equals the area under the curve between initial position $s_0$ and final position $s_f$.