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}x dx + \int extbf{F}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 (yf - yi)

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}x dx + \int extbf{F}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}x dx + \int \mathbf{F}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.