Energy and Work: Non-Constant Forces and Spring Potential Energy Studies

Generalizing Work and Non-Constant Forces

• The concept of "swerve" (work) is defined as force multiplied by distance, specifically the component of the force in the direction of motion: $W = F \times d$.

• Standard work formulas often assume a constant force, such as gravitational force on a human scale. While gravity weakens as distance from Earth increases, it is approximated as a constant force in university physics for human-scale objects.

• The constant gravitational force is expressed as $F_G = m \times g$, where $m$ is the mass and $g = 9.8\,m/s^2$. This leads to the expression for gravitational potential energy: $U_G = mg \times h$, where $h$ is height.

• A non-constant force varies with time or, more specifically for this discussion, with spatial position. This means the force has different values at different coordinates.

• Example of a non-constant force: Compressing a spring. As the spring is compressed, the force required to keep it compressed changes depending on its position in space.

• Mathematically, if force $F$ is not a single value but changes throughout a process, a simple multiplication ($F \times d$) cannot be performed because there is no single $F$ to input into the equation.

Graphical Interpretation of Work

• To solve for work done by a non-constant force, a graphical interpretation is used by plotting Force ($F_x$) vs. Position ($x$).

• For a constant force, the plot is a horizontal line. Moving an object from position $0$ to distance $d$ creates a rectangle on the graph.

• The area of this rectangle is calculated as $\text{height} \times \text{width}$, which corresponds to $F_x \times d$. Therefore, in a Force vs. Distance graph, the area under the curve is equal to the work done.

• Note on units: This is "graphical area" (Newtons $\times$ meters), not a physical area of a surface (like square inches or square meters). The value of this graphical area equals the work in Joules.

• For non-constant forces, the work done throughout a process is the total area of the region bounded by the $x$-axis, the vertical axes (start and end points), and the force function curve.

• Calculus connection: An integral is the most direct way to find the area under a curve. An integral transforms the graphical interpretation into manageable equations via shortcuts and rules not required for this specific algebra-based course.

• The "Brute Force" or Computer Method: To find the area of an arbitrary complex curve without calculus, the distance can be broken into small sections (increments).

  • Total work is the sum of small work increments: $W_{total} = w_1 + w_2 + w_3 + \dots$
  • Each small section of a smooth curve can be approximated as a rectangle if the interval (delta $x$) is small enough.
  • The formula for this sum is: $W = F_{x1} \times \Delta x_1 + F_{x2} \times \Delta x_2 + \dots$
  • As the boxes (increments) become infinitesimally small, this summation becomes a calculus integral. Computers use this method by breaking a curve into thousands of tiny boxes for high accuracy.

Springs and Elasticity

• Springs refer to any object with an elastic nature, including metal coils, rubber bands, bungee cords, and trampolines.

• Broad Scientific Applications:

  • Chemical bonds between atoms mathematically behave like springs with specific bouncing natures and spring constants.
  • Electrons in atoms or quantum wells (as seen in laser research) bounce in a manner that can be described by an effective spring constant.

• The Equilibrium Point: The position where the spring is fully relaxed and motionless ($x = 0$).

  • This is the "ground level" equivalent for spring potential energy.
  • "Equilibrium" means a balancing of forces where the net force is zero ($F_{net} = 0$).
  • According to Newton's First Law, an object at rest at the equilibrium point will stay at rest unless acted upon.

• Displacement from Equilibrium ($x$): This is the position vector pointing from the equilibrium point to the object at the end of the spring. It describes the change in position due to compression or stretching.

Hooke's Law and Spring Force

• Hooke's Law: The force exerted by a spring ($F_s$) is proportional to the displacement from equilibrium.

  • Vector Equation: $\mathbf{F}_s = -k \times \mathbf{x}$
  • Magnitude Only Equation: $F_s = k \times x$

• Components of the Law:

  • $k$ (Spring Constant): Describes the stiffness of the spring.
  • Units are Newtons per meter ($N/m$).
  • It is always a positive value.
  • Example: A spring with $k = 3\,N/m$ requires $3\,N$ of force to compress it $1\,m$. This is equivalent to $3\,mN$ per millimeter ($mm$).
  • $-$ (Negative Sign): Indicates directionality. The spring force is always in the opposite direction of the displacement. If you compress it left, it pushes right; if you stretch it right, it pulls left.

• Hooke's Law is a "rule of thumb" or an empirical law, not a fundamental universal law like gravity. It fails if the spring is stretched or compressed beyond its elastic limit.

Spring Potential Energy Derivation

• Using the Work-Energy Theorem: $W = \Delta E$.

• To isolate potential energy, we assume a scenario where an object starts and ends motionless ($K_i = 0$, $K_f = 0$) and begins at the equilibrium point ($U_i = 0$).

• Consequently, all work done to compress the spring becomes spring potential energy ($U_s$).

• Since spring force ($F_s = kx$) is non-constant, we find the work by calculating the area under the Force vs. Distance curve.

  • The plot of $F_s = kx$ is a linear line starting at the origin ($0,0$) with a slope of $k$.
  • The shape under the line is a triangle.
  • Area of a triangle: $\frac{1}{2} \times \text{base} \times \text{height}$.
  • Base = $x$ (displacement).
  • Height = $kx$ (the force at that displacement).
  • Resulting formula for Spring Potential Energy: $U_s = \frac{1}{2} k \times x^2$.

Sample Problem: Energy Conservation with Springs and Gravity

• Scenario: A spring-loaded block is compressed and released to shoot up a frictionless ramp.

• Given Data:

  • Spring Constant ($k$): $10,000\,N/m$
  • Mass of Block ($m$): $10\,kg$
  • Final Height ($h$): $6\,m$
  • Final Velocity ($v_f$): $5\,m/s$
  • Acceleration due to gravity ($g$): $9.8\,m/s^2$

• Conservation of Energy Equation:

  • $E_i = E_f$
  • $K_i + U_{si} + U_{gi} = K_f + U_{sf} + U_{gf}$

• Identification of States:

  • Initial state: Compressed spring, block at rest ($K_i = 0$), at ground level ($U_{gi} = 0$).
  • Final state: Spring is relaxed ($U_{sf} = 0$), block is moving at height $h$.
  • Equation simplifies to: $U_{si} = K_f + U_{gf}$

• Substitution of Equations:

  • $\frac{1}{2} k x_i^2 = \frac{1}{2} m v_f^2 + mgh$

• Solving for Compression Distance ($x_i$):

  • Multiply by $2$ to clear fractions: $k x_i^2 = m v_f^2 + 2mgh$
  • Isolate $x_i$: $x_i = \sqrt{\frac{m v_f^2 + 2mgh}{k}}$

• Calculation:

  • $x_i = \sqrt{\frac{10 \times (5)^2 + 2 \times 10 \times 9.8 \times 6}{10,000}}$
  • $x_i \approx 0.38\,m$
  • The spring must be compressed $38\,cm$ to achieve the desired height and speed.

Questions & Discussion

• Q: So you're saying that $x = 0$, the equilibrium point, is when the spring is fully relaxed?

• A: Yes. In the diagram shown, if the spring is coiled or stretched, it is under displacement. We mark the location of the center of the object when the spring is relaxed as our origin ($x = 0$) to simplify calculations. If we chose another point, the equations would be twice as complicated.

• Q: What if the spring is not a meter long but the constant is in Newtons per meter?

• A: The spring constant is a ratio. $10,000\,N/m$ is the same ratio as $10\,N$ per millimeter. You do not need a meter of spring to use the constant; it simply scales the force to the distance moved.