Lecture Notes on Kinetic Energy and Work

Today's Learning Goals

• Understand the concept of Work.

• Learn about the Work-Kinetic Energy Theorem.

Recap on Kinetic Energy

• Equivalent Units of Kinetic Energy:

  • Newtons * m

  • Joules

  • kg * m² * s⁻²

  • All of the above.

Work

• Definition:

  • Work is the energy transferred to or from an object by a force acting on it.

• Positive vs Negative Work:

  • Positive work: Energy is transferred to the object.

  • Negative work: Energy is transferred from the object.

• Mathematical Expression: $W = rac{dW}{dr} F \cdot dr$

  • Where $F$ is the force vector and $dr$ is the displacement vector.

Dot Product

• Definition: The dot product (or scalar product) results in a scalar from two vectors.

• Mathematical Representation:
  $a \cdot b = |a||b| \cos(\theta)$

• Calculating Dot Product: $a \cdot b = a<em>xb</em>x + a<em>yb</em>y + a<em>zb</em>z$

  • The sign of $a \cdot b$ depends on the angle $\theta$ between the vectors.

Work by a Constant Force

• When force is constant, work can be calculated as: $W = F \cdot \Delta r$

  • For a constant force $F$:
    $W = F<em>x \Delta x + F</em>y \Delta y + F_z \Delta z$

Gravitational Force and Work

• Work done by gravitational force can be determined by the direction of motion:

  • Moving Up the Slope: Work done by gravitational force is negative.

  • Moving Down the Slope: Work done by gravitational force is positive.

  • Mathematical expression for work done by gravity when moving down an incline of angle $\alpha$:

  • $W = -mg|\Delta r| \sin(\alpha)$

Work-Kinetic Energy Theorem

• Statement: The net work done by all forces on an object equals the change in its kinetic energy.

• Expressed as: $W<em>{NET} = K</em>f - K_i$

  • Where $K<em>f$ is final kinetic energy and $K</em>i$ is initial kinetic energy.

  • The total work done can also be calculated as:
    $W<em>{NET} = \sum W</em>i = W<em>1 + W</em>2 + … + W_n$

Example Problems

1. Baseball Example:

   • A pitcher throws a 0.5 kg baseball from rest to an exit speed of 50 m/s. The net work done on the baseball can be calculated using the Work-Kinetic Energy theorem.

2. Gravity Work Calculation:

   • For a climber of mass $m$ climbing a wall, work done by gravity as they climb from $r<em>1$ to $r</em>2$ is given by:
     $W = -mgΔy$

3. Displacement Example:

   • For a climber with $65$ kg mass moving a displacement of $2m$ horizontally and $5m$ vertically, compute the work done against gravity.

4. Kinetic Energy Comparison:

   • Determine which of three slides results in the highest kinetic energy at the bottom based on conservation of energy principles (potential energy converted to kinetic energy).

Conclusions

• Work and kinetic energy are interrelated concepts in mechanics.

• Understanding the mathematical expressions and real-world implications of these concepts is critical for problem-solving in physics.

• The sign of work done (positive, negative, or zero) is determined by the direction of force relative to the displacement.

Key Equations and Concepts

Work

• Definition: Work is the energy transferred to or from an object by a force acting on it.

• Mathematical Expression: $W = \frac{dW}{dr} F \cdot dr$

  Where $F$ is the force vector and $dr$ is the displacement vector.

Dot Product

• Definition: The dot product results in a scalar from two vectors.

• Mathematical Representation: $a \cdot b = |a||b| \cos(\theta)$

• Calculating Dot Product: $a \cdot b = axbx + ayby + azbz$

Work by a Constant Force

• When force is constant, work can be calculated as: $W = F \cdot \Delta r$

  • For a constant force $F$: $W = Fx \Delta x + Fy \Delta y + F_z \Delta z$

Gravitational Force and Work

• Direction Matters:

  • Moving Up the Slope: Work done by gravitational force is negative.

  • Moving Down the Slope: Work done by gravitational force is positive.

• Expression for Work Done by Gravity (Moving Down an Incline of Angle $\alpha$): $W = -mg|\Delta r| \sin(\alpha)$

Work-Kinetic Energy Theorem

• Statement: The net work done by all forces on an object equals the change in its kinetic energy.

• Expressed as: $W<em>{NET} = Kf - K</em>i$

• Total Work Done: $W<em>{NET} = \sum W</em>i = W<em>1 + W</em>2 + … + W_n$

Conclusions

• Work and kinetic energy are interrelated concepts in mechanics.

• Understanding the mathematical expressions and real-world implications of these concepts is critical for problem-solving in physics.