Model Energy Transfer Computationally - Physics

1. What is the elastic potential energy formula for a spring in computational energy tracking?

   $U_s=\frac{1}{2}kx^2$. Elastic PE is proportional to spring constant and displacement squared.

2. What is the work done by a constant force parallel to the displacement?

   $W=Fd$. Work equals force times distance when parallel.

3. What is the net work–kinetic energy relation used to compute speed changes?

   $W_{\text{net}}=\Delta K$. Work-energy theorem: net work changes kinetic energy.

4. What is the power definition used to model energy transfer rate?

   $P=\frac{\Delta E}{\Delta t}$. Power is energy transfer rate.

5. Find spring energy $U_s$ for $k=200\,\text{N/m}$ and $x=0.10\,\text{m}$.

   $1\,\text{J}$. $U_s = \frac{1}{2}(200)(0.1)^2 = \frac{1}{2}(200)(0.01) = 1\,\text{J}$

6. What is the work formula for a constant force parallel to displacement?

   $W=Fd$. Work equals force times distance when force is parallel to motion.

7. What equation relates energy transferred by heating to mass, specific heat, and temperature change?

   $Q=mc\Delta T$. Heat capacity relates thermal energy to temperature change.

8. What is the thermal energy transfer from kinetic friction over distance $d$ on a level surface?

   $E_{\text{th}}=f_k d=\mu_k N d$. Friction converts mechanical energy to thermal over distance.

9. Find heat $Q$ for $m=0.50\,\text{kg}$, c=4000\,\text{J/(kg\cdot ^\circ C)}, $\Delta T=2\,^\circ\text{C}$.

   $4000\,\text{J}$. $Q=(0.50)(4000)(2)=4000\,\text{J}$.

10. Find work $W$ for $F=10\,\text{N}$, $d=5\,\text{m}$, and $\theta=60^\circ$.

    $25\,\text{J}$. $W = (10)(5)\cos(60°) = 50(0.5) = 25\,\text{J}$

11. Find average power for $\Delta E=600\,\text{J}$ transferred in $\Delta t=3\,\text{s}$.

    $200\,\text{W}$. $P = \frac{600}{3} = 200\,\text{W}$

12. What is the conservation of energy equation for a closed system in computational models?

    $E_{\text{initial}}=E_{\text{final}}$. Energy is conserved in closed systems with no external work or heat transfer.

13. What is the work done by a constant force at angle $\theta$ to the displacement?

    $W=Fd\cos\theta$. Cosine accounts for the component of force along displacement.

14. What is the kinetic energy formula used in energy-transfer calculations?

    $K=\frac{1}{2}mv^2$. Kinetic energy depends on mass and velocity squared.

15. Identify the computational update rule for speed from kinetic energy $K$ and mass $m$.

    $v=\sqrt{\frac{2K}{m}}$. Rearranged kinetic energy formula to solve for speed.

16. Find work $W$ for $F=10\,\text{N}$, $d=4\,\text{m}$, and $\theta=60^\circ$.

    $20\,\text{J}$. $W=(10)(4)\cos(60^\circ)=(10)(4)(0.5)=20\,\text{J}$.

17. What is the thermal energy change formula for heating with no phase change?

    $Q=mc\Delta T$. Heat transfer depends on mass, specific heat, and temperature change.

18. What is the work formula for a constant force at angle $\theta$ to displacement?

    $W=Fd\cos\theta$. Work depends on the component of force in the direction of motion.

19. Identify the correct discrete update for speed from kinetic energy $K$ and mass $m$.

    $v=\sqrt{\frac{2K}{m}}$. Rearranging $K=\frac{1}{2}mv^2$ to solve for velocity.

20. What is the energy balance statement including nonconservative work $W_{\text{nc}}$?

    $\Delta K+\Delta U=W_{\text{nc}}$. Total mechanical energy change equals work done by nonconservative forces.

21. What is the conservation of energy statement used in computational energy-transfer models?

    $E_{\text{in}}-E_{\text{out}}=\Delta E_{\text{system}}$. Energy conservation: input minus output equals system change.

22. Find efficiency $\eta$ if $E_{\text{in}}=200\,\text{J}$ and $E_{\text{out}}=150\,\text{J}$.

    $0.75$. $\eta=\frac{150}{200}=0.75$ or 75% efficient.

23. What is the mechanical power relation connecting energy transfer and time?

    $P=\frac{\Delta E}{\Delta t}$. Power is the rate of energy transfer.

24. Find $K$ for $m=2\,\text{kg}$ and $v=3\,\text{m/s}$ using $K=\frac{1}{2}mv^2$.

    $9\,\text{J}$. $K = \frac{1}{2}(2)(3^2) = \frac{1}{2}(2)(9) = 9\,\text{J}$

25. What is the efficiency formula for an energy-transfer process in computational models?

    $\eta=\frac{E_{\text{out}}}{E_{\text{in}}}$. Efficiency is the ratio of useful output to total input energy.

26. What is the gravitational potential energy function for two masses separated by distance $r$?

    $U=-\frac{Gm_1m_2}{r}$. Negative sign indicates attractive force lowers PE as $r$ decreases.

27. Find $\Delta U_g$ for $m=1\,\text{kg}$, $\Delta h=5\,\text{m}$, $g=10\,\text{m/s}^2$.

    $50\,\text{J}$. $\Delta U_g=(1)(10)(5)=50\,\text{J}$.

28. Find $\Delta U_g$ for $m=4\,\text{kg}$, $\Delta h=2\,\text{m}$, $g=10\,\text{m/s}^2$.

    $80\,\text{J}$. $\Delta U_g = (4)(10)(2) = 80\,\text{J}$

29. Identify the correct discrete update for energy using constant power $P$ over time step $\Delta t$.

    $E_{n+1}=E_n+P\Delta t$. Energy at next time step equals current energy plus power times time interval.

30. What is the change in gravitational potential energy near Earth?

    $\Delta U_g=mg\Delta h$. Gravitational PE change is weight times height change.

31. What is the average power in terms of work done over a time interval?

    $P=\frac{W}{\Delta t}$. Power equals work divided by time interval.

32. What equation updates system energy when heat $Q$ is added and work $W$ is done by the system?

    $\Delta E=Q-W$. First law of thermodynamics: energy change equals heat added minus work done by system.

33. What is the gravitational potential energy change near Earth used in energy models?

    $\Delta U_g=mg\Delta h$. Gravitational PE change is weight times height change.

34. Identify the computational update rule for energy using constant power over time step $\Delta t$.

    $E_{\text{new}}=E_{\text{old}}+P\Delta t$. Energy increases by power times elapsed time.

35. What is the elastic potential energy stored in an ideal spring compressed or stretched by $x$?

    $U_s=\frac{1}{2}kx^2$. Spring PE is quadratic in displacement from equilibrium.

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