Notes on Kinetic Energy and Work

Learning Goals

• Understand work done by a general variable force.

• Define and calculate power.

• Review concepts from Chapters 5-7.

• Solve more practice examples.

Definition of Work

• Work (W) is defined for a variable force $ extbf{F}(r)$. The general expression for work done is:
  W = extbf{F} ullet d extbf{r} = orall ext{Force Components} = extstyle{ extstyle rac{W}{ ext{Net Work}} o extstyle extstyle orall_{i,j,k} ext{Variable}}

• Expanded formula:
  $W = \int extbf{F}<em>x dx + \int extbf{F}</em>y dy + \int extbf{F}_z dz$

Example of Work Calculation

• Particle position $ extbf{r} = x extbf{i} + y extbf{j} + z extbf{k}$ in meters.

• Variable force $ extbf{F} = (3.00 ext{ N/m}) x extbf{i} + 5.00 ext{ N} extbf{j}$.

• Movement from:

  • Initial position: $ extbf{r}_i = 7.00 ext{ m} extbf{i} + 6.00 ext{ m} extbf{j}$

  • Final position: $ extbf{r}_f = -4.00 ext{ m} extbf{i} - 3.00 ext{ m} extbf{j}$

• Work done can be calculated using:
  W = extbf{F} ullet d extbf{r} = extbf{F} ullet ( extbf{r}2 - extbf{r}1)
  or $W = -m g (y<em>f - y</em>i)$

Power

• Power (P) is defined as the rate of energy transfer or rate of doing work.

• Average power can be calculated as:
  $P = rac{W}{ riangle t}$

• Instantaneous power can be expressed as:
  $P = rac{dW}{dt}$ or P = extbf{F} ullet extbf{v}

• Units of power:

  • Watt (W), where $1 ext{ W} = 1 ext{ J/s}$.

Average Power Example

• Mike performs $5 ext{ J}$ of work in $10 ext{ s}$.

• Joe performs $3 ext{ J}$ in $5 ext{ s}$.

• To determine who produced greater average power, calculate:

  • Mike: $P_{Mike} = rac{5 ext{ J}}{10 ext{ s}} = 0.5 ext{ W}$

  • Joe: $P_{Joe} = rac{3 ext{ J}}{5 ext{ s}} = 0.6 ext{ W}$

  • Conclusion: Joe produced more power.

Work and Energy in Uniform Circular Motion

• In cases where Tidal friction is neglected, such as Earth orbiting the sun, the work done on Earth by the sun is:

  • $W = 0$ (No net work in uniform circular motion as force direction is perpendicular to displacement).

Group Activity Example

1. Particle with mass $m = 5 ext{ kg}$ is pulled by a force $ extbf{F} = 10 ext{ N} extbf{i} + 10 ext{ N} extbf{j}$ from rest for a time $ riangle t = 5 ext{ s}$.

   • Calculate work done:
     W = extbf{F} ullet extbf{d}

   • Calculate average power:
     $P = rac{W}{ riangle t}$

Summary of Relevant Equations

• Work in general is given by:
  $W = \int extbf{F}<em>x dx + \int extbf{F}</em>y dy + \int extbf{F}_z dz$

• Work-kinetic energy theorem states:
  $W_{net} = riangle K = rac{1}{2} mv^2$

• Power in terms of force and velocity:
  P = extbf{F} ullet extbf{v}

Important Equations and Concepts

1. Work (W)

   • General expression for work done by a variable force:
     $W = \mathbf{F} \bullet d \mathbf{r}$

   • Expanded formula:
     $W = \int \mathbf{F}<em>x dx + \int \mathbf{F}</em>y dy + \int \mathbf{F}_z dz$

2. Work-Kinetic Energy Theorem

   • States that the net work done on an object is equal to the change in kinetic energy:
     $W_{net} = \Delta K = \frac{1}{2} mv^2$

3. Power (P)

   • Power is the rate at which work is done:
     $P = \frac{W}{\Delta t}$

   • Instantaneous power can be expressed as:
     $P = \frac{dW}{dt}$
     or
     $P = \mathbf{F} \bullet \mathbf{v}$

   • Units of power: 1 Watt (W) = 1 Joule/second (J/s)

4. Work in Uniform Circular Motion

   • The work done is zero because the force is perpendicular to the displacement:
     $W = 0$

5. Average Power Examples

   • Example calculations for average power, comparing different scenarios to see who produces more power.