Revision/fast recall

1. TITLE

Spring Force vs. Potential Energy Formulas cheat sheet — Everything for NEET on one page

2. QUICK FACTS BOX

SPRING FORCE & POTENTIAL ENERGY — QUICK FACTS

  • Spring force is an ideal, conservative, variable force.
  • Force is directly proportional to displacement (Hooke's Law).
  • Spring constant ($k$) measures stiffness; large $k$ means stiffer spring.
  • Spring force and displacement always have opposite signs.
  • Work done by a spring in a cyclic process is zero.
  • Potential energy is defined as zero at the equilibrium position ($x=0$).
  • Total mechanical energy ($E = K + V$) remains constant without friction.
  • Kinetic energy is maximum at equilibrium and zero at extreme positions.

3. FORMULA SHEET

FORMULAS:

  • Spring Force: $F_s = -kx$ (Unit: $\text{N}$)
    → Use when: Calculating the restoring force exerted by an ideal spring for a displacement $x$ from equilibrium.
  • Potential Energy of a Spring: $V(x) = \frac{1}{2}kx^2$ (Unit: $\text{J}$)
    → Use when: Finding the stored energy in a spring compressed or extended by $x$.
  • Work Done by Spring: $W_s = -\frac{1}{2}kx_m^2$ (Unit: $\text{J}$)
    → Use when: Calculating the work done by the spring force from equilibrium ($0$) to a maximum extension/compression ($x_m$).
  • Conservation of Mechanical Energy: $\frac{1}{2}kx_m^2 = \frac{1}{2}kx^2 + \frac{1}{2}mv^2$ (Unit: $\text{J}$)
    → Use when: Relating the speed $v$ of a mass $m$ to its position $x$ during oscillation.
  • Maximum Speed: $v_m = \sqrt{\frac{k}{m}}x_m$ (Unit: $\text{m s}^{-1}$)
    → Use when: Finding the maximum velocity of an oscillating mass as it passes through the equilibrium position.

4. MUST-REMEMBER VALUES/CONSTANTS

KEY VALUES:

  • Spring Constant ($k$) Dimensions = $[MT^{-2}]$
  • Potential Energy ($V$) Dimensions = $[ML^2T^{-2}]$
  • SI Unit of $k$ = $\text{N m}^{-1}$

5. COMPARISON SUMMARY

Feature Spring Force ($F_s$) Potential Energy ($V$)
Formula $-kx$ $\frac{1}{2}kx^2$
Nature Vector (has direction opposing displacement) Scalar (stored energy)
Sign Can be positive or negative Always positive (for any non-zero $x$)
Value at Equilibrium ($x=0$) Zero Zero
Value at Max Displacement ($x_m$) Maximum magnitude Maximum

6. COMMON MISTAKES — QUICK LIST

DON'T:

  • Forget the negative sign in Hooke's Law ($F_s = -kx$); the force always opposes displacement.
  • Confuse the work done by the spring (which is negative when stretched/compressed) with the work done by an external pulling force (which is positive).
  • Use kinematic equations for constant acceleration; spring force is a variable force.

DO:

  • Remember that at maximum compression/extension ($\pm x_m$), kinetic energy is zero.
  • Set Potential Energy ($V=0$) at the equilibrium position consistently to simplify calculations.

7. MNEMONICS & MEMORY TRICKS

REMEMBER:
* "Springs push back": The negative sign in $F_s = -kx$ reminds you that restoring force is always opposite to the stretch/compression.
* "Zero at the Center, Max at the Edges": Potential energy is zero at equilibrium and maximum at extreme ends.
* "To find max speed, trade all $V$ for $K$ at the equilibrium line" ($v_m$ happens when all stored potential energy converts to kinetic energy).

8. HIGH-YIELD NEET POINTS

NEET LOVES TO ASK:

  • The relationship between kinetic and potential energy during oscillation: they continuously convert into each other, but $E = K + V$ remains constant.
  • Identifying that the spring force is a conservative variable force, meaning work done depends only on endpoints.
  • Where maximum speed occurs: Speed and kinetic energy are maximum strictly at the equilibrium position ($x=0$).
  • Identifying the dimensional formula for the spring constant $k$, which is $[MT^{-2}]$.

9. ONE MCQ — SELF-TEST

Question: A block oscillates on a spring between $x = -x_m$ and $x = +x_m$. Where is its speed maximum?
(a) At $x = +x_m$
(b) At $x = -x_m$
(c) At $x = 0$
(d) Same at all positions
Answer: (c) At $x = 0$. At maximum compression/extension ($\pm x_m$), $V(x)$ is maximum and kinetic energy is zero. At $x = 0$, $V(x)$ is zero, so Kinetic Energy (and speed) must be maximum.

10. NCERT PAGE REFERENCES

Revise from:
- Page 80: The potential energy of a spring & Hooke's Law
- Page 81: Conservation of mechanical energy in a spring system & parabolic plots
- Page 87: Dimensions and Units summary table for Work, Energy, and Power

11. CTA

Detailed explanation → 4-Work, Energy and Power Guide
Quick test → [Revision Test: 4-Work, Energy and Power]