Work by Non-Constant Forces and Spring Potential Energy

Generalizing Work for Non-Constant Forces

• Conceptual Overview of Work:

  • Work describes a process resulting from the total effect of a force over time or space, rather than an instantaneous state.
  • In simple scenarios with a constant force, work is calculated as:     $W = \text{force} \times \text{distance}$
  • Specifically, this involves the component of the force acting in the direction of motion.

• Approximation of Gravity as a Constant Force:

  • Technically, gravity is non-constant; it weakens as an object moves farther from the Earth's center.
  • On a human scale, gravity is treated as a constant force ($F = m \times g$) because the change in position is negligible compared to the size of the Earth.
  • The gravitational potential energy ($U_g$) on this scale is defined as:     $U_g = m \times g \times h$
  • Here, $g$ is approximately $9.8\,m/s^2$.

• Limitations of the Constant Force Formula:

  • If a force ($F$) changes value from one second to the next (being non-constant in time) or from one meter to the next (being non-constant in space), a single value cannot be plugged into the equation $W = F \times d$.
  • An example of a spatially non-constant force is compressing a spring: the further the spring is moved, the more it pushes back.

Graphical Interpretation of Work

• Work as Area Under a Curve:

  • If you plot Force ($F_x$) vs. Position ($x$), the relationship between work and area becomes clear.
  • For Constant Force: The plot is a horizontal line. Moving an object from position $0$ to distance $d$ creates a rectangle on the graph. The height of the rectangle is the force ($f_x$) and the width is the distance ($d$). Since the area of a rectangle is $\text{height} \times \text{width}$, the area on the graph ($f_x \times d$) directly equals the calculated work.
  • For Non-Constant Force: The "Area Under the Curve" is still the correct physical interpretation for the work done. No matter how the force fluctuates, finding the total area of the region bounded by the x-axis, the y-axis, the curve itself, and a vertical reference line at the final position gives the total work done.

• Units of Graphical Area:

  • This is not a physical area (measured in square meters or square inches).
  • The units of this "area" are $\text{newtons} \times \text{meters}$ (Joules).

• Calculus Connection:

  • The most precise way to find the area under a curve is through an integral. An integral effectively converts graphical interpretations into manageable equations.
  • In algebra-based courses without calculus, we rely on the graphical interpretation or simple geometric shapes (like triangles) to find work for non-constant forces.

Numerical and Brute Force Methods for Calculating Work

• Summation of Small Work Bit:

  • Since work is just energy transfer, and energy is additive, the total work done is the sum of small bits of work done over tiny intervals of distance:     $W_{\text{total}} = W_1 + W_2 + W_3 + \dots$
  • Graphically, this involves breaking the large, irregular area under a curve into many thin vertical strips.

• The Rectangular Approximation:

  • If a curve is smooth (meaning it lacks random spikes), zooming in sufficiently makes any small segment appear as a straight, tilted line.
  • By drawing a rectangle that passes through the middle of that tilted line, the area above the line and the empty space below it effectively cancel out.
  • The area of one small section ($W_1$) is approximated as:     $W_1 = F_x(x_1) \times \Delta x_1$

• Applications in Research and Computing:

  • Computers cannot perform abstract calculus on arbitrary shapes; they use brute force to calculate work by breaking a curve into thousands of tiny boxes (infinitesimals) and summing their areas.
  • This method is used in high-tech research, such as studying quantum well lasers, where code is written to calculate energy transfer from one electron to the next.

Properties of Springs and Elasticity

• Broad Applications of Spring Concepts:

  • While often illustrated with metal coils, spring physics applies to anything with an elastic nature: rubber bands, bungee cords, trampolines, and even chemical bonds.
  • Chemical Bonds: Mathematically, the bonds between atoms act like springs with their own bouncing nature and spring constants.
  • Quantum Mechanics: Electrons trapped in quantum wells transition between states and oscillate as if they are at the end of a spring.

• The Equilibrium Point:

  • Definition: The location where the spring is fully relaxed and motionless. It is designated as the origin ($x = 0$).
  • Setting the origin at the relaxed length simplifies the resulting physics equations significantly.
  • At the equilibrium point, the net force on the object is zero ($\sum F = 0$), following Newton's First Law.

• Displacement from Equilibrium:

  • The variable ($x$) refers to the position vector of the object relative to the origin ($x = 0$).
  • It is formally called displacement from equilibrium because it measures how far and in what direction the object has moved from its relaxed state.
  • Both stretching (expanding) and compressing (shortening) a spring store potential energy. When released, this potential energy converts into kinetic energy.

