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Work and Power | Study Mondo
Topics / Work and Energy / Work and Power Work and Power Calculating work using line integrals and instantaneous power
๐ฏ โญ INTERACTIVE LESSON
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Start Interactive Lesson โ Work and Power
Work as a Line Integral
For a variable force F โ \vec{F} F along a path from point A to B:
W = โซ A B F โ โ
d r โ W = \int_A^B \vec{F} \cdot d\vec{r} W
๐ Practice ProblemsNo example problems available yet.
Explain using: ๐ Simple words ๐ Analogy ๐จ Visual desc. ๐ Example ๐ก Explain
๐งช Practice Lab Interactive practice problems for Work and Power
โพ ๐ Related Topics in Work and Energyโ Frequently Asked QuestionsWhat is Work and Power?โพ Calculating work using line integrals and instantaneous power
How can I study Work and Power effectively?โพ Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Regular review and active practice are key to retention.
Is this Work and Power study guide free?โพ Yes โ all study notes, flashcards, and practice problems for Work and Power on Study Mondo are 100% free. No account is needed to access the content.
What course covers Work and Power?โพ Work and Power is part of the AP Physics C: Mechanics course on Study Mondo, specifically in the Work and Energy section. You can explore the full course for more related topics and practice resources.
๐ก Study Tipsโ Work through examples step-by-step โ Practice with flashcards daily โ Review common mistakes =
In one dimension:
W = โซ x 1 x 2 F x โ d x W = \int_{x_1}^{x_2} F_x \, dx W = โซ x 1 โ x 2 โ โ F x โ d x
In three dimensions:
W = โซ A B ( F x โ d x + F y โ d y + F z โ d z ) W = \int_A^B (F_x \, dx + F_y \, dy + F_z \, dz) W = โซ A B โ ( F x โ d x + F y โ d y + F z โ d z )
Work-Energy Theorem From Newton's second law:
F = m d v d t = m d v d x d x d t = m v d v d x F = m\frac{dv}{dt} = m\frac{dv}{dx}\frac{dx}{dt} = mv\frac{dv}{dx} F = m d t d v โ = m d x d v โ d t d x โ = m v d x d v โ
F โ d x = m v โ d v F \, dx = mv \, dv F d x = m v d v
Integrating:
โซ x 1 x 2 F โ d x = โซ v 1 v 2 m v โ d v \int_{x_1}^{x_2} F \, dx = \int_{v_1}^{v_2} mv \, dv โซ x 1 โ x 2 โ โ F d x = โซ v 1 โ v 2 โ โ m v d v
W = 1 2 m v 2 2 โ 1 2 m v 1 2 = ฮ K E W = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 = \Delta KE W = 2 1 โ m v 2 2 โ โ 2 1 โ m v 1 2 โ = ฮ K E
Work-energy theorem: W n e t = ฮ K E W_{net} = \Delta KE W n e t โ = ฮ K E
Common Force Examples
Constant Force W = F โ
d = F d cos โก ฮธ W = F \cdot d = Fd\cos\theta W = F โ
d = F d cos ฮธ
where ฮธ \theta ฮธ is angle between force and displacement.
Spring Force W = โซ 0 x ( โ k x โฒ ) โ d x โฒ = โ 1 2 k x 2 W = \int_0^x (-kx') \, dx' = -\frac{1}{2}kx^2 W = โซ 0 x โ ( โ k x โฒ ) d x โฒ = โ 2 1 โ k x 2
(Work done by spring; work done on spring is positive)
Gravity Near Earth's Surface W = โซ y 1 y 2 ( โ m g ) โ d y = โ m g ( y 2 โ y 1 ) = โ m g ฮ h W = \int_{y_1}^{y_2} (-mg) \, dy = -mg(y_2 - y_1) = -mg\Delta h W = โซ y 1 โ y 2 โ โ ( โ m g ) d y = โ m g ( y 2 โ โ y 1 โ ) = โ m g ฮ h
Inverse Square Force (Gravity/Electrostatic) F = โ k r 2 F = -\frac{k}{r^2} F = โ r 2 k โ
W = โซ r 1 r 2 ( โ k r 2 ) d r = k ( 1 r 2 โ 1 r 1 ) W = \int_{r_1}^{r_2} \left(-\frac{k}{r^2}\right) dr = k\left(\frac{1}{r_2} - \frac{1}{r_1}\right) W = โซ r 1 โ r 2 โ โ ( โ r 2 k โ ) d r = k ( r 2 โ 1 โ โ r 1 โ
Power Instantaneous power:
P = d W d t P = \frac{dW}{dt} P = d t d W โ
Since d W = F โ โ
d r โ dW = \vec{F} \cdot d\vec{r} d W = F โ
d r :
P = d W d t = F โ โ
d r โ d t = F โ โ
v โ P = \frac{dW}{dt} = \vec{F} \cdot \frac{d\vec{r}}{dt} = \vec{F} \cdot \vec{v} P = d t d W โ = F โ
d t d r โ = F โ
v
In one dimension:
P = F v P = Fv P = F v
Average Power P a v g = W ฮ t P_{avg} = \frac{W}{\Delta t} P a vg โ = ฮ t W โ
Power and Kinetic Energy P = d K E d t = d d t ( 1 2 m v 2 ) = m v d v d t = m v a = F v P = \frac{dKE}{dt} = \frac{d}{dt}\left(\frac{1}{2}mv^2\right) = mv\frac{dv}{dt} = mva = Fv P = d t d K E โ = d t d โ ( 2 1 โ m v 2 ) = m v d t d v โ = m v a = F v
Work on Variable Mass Systems For rocket with exhaust velocity v e v_e v e โ relative to rocket:
Power delivered by engine:
P = F t h r u s t โ
v = v r e l โฃ d m d t โฃ โ
v P = F_{thrust} \cdot v = v_{rel}\left|\frac{dm}{dt}\right| \cdot v P = F t h r u s t โ โ
v = v re l โ โ d t d m โ โ โ
v
Example: Work Against Drag Object moving through fluid with drag F d = โ b v 2 F_d = -bv^2 F d โ = โ b v 2 :
W = โซ 0 d F d โ d x = โ b โซ 0 d v 2 โ d x W = \int_0^d F_d \, dx = -b\int_0^d v^2 \, dx W = โซ 0 d โ F d โ d x = โ b โซ 0 d โ v 2 d x
If v ( t ) v(t) v ( t ) is known, use d x = v โ d t dx = v \, dt d x = v d t :
W = โ b โซ 0 T v 3 โ d t W = -b\int_0^T v^3 \, dt W = โ b โซ 0 T โ v 3 d t
This work is dissipated as heat.
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