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 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

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}2 - \mathbf{r}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}1 to \mathbf{r}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:

    • Ws = \frac{1}{2} k (xi^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: Ws = \frac{1}{2} k (xi^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 (xi^2 - xf^2)