Notes on Work by a Variable Force

Work by a Variable Force

• Key Concept: Work done by a variable force can be expressed mathematically as:

  • $W = \int \mathbf{F} \cdot d\mathbf{r} = \int F<em>x dx + \int F</em>y dy + \int F_z dz$

• Work by Constant Force:

  • Work by constant force can be simplified as:

  • $W = F \cdot d$

Recap on Work and Gravitational Force

• Consider a climber of mass $m$ climbing on a vertical wall:

  • Displacement can be represented as:

  • $\Delta \mathbf{r} = \mathbf{r}<em>2 - \mathbf{r}</em>1 = (\Delta x \hat{i} + \Delta y \hat{j})$

  • Work done on the climber by gravitational force ($W$):

  • $W = mgh$

  • For an upward displacement:

    • $W = -mg\Delta y$

Example: Work Done by Gravitational Force

• Example 3:

  • A climber of mass $65 ext{ kg}$ climbs from position $\mathbf{r}<em>1$ to $\mathbf{r}</em>2$:

  • Displacement: $\Delta \mathbf{r} = (2 ext{ m} \hat{i} + 5 ext{ m} \hat{j})$

  • Work done: Determine $W = mgh$ considering the change in height.

Kinetic Energy and Work-Energy Theorem

• Clicker Question 8: Ranking of slides for kinetic energy:

  • Hint: Work-Kinetic energy theorem states:

  • $W_{net} = \Delta KE$

  • Possible answers for piglet’s kinetic energy:

  • a > b > c, or others.

Work by Non-Constant Forces

• Work Calculation: For a non-constant force:

  • $W = \int \mathbf{F} \cdot d\mathbf{l}$

• Graphical interpretation involves integrating across the path of the force.

Spring Force and Hooke’s Law

• Hooke's Law:

• Spring Force given by:

  • $F_s = -kx$

  • Where $k$ is the spring constant and $x$ is the displacement from the relaxed position.

  • The force is linear with displacement.

Work Done by Spring Force

• Formula for Work by Spring:

  • $W<em>s = \frac{1}{2} k (x</em>i^2 - x_f^2)$

  • Work only depends on initial and final positions relative to the relaxed spring length.

Box Example with Spring

• Consider a box attached to a spring:

  • When compressed by distance $D$, determine if the spring does positive, negative, or zero work during these motions:

  • Pushing box compresses spring (work done by spring)? (Select from [A, B, C])

  • Releasing box back to relaxed position (work done by spring)?

• Example 4:

  • A box is attached to a spring with spring constant $k = 1.0 ext{ N/m}$:

  • If compressed by distance $D = 0.5 ext{ m}$, calculate work done by the spring:

    • Use formula: $W<em>s = \frac{1}{2} k (x</em>i^2 - x_f^2)$

Functions of Displacement from Relaxed Position

• It is crucial to understand how the displacement affects work done by a spring, since:

  • Work depends on the initial and final spring lengths, not the path taken.

Key Concept: Work done by a variable force can be expressed mathematically as:

$W = \int \mathbf{F} \cdot d\mathbf{r} = \int Fx dx + \int Fy dy + \int F_z dz$

Work by Constant Force:
Work by constant force can be simplified as:

$W = F \cdot d$

Work done on the climber by gravitational force ($W$):
$W = mgh$
For an upward displacement:
$W = -mg\Delta y$

Kinetic Energy and Work-Energy Theorem:
Work-Kinetic energy theorem states:

$W_{net} = \Delta KE$

Work Calculation for non-constant forces:
$W = \int \mathbf{F} \cdot d\mathbf{l}$

Spring Force given by Hooke's Law:
$F_s = -kx$

Work Done by Spring:
$Ws = \frac{1}{2} k (x<em>i^2 - x</em>f^2)$