Work Energy AND Power flash cards

1. 1.Define work done by a constant force $\vec{F}$ over displacement $\vec{d}$.

   $W = \vec{F}\cdot\vec{d} = Fd\cos\theta$, where $\theta$ is the angle between force and displacement. Work is a scalar with SI unit joule (J).

   Hint: Dot product of force and displacement.

2. 2.For what angle between force and displacement is the work done (i) maximum positive, (ii) zero, (iii) maximum negative?

   (i) $\theta = 0^\circ$ gives $W = +Fd$; (ii) $\theta = 90^\circ$ gives $W = 0$; (iii) $\theta = 180^\circ$ gives $W = -Fd$.

   Hint: $\cos\theta$ controls the sign.

3. 3.Is work a scalar or a vector? What are its dimensions?

   Work is a scalar. Dimensions: $[M L^2 T^{-2}]$. SI unit joule, $1\text{ J} = 1\text{ N·m}$.

   Hint: Force × distance.

4. 4.Give the general integral expression for work done by a variable force along a path.

   $W = \int \vec{F}\cdot d\vec{r}$. For 1-D, $W = \int_{x_1}^{x_2} F(x)\,dx$, which equals the area under the $F$ vs $x$ graph.

   Hint: Sum of infinitesimal $\vec{F}\cdot d\vec{r}$.

5. 5.How do you read work done from a force–displacement ($F$–$x$) graph?

   Work equals the signed area between the curve and the $x$-axis. Area above the axis is positive work; area below is negative work.

   Hint: Integral = area.

6. 6.State the work–energy theorem.

   The net work done by all forces on a particle equals the change in its kinetic energy: $W_{net} = \Delta K = \tfrac12 mv^2 - \tfrac12 mu^2$.

   Hint: Net work = change in KE.

7. 7.Does the work–energy theorem hold for variable forces and curved paths?

   Yes. It is a general result derived from Newton's second law and holds for constant or variable forces, straight or curved paths, in an inertial frame.

   Hint: General consequence of $F=ma$.

8. 8.Define kinetic energy and write its formula.

   Kinetic energy is the energy possessed by a body due to its motion: $K = \tfrac12 mv^2$. It is always non-negative and a scalar.

   Hint: Half m v squared.

9. 9.Relate kinetic energy to linear momentum $p$.

   $K = \dfrac{p^2}{2m}$, equivalently $p = \sqrt{2mK}$. Useful when momentum is known instead of velocity.

   Hint: Substitute $v = p/m$.

10. 10.Two bodies have equal momentum. Which has greater kinetic energy?

    Since $K = p^2/2m$ with equal $p$, the lighter body (smaller $m$) has greater kinetic energy.

    Hint: $K \propto 1/m$ at fixed $p$.

11. 11.Two bodies have equal kinetic energy. Which has greater momentum?

    Since $p = \sqrt{2mK}$ with equal $K$, the heavier body (larger $m$) has greater momentum.

    Hint: $p \propto \sqrt{m}$ at fixed $K$.

12. 12.If a body's speed is doubled, how does its kinetic energy change?

    KE becomes 4 times, since $K \propto v^2$. Tripling speed makes KE 9 times.

    Hint: $K \propto v^2$.

13. 13.Can kinetic energy be negative? Can work be negative?

    Kinetic energy can never be negative ($\tfrac12 mv^2 \ge 0$). Work can be negative when force opposes displacement ($\cos\theta < 0$).

    Hint: KE always $\ge 0$; work can be $-$.

14. 14.Define potential energy.

    Potential energy is energy stored in a system by virtue of the configuration/position of its parts in a conservative force field. Only changes in PE (not absolute value) are physically meaningful.

    Hint: Energy of configuration.

15. 15.Write the gravitational PE near Earth's surface and state the reference dependence.

    $U = mgh$ measured from a chosen reference level. The zero of $U$ is arbitrary; only differences $\Delta U$ matter.

    Hint: $mgh$ above a datum.

16. 16.Define a conservative force.

    A force is conservative if the work it does is path-independent (depends only on endpoints), equivalently the work done in any closed loop is zero. A PE function can then be defined.

    Hint: Path-independent; zero round-trip work.

17. 17.Give three examples each of conservative and non-conservative forces.

    Conservative: gravity, spring (elastic) force, electrostatic force. Non-conservative: friction, air drag, viscous force (and applied/normal forces in general).

    Hint: Fields vs dissipative.

18. 18.How is a conservative force related to its potential energy in 1-D?

    $F(x) = -\dfrac{dU}{dx}$. The force points toward decreasing potential energy.

    Hint: Negative slope of $U(x)$.

19. 19.Write the 3-D relation between a conservative force and potential energy.

    $\vec{F} = -\nabla U = -\left(\dfrac{\partial U}{\partial x}\hat{i} + \dfrac{\partial U}{\partial y}\hat{j} + \dfrac{\partial U}{\partial z}\hat{k}\right)$.

    Hint: Force = minus gradient of U.

20. 20.Express the change in potential energy in terms of the conservative force.

    $\Delta U = U_f - U_i = -\int_i^f \vec{F}\cdot d\vec{r}$, i.e. $U(x) = -\int F\,dx + C$.

    Hint: PE = minus work by conservative force.

21. 21.What does a positive slope of the $U(x)$ curve tell you about the force?

    If $dU/dx > 0$, then $F = -dU/dx < 0$, so the force points in the $-x$ direction (toward lower $U$).

    Hint: Force pushes downhill on the U-curve.

22. 22.Define stable, unstable, and neutral equilibrium in terms of $U(x)$.

    Equilibrium requires $dU/dx = 0$. Stable: $U$ minimum ($d^2U/dx^2 > 0$). Unstable: $U$ maximum ($d^2U/dx^2 < 0$). Neutral: $U$ constant around the point.

    Hint: Minimum, maximum, flat of U.

23. 23.State the principle of conservation of mechanical energy.

    If only conservative forces do work, the total mechanical energy $E = K + U$ remains constant: $\Delta K + \Delta U = 0$.

    Hint: K + U constant with only conservative forces.

24. 24.How is the work–energy theorem modified when non-conservative forces act?

    $W_{nc} = \Delta K + \Delta U = \Delta E$. The work by non-conservative forces equals the change in total mechanical energy (usually negative for friction).

    Hint: $W_{nc} = \Delta E$.

25. 25.A block slides down a rough incline. Where does the lost mechanical energy go?

    It is converted mostly into heat (and some sound) due to friction: $|W_{friction}| =$ mechanical energy lost. Total energy (including thermal) is still conserved.

    Hint: Friction → heat.

26. 26.State the general (broad) law of conservation of energy.

    Energy can neither be created nor destroyed, only transformed from one form to another; the total energy of an isolated system is constant.

    Hint: Total energy is invariant.

27. 27.Define power. Give its SI unit and dimensions.

    Power is the rate of doing work: $P = \dfrac{dW}{dt}$. SI unit watt (W), $1\text{ W}=1\text{ J/s}$; dimensions $[M L^2 T^{-3}]$.

    Hint: Rate of work.

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