Spring Forces - AP Physics 1

1. State the formula for potential energy stored in a spring.

   $U = \frac{1}{2}kx^2$. Energy stored when spring is displaced from equilibrium.

2. What type of motion does a mass undergo when attached to a spring?

   Simple harmonic motion. Oscillatory motion with restoring force proportional to displacement.

3. What does the negative sign in Hooke's Law represent?

   Opposite direction to displacement. Spring force always acts to restore equilibrium position.

4. Which physical law describes the force in a spring?

   Hooke's Law. Named after Robert Hooke, describes elastic behavior.

5. What is the maximum force a spring can exert when fully stretched?

   Force at maximum displacement. Occurs at maximum displacement from equilibrium.

6. What is the restoring force in a spring system?

   Force returning system to equilibrium. Force that opposes displacement from equilibrium.

7. Find the displacement if $F_s = -10 \, N$ and $k = 50 \, N/m$.

   $x = 0.2 \, m$. From $x = \frac{F_s}{-k} = \frac{-10}{-50} = 0.2$ m.

8. Calculate the displacement if $F_s = -40 \, N$ and $k = 80 \, N/m$.

   $x = 0.5 \, m$. Using $x = \frac{|F_s|}{k} = \frac{40}{80} = 0.5$ m.

9. What is the potential energy of a spring at maximum stretch?

   Maximum potential energy. Energy is maximum when displacement is maximum.

10. What is the potential energy for a spring at its equilibrium position?

    $0 \, J$. No displacement means no stored energy in the spring.

11. Identify the variable representing displacement in Hooke's Law.

    $x$. Distance from equilibrium position, positive or negative.

12. Identify the proportionality constant in Hooke's Law.

    Spring constant $k$. The coefficient relating force to displacement.

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

    Joule (J). Standard SI unit for all forms of energy.

14. What is the force exerted by a spring called?

    Restoring force. Force that acts to restore system to equilibrium.

15. Calculate the force if $k = 150 \, N/m$ and $x = -0.3 \, m$.

    $F_s = 45 \, N$. Force magnitude: $|F_s| = kx = 150 \times 0.3 = 45$ N.

16. Calculate spring force with $k = 250 \, N/m$ and $x = -0.04 \, m$.

    $F_s = 10 \, N$. Force magnitude: $|F_s| = k|x| = 250 \times 0.04 = 10$ N.

17. What is the effect of increasing spring constant on stiffness?

    Increases stiffness. Higher $k$ means greater resistance to deformation.

18. What happens to potential energy if displacement is tripled?

    Increases ninefold. Potential energy depends on $x^2$, so $(3x)^2 = 9x^2$.

19. What is Hooke's Law formula for spring force?

    $F_s = -kx$. The fundamental equation showing force is proportional to displacement.

20. What does $k$ represent in Hooke's Law?

    Spring constant. Measure of spring stiffness in Hooke's Law.

21. Identify the variable representing potential energy in spring systems.

    $U$. Symbol for potential energy in spring equations.

22. Determine potential energy if $k = 50 \, N/m$ and $x = 0.4 \, m$.

    $U = 4 \, J$. Using $U = \frac{1}{2}kx^2 = \frac{1}{2}(50)(0.4)^2 = 4$ J.

23. Identify the type of force acting when a spring is neither compressed nor stretched.

    No force (equilibrium). Spring is at its natural length with zero displacement.

24. Identify the nature of the graph between force and displacement for springs.

    Linear. Hooke's Law creates a straight-line relationship.

25. What is the equilibrium position of a spring?

    Position where net force is zero. Natural length where spring exerts no force.

26. What is the potential energy at maximum compression for a spring?

    Maximum potential energy. All kinetic energy converted to potential energy.

27. Find the spring constant for $F_s = 30 \, N$ and $x = 0.15 \, m$.

    $k = 200 \, N/m$. From $k = \frac{F_s}{x} = \frac{30}{0.15} = 200$ N/m.

28. Find the spring force if $k = 200 \, N/m$ and $x = 0.1 \, m$.

    $F_s = -20 \, N$. Using $F_s = -kx = -200 \times 0.1 = -20$ N.

29. Which type of energy is stored in a compressed or stretched spring?

    Elastic potential energy. Energy stored due to elastic deformation of the spring.

30. Find the spring constant if $F_s = 50 \, N$ and $x = 0.25 \, m$.

    $k = 200 \, N/m$. Using $k = \frac{F_s}{x} = \frac{50}{0.25} = 200$ N/m.

31. What is the potential energy if $k = 100 \, N/m$ and $x = 0.1 \, m$?

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

32. What happens to spring force if the spring constant is doubled?

    Force doubles. Force is directly proportional to spring constant.

33. Determine the spring constant if $U = 2 \, J$ and $x = 0.1 \, m$.

    $k = 400 \, N/m$. From $k = \frac{2U}{x^2} = \frac{2(2)}{(0.1)^2} = 400$ N/m.

34. Calculate potential energy for $k = 300 \, N/m$ and $x = 0.2 \, m$.

    $U = 6 \, J$. Using $U = \frac{1}{2}kx^2 = \frac{1}{2}(300)(0.2)^2 = 6$ J.

35. Identify the unit of spring constant $k$ in SI units.

    Newton per meter (N/m). Force per unit length, derived from $F = kx$.

36. What is the relationship between spring force and displacement?

    Directly proportional. Larger displacement produces proportionally larger force magnitude.

37. Find the spring constant when $U = 8 \, J$ and $x = 0.2 \, m$.

    $k = 400 \, N/m$. From $k = \frac{2U}{x^2} = \frac{2(8)}{(0.2)^2} = 400$ N/m.

38. Identify the direction of spring force when a spring is compressed.

    Opposite to compression. Spring pushes back against the compression direction.

39. Calculate spring force for $k = 100 \, N/m$ and $x = 0.05 \, m$.

    $F_s = -5 \, N$. Using $F_s = -kx = -100 \times 0.05 = -5$ N.

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