Analyze Energy Using Conservation Laws - Physics

1. Identify the correct energy equation for a pendulum bob between two heights when air resistance is negligible.

   $mgh_i+\frac{1}{2}mv_i^2=mgh_f+\frac{1}{2}mv_f^2$. Conservation of mechanical energy for pendulum motion.

2. Find the speed $v$ of a mass $m$ after descending height $h$ with kinetic friction doing work $-f_k d$.

   $v=\sqrt{2gh-\frac{2f_k d}{m}}$. Apply energy conservation with friction work included.

3. What is the gravitational potential energy formula near Earth for mass $m$ at height $h$?

   $U_g=mgh$. Weight ($mg$) times height gives gravitational potential energy.

4. State the conservation of mechanical energy equation when only conservative forces act.

   $K_i+U_i=K_f+U_f$. Total mechanical energy remains constant with only conservative forces.

5. Find the speed $v$ after dropping from rest through height $h$ with no losses.

   $v=\sqrt{2gh}$. From $mgh = \frac{1}{2}mv^2$ with energy conservation.

6. Identify the correct energy equation for a block starting from rest: slide down height $h$ with friction work $W_f$.

   $mgh+W_f=\frac{1}{2}mv^2$. Initial PE plus friction work equals final KE.

7. What is the general energy accounting equation including nonconservative work?

   $K_i+U_i+W_{nc}=K_f+U_f$. Includes nonconservative work in energy conservation.

8. What is the general energy accounting equation including nonconservative work $W_{\text{nc}}$?

   $K_i+U_i+W_{\text{nc}}=K_f+U_f$. Nonconservative work accounts for energy not conserved.

9. What is the relationship between nonconservative work and mechanical energy change?

   $W_{nc}=\Delta E_{mech}$. Nonconservative work equals the change in total mechanical energy.

10. Find the speed at the bottom after dropping from rest through height $h$, ignoring air resistance.

    $v=\sqrt{2gh}$. From $mgh=\frac{1}{2}mv^2$, solving for $v$ gives $\sqrt{2gh}$.

11. Find the speed $v$ at height $h$ for an object launched upward with speed $v_0$ (no air drag).

    $v=\sqrt{v_0^2-2gh}$. Energy conservation: $\frac{1}{2}mv_0^2=\frac{1}{2}mv^2+mgh$.

12. Find the spring compression $x$ needed to stop a mass $m$ moving at speed $v$ on a frictionless surface.

    $x=v\sqrt{\frac{m}{k}}$. All kinetic energy converts to spring PE: $\frac{1}{2}mv^2=\frac{1}{2}kx^2$.

13. What is the work–energy theorem relating net work and kinetic energy change?

    $W_{net}=\Delta K$. Net work equals the change in kinetic energy.

14. State the work–energy theorem relating net work and kinetic energy change.

    $W_{\text{net}}=\Delta K$. Net work equals the change in kinetic energy.

15. What is the kinetic friction magnitude formula using coefficient $\mu_k$ and normal force $N$?

    $f_k=\mu_k N$. Kinetic friction equals coefficient times normal force.

16. Find the minimum initial speed $v_0$ needed to reach height $h$ if nonconservative work is $W_{nc}<0$.

    $v_0=\sqrt{2gh-\frac{2W_{nc}}{m}}$. Rearrange energy equation with negative nonconservative work.

17. Find the final speed $v_f$ if net work on a mass $m$ is $W_{net}$ and initial speed is $v_i$.

    $v_f=\sqrt{v_i^2+\frac{2W_{net}}{m}}$. From work-energy theorem: $W_{net} = \frac{1}{2}m(v_f^2 - v_i^2)$.

18. Find the work done by friction $W_f$ if mechanical energy decreases from $E_i$ to $E_f$.

    $W_f=E_f-E_i$. Friction work equals the mechanical energy loss.

19. Find the maximum height $h$ reached by an object launched upward with initial speed $v_0$ (no air drag).

    $h=\frac{v_0^2}{2g}$. From $\frac{1}{2}mv_0^2=mgh$, solving for $h$ gives $\frac{v_0^2}{2g}$.

