Hooke's Law and Spring Force — Video Notes

Setup and fundamental idea

• System: a mass m is connected to a wall on the left by a spring. The mass rests on a flat surface which defines the x-axis (horizontal).
• Natural length: when the spring is neither stretched nor compressed, it is at its natural length. The mass position at this state is defined as x = 0.
• Displacement and force directions (one-dimensional):
  • If the mass is displaced to the right (x > 0), the spring stretches and exerts a force to the left.
  • If the mass is displaced to the left (x < 0), the spring compresses and exerts a force to the right.
  • Thus displacements and forces point in opposite directions: a positive x gives a negative force, and a negative x gives a positive force.
• Hooke’s law (vector form): the force vector f is proportional to the displacement vector x and oppositely directed:
  $\vec{\mathbf{f}} = -k \, \vec{\mathbf{x}}$
  where the spring constant k > 0 is called Hooke’s constant.
• Magnitude relation (within elastic limit): the magnitude of the force is proportional to the magnitude of the displacement:
  $|\vec{f}| = k \; |\vec{\mathbf{x}}|$
• Sign conventions and practical note: the minus sign encodes direction, but when solving problems it is often convenient to work with magnitudes and reason about directions separately.

Historical context: Hooke and Newton

• Robert Hooke and Isaac Newton lived in roughly the same period around London, England; they were rivals with competing theories about different phenomena.
• On light: Hooke proposed the wave nature of light, Newton proposed the particle nature of light.
• Newton ultimately became more famous; Hooke less so, potentially due to competitive suppression by Newton.
• Portrait anecdote: a portrait from that era is uncertain in identity; some conjecture Newton suppressed the identity of the portrait of Hooke (information cited as something you can read on Wikipedia).

Practical usage: magnitudes vs vectors

• In many problem setups, it is convenient to use Hooke’s law for magnitudes and handle directions with logic about the setup, rather than carrying the vector minus sign throughout the calculation.

Example 1: Mass between two springs attached to opposite walls

• Setup: A mass (red ball) is connected to the left wall by spring with constant $k1$ and to the right wall by spring with constant $k2$; both springs have their natural lengths (no initial stretch).
• Geometry/notation:
  • Move the right wall to the right by distance $x$.
  • The mass moves to the right by a displacement $x_1$.
  • The left spring is stretched by $x1$; the right spring is stretched by $x2$.
  • The total wall displacement satisfies $x = x<em>1 + x</em>2$
• Force balance (equilibrium): The mass is at rest, so the leftward force from spring 1 equals the rightward force from spring 2:
  $F<em>1 = F</em>2 \quad \Rightarrow\quad k<em>1 x</em>1 = k<em>2 x</em>2$
• Solving the two equations:
  • From $k1 x1 = k2 x2$, express $x2$ in terms of $x1$: $x<em>2 = \frac{k</em>1}{k<em>2} x</em>1$
  • Substitute into $x = x1 + x2$:
    $x = x<em>1 + \frac{k</em>1}{k<em>2} x</em>1 = x<em>1 \left(1 + \frac{k</em>1}{k<em>2}\right) = x</em>1 \frac{k<em>1 + k</em>2}{k_2}$
  • Solve for $x1$: $x</em>1 = \frac{k<em>2}{k</em>1 + k_2}\, x$
  • Then $x2 = \frac{k1}{k1 + k2}\, x$
• Result: The displacement of the mass to the right is
  $x<em>1 = \frac{k</em>2}{k<em>1 + k</em>2}\; x$
  and the extension of the right spring is
  $x<em>2 = \frac{k</em>1}{k<em>1 + k</em>2}\; x$
• Notes:
  • The geometry is set up so that neither spring is initially stretched.
  • The analysis uses two equations: a geometric constraint ($x = x1 + x2$) and a force-balance constraint ($k1 x1 = k2 x2$).

Example 2: Mass hanging from a vertical spring

• Setup: A mass $m$ is suspended from the ceiling by a spring with spring constant $k$ (vertical orientation).
• Forces: Gravity downward with magnitude $mg$, spring force upward with magnitude $k x$ (where $x$ is the downward displacement from the natural length).
• Equilibrium (no acceleration): The upward spring force balances the downward weight:
  $k x = m g$
• Solve for the extension $x$:
  $x = \frac{m g}{k}$
• Notes:
  • The minus sign is not required when using magnitudes; the actual directions are downward for displacement and upward for the spring force.
  • This demonstrates that Hooke’s law can be applied in the vertical direction as well as horizontally, with force balance providing the required relation.

Key takeaways and formulas

• Hooke’s law in vector form:
  \vec{\mathbf{f}} = -k \, \vec{\mathbf{x}},\quad k > 0
• Magnitude form (often convenient in problem solving):
  $|\vec{\mathbf{f}}| = k \; |\vec{\mathbf{x}}|$
• For a mass attached to two springs in equilibrium, if the left spring has stiffness $k1$ and the right spring has stiffness $k2$ and the right wall is displaced by $x$, then the displacements satisfy:
  • $x = x<em>1 + x</em>2$
  • $k<em>1 x</em>1 = k<em>2 x</em>2$
  • Solutions: $x<em>1 = \frac{k</em>2}{k<em>1 + k</em>2} \; x, \quad x<em>2 = \frac{k</em>1}{k<em>1 + k</em>2} \; x$
• For a vertical spring with a hanging mass:
  • $k x = m g \Rightarrow x = \frac{m g}{k}$
• Physical interpretation:
  • The minus sign in the vector form encodes opposite directions of force and displacement.
  • In many problems, handling magnitudes first and then assigning directions via free-body analysis simplifies calculations.
• Real-world context notes:
  • Hooke and Newton were contemporaries with rival theories about light; Newton’s success and scientific impact outpaced Hooke, partially due to competitive suppression.
  • Historical portrait ambiguities reflect the complex history of science; see discussions in popular encyclopedias such as Wikipedia for more context.