Energy of Simple Harmonic Oscillators - AP Physics 1

1. What is the unit of potential energy in the SI system?

   Joule (J). Energy is measured in joules in the SI system.

2. Determine the factor by which energy increases if amplitude is tripled.

   Energy increases by a factor of 9. Energy is proportional to $A^2$, so tripling A increases E by $3^2 = 9$.

3. What is the relationship between total energy, kinetic energy, and potential energy?

   $E = KE + PE$. Conservation of energy states total equals sum of parts.

4. What is the unit of amplitude in the SI system?

   Meter (m). Distance is measured in meters in the SI system.

5. Determine the energy type at maximum speed in a harmonic oscillator.

   Kinetic energy. Maximum speed occurs at equilibrium where all energy is kinetic.

6. What happens to the total energy if the amplitude is doubled in a harmonic oscillator?

   Total energy quadruples. Energy is proportional to $A^2$, so doubling A increases E by $2^2 = 4$.

7. Calculate the kinetic energy if $m = 2$ kg and $v = 3$ m/s.

   $KE = 9$ J. Using $KE = \frac{1}{2}mv^2 = \frac{1}{2}(2)(3)^2 = 9$ J.

8. Which type of energy is zero at the maximum displacement of a harmonic oscillator?

   Kinetic energy. At maximum displacement, velocity is zero making KE zero.

9. If $E = 10$ J, $PE = 6$ J at a point, find the kinetic energy.

   $KE = 4$ J. Energy conservation: $KE = E - PE = 10 - 6 = 4$ J.

10. State the expression for kinetic energy in a simple harmonic oscillator.

    $KE = \frac{1}{2} m v^2$. Standard kinetic energy formula with mass and velocity.

11. What is the formula for the total mechanical energy in a simple harmonic oscillator?

    $E = \frac{1}{2} k A^2$. Total energy equals half the spring constant times amplitude squared.

12. What is the unit of displacement in the SI system?

    Meter (m). Distance is measured in meters in the SI system.

13. What is the unit of kinetic energy in the SI system?

    Joule (J). Energy is measured in joules in the SI system.

14. What remains constant for a simple harmonic oscillator in the absence of non-conservative forces?

    Total mechanical energy. Energy conservation applies when no energy is lost to friction.

15. Find the total mechanical energy if $k = 200$ N/m and $A = 0.5$ m.

    $E = 25$ J. Using $E = \frac{1}{2}kA^2 = \frac{1}{2}(200)(0.5)^2 = 25$ J.

16. If $E = 15$ J, $KE = 9$ J at a point, find the potential energy.

    $PE = 6$ J. Energy conservation: $PE = E - KE = 15 - 9 = 6$ J.

17. State the expression for potential energy in a simple harmonic oscillator.

    $PE = \frac{1}{2} k x^2$. Energy stored in spring proportional to displacement squared.

18. Find the spring constant if the total energy is $50$ J and $A = 0.4$ m.

    $k = 625$ N/m. Solving $50 = \frac{1}{2}k(0.4)^2$ gives $k = 625$ N/m.

19. Which type of energy is zero at the equilibrium position of a harmonic oscillator?

    Potential energy. At equilibrium, displacement is zero making PE zero.

20. What does the variable $k$ represent in the energy formula of a harmonic oscillator?

    Spring constant. The proportionality constant relating force to displacement.

21. Identify the variable $x$ in the potential energy formula of a harmonic oscillator.

    Displacement from equilibrium. Distance from the equilibrium position of the oscillator.

22. Determine the amplitude if the total energy is $32$ J and $k = 8$ N/m.

    $A = 2$ m. Solving $32 = \frac{1}{2}(8)A^2$ gives $A = 2$ m.

23. If $k = 250$ N/m and $A = 0.1$ m, calculate the total energy.

    $E = 1.25$ J. Using $E = \frac{1}{2}(250)(0.1)^2 = 1.25$ J.

24. Find $v$ if $KE = 18$ J and $m = 3$ kg.

    $v = 4$ m/s. Solving $18 = \frac{1}{2}(3)v^2$ gives $v = 4$ m/s.

25. What is the form of energy when the displacement of a harmonic oscillator is maximum?

    Potential energy. At maximum displacement, velocity is zero so only PE exists.

26. Find the spring constant if $E = 18$ J and $A = 0.3$ m.

    $k = 400$ N/m. Solving $18 = \frac{1}{2}k(0.3)^2$ gives $k = 400$ N/m.

27. What is the unit of the spring constant $k$ in the SI system?

    Newton per meter (N/m). Force per unit displacement has units of N/m.

28. What is the form of energy when the velocity of a harmonic oscillator is maximum?

    Kinetic energy. At equilibrium, displacement is zero so only KE exists.

29. If $m = 1.5$ kg and $v = 2$ m/s, find the kinetic energy.

    $KE = 3$ J. Using $KE = \frac{1}{2}(1.5)(2)^2 = 3$ J.

30. Calculate the total energy if $k = 150$ N/m and $A = 0.3$ m.

    $E = 6.75$ J. Using $E = \frac{1}{2}(150)(0.3)^2 = 6.75$ J.

31. Calculate the mass if $v = 4$ m/s and $KE = 32$ J.

    $m = 4$ kg. Solving $32 = \frac{1}{2}m(4)^2$ gives $m = 4$ kg.

32. Calculate the potential energy if $k = 100$ N/m and $x = 0.1$ m.

    $PE = 0.5$ J. Using $PE = \frac{1}{2}kx^2 = \frac{1}{2}(100)(0.1)^2 = 0.5$ J.

33. Find the displacement if $k = 50$ N/m and $PE = 4$ J.

    $x = 0.4$ m. Solving $4 = \frac{1}{2}(50)x^2$ gives $x = 0.4$ m.

34. Find $x$ if $PE = 5$ J and $k = 125$ N/m.

    $x = 0.2$ m. Solving $5 = \frac{1}{2}(125)x^2$ gives $x = 0.2$ m.

35. If $m = 0.5$ kg and $v = 5$ m/s, calculate the kinetic energy.

    $KE = 6.25$ J. Using $KE = \frac{1}{2}(0.5)(5)^2 = 6.25$ J.

36. What does the variable $A$ represent in the energy formula of a harmonic oscillator?

    Amplitude. Maximum displacement from equilibrium position.

37. If $k = 200$ N/m and $x = 0.2$ m, find the potential energy.

    $PE = 4$ J. Using $PE = \frac{1}{2}(200)(0.2)^2 = 4$ J.

38. What is the unit of total mechanical energy in the SI system?

    Joule (J). Energy is measured in joules in the SI system.

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