20. What conservation equation applies when only conservative forces do work on a system?

    $K_i+U_i=K_f+U_f$. Mechanical energy is conserved when only conservative forces act.

21. Identify the correct expression for mechanical energy lost to friction over distance $d$ on level ground.

    $\Delta E_{mech}=-\mu_k N d$. Friction force times distance gives energy lost.

22. What is the elastic potential energy formula for a spring with constant $k$ stretched by $x$?

    $U_s=\frac{1}{2}kx^2$. Spring energy equals half the spring constant times displacement squared.

23. What is the definition of total mechanical energy $E_{mech}$ in a system?

    $E_{mech}=K+U$. Total mechanical energy is the sum of kinetic and potential energies.

24. Identify the sign of $\Delta U_g$ when an object moves downward by $\Delta h<0$.

    $\Delta U_g<0$. Moving down means $\Delta h<0$, so $mg\Delta h<0$.

25. Find the height $h$ reached if an object launched upward with speed $v_0$ stops at the top (no losses).

    $h=\frac{v_0^2}{2g}$. From $\frac{1}{2}mv_0^2 = mgh$ at maximum height.

26. What is the work done by a constant force $F$ over displacement $d$ at angle $\theta$?

    $W=Fd\cos\theta$. Work equals force times displacement times cosine of angle between them.

27. Find the speed $v$ of mass $m$ after sliding down height $h$ with kinetic friction $\mu_k$ over distance $d$.

    $v=\sqrt{2g(h-\mu_k d)}$. Initial PE minus friction work equals final KE.

28. Choose the correct statement: with only conservative forces, mechanical energy is (constant) or (decreases).

    Mechanical energy is constant. Conservative forces don't dissipate energy.

29. What is the kinetic energy formula for a mass $m$ moving at speed $v$?

    $K=\frac{1}{2}mv^2$. Energy of motion equals half the mass times velocity squared.

30. What is the gravitational potential energy change when an object drops by height $h$?

    $\Delta U_g=-mgh$. Negative because potential energy decreases when falling.

31. Identify the correct condition for using $mgh$ for gravitational potential energy.

    Near Earth with approximately constant $g$. $mgh$ assumes constant gravitational field strength.

32. Find the minimum initial speed $v_0$ to reach height $h$ if a constant friction force $f_k$ acts over distance $d$.

    $v_0=\sqrt{2gh+\frac{2f_k d}{m}}$. Initial KE must overcome both PE gain and friction work.

33. What is the gravitational potential energy change near Earth for height change $\Delta h$?

    $\Delta U_g=mg\Delta h$. Weight $mg$ times height change gives gravitational PE change.

34. Find the final speed $v$ if net work done on mass $m$ is $W_{\text{net}}$ and initial speed is $v_i$.

    $v=\sqrt{v_i^2+\frac{2W_{\text{net}}}{m}}$. From work-energy theorem: $W_{\text{net}}=\frac{1}{2}m(v^2-v_i^2)$.

35. Find the compression $x$ of a spring that stops a mass $m$ moving at speed $v$ on a frictionless surface.

    $x=v\sqrt{\frac{m}{k}}$. From $\frac{1}{2}mv^2 = \frac{1}{2}kx^2$ by energy conservation.

36. Which option is correct for a conservative force: $W_c=-\Delta U$ or $W_c=+\Delta U$?

    $W_c=-\Delta U$. Conservative force work equals negative of potential energy change.

37. What is the sign of work done by kinetic friction on a moving object over distance $d$?

    $W_f=-f_k d$. Friction opposes motion, so work is negative.

38. What is the elastic potential energy stored in a spring with constant $k$ stretched by $x$?

    $U_s=\frac{1}{2}kx^2$. Spring PE equals half the spring constant times displacement squared.

39. What is the work done by kinetic friction with coefficient $\mu_k$ over distance $d$ on level ground?

    $W_f=-\mu_kmgd$. Friction opposes motion, so work is negative.

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