Hooke's Law and the Spring Constant

• Hooke's Law Equation:$F_s = -k \times x$

  • $F_s$ (Spring Force): Measured in newtons ($N$). It is an empirical rule of thumb rather than a fundamental law like gravity; it remains accurate only if the spring is not stretched or compressed to an extreme.
  • $k$ (Spring Constant): Represents the stiffness of the spring.
    • It is always a positive value.
    • Units: $\text{newtons per meter}$ ($N/m$) or $\text{millinewtons per millimeter}$ ($mN/mm$).
    • Example: A constant of $10,000\,N/m$ sounds high, but as a ratio, it equals $10\,N/mm$ (about $2\,\text{pounds}$ of force to move it $1\,mm$).
  • The Linear Relationship: The spring force is linear. If the displacement matches twice the original distance, the force pushes back twice as hard.

• Directionality and Magnitude:

  • Direction: The negative sign in Hooke's Law signifies that the force is a restoring force, always pointing in the opposite direction of the displacement. If you pull it right ($+x$), the spring pulls left ($-F$).
  • Magnitude only: For simple energy calculations, we often ignore the sign and direction, using the magnitude:     $F_s = k \times x$

Derivation of Spring Potential Energy

• Work-Energy Connection:

  • To find the expression for potential energy, we analyze the work required to compress a spring from equilibrium to a distance ($x$).
  • We assume the object starts and ends motionless to ensure all transferred energy goes into potential storage ($U_s$).

• Graphical Calculation:

  • When plotting $F_s = k \times x$, the result is a straight line starting at the origin ($0,0$) with a slope equal to $k$
  • The area under this line is the shape of a triangle.
  • Area of a triangle: $\text{Area} = \frac{1}{2} \text{base} \times \text{height}$
  • Base: The displacement distance ($x$).
  • Height: The value of the force at that distance ($k \times x$).

• Final Expression for Spring Potential Energy:$U_s = \frac{1}{2} k x^2$

  • Note: In problems involving both gravity and springs, it is helpful to distinguish them as $U_g$ (gravitational) and $U_s$ (spring).

Questions & Discussion

• Student Question: So you're saying that $x = 0$, the equilibrium point, is when the spring is fully relaxed? In the picture, it looks coiled right now.
• Professor Response: Yes, that is correct. In the diagram shown, the spring is depicted as compressed. The position designated as equilibrium refers specifically to the location of the center of the block when the spring is in its fully relaxed state. This serves as the reference point for all displacement measurements.

Sample Problem: Spring-Loaded Block on a Ramp

• Scenario: A block of mass ($m$) is compressed against a spring with constant ($k$). It is released (initial velocity is zero), shoots across a frictionless surface, and goes up a ramp of height ($h$). We want the block to reach the top of the ramp with a specific final speed ($V_f$).

• Given Data:

  • Spring Constant ($k$): $10,000\,N/m$
  • Mass ($m$): $10\,kg$
  • Gravity ($g$): $9.8\,m/s^2$
  • Ramp Height ($h$): $6\,m$
  • Final Velocity ($V_f$): $5\,m/s$

• Step 1: Energy Conservation Equation:$E_i = E_f$$K_i + U_{si} + U_{gi} = K_f + U_{sf} + U_{gf}$

• Step 2: Identify Zero Values:

  • Initial Kinetic Energy ($K_i$) is $0$ (starts motionless).
  • Initial Gravitational Potential ($U_{gi}$) is $0$ (starts at ground level).
  • Final Spring Potential ($U_{sf}$) is $0$ (spring is relaxed after release).

• Step 3: Substitute Expressions:$\frac{1}{2} k x_i^2 = \frac{1}{2} m V_f^2 + m g h$

• Step 4: Solve for Compression Distance ($x_i$):

  • Multiply by $2$ to clear fractions: $k x_i^2 = m V_f^2 + 2 m g h$
  • Rearrange: $x_i = \sqrt{\frac{m V_f^2 + 2 m g h}{k}}$
  • Plug in values:     $x_i = \sqrt{\frac{(10 \times 5^2) + (2 \times 10 \times 9.8 \times 6)}{10,000}}$$x_i = \sqrt{\frac{250 + 1176}{10,000}}$$x_i = \sqrt{\frac{1426}{10,000}}$$x_i \approx 0.38\,m$

• Result: You must compress the spring approximately $38\,cm$ to achieve the desired speed at the top of the ramp.