Work done by a variable force Knowledge-Based Study

Study notes

1. Work Done by a Non-Constant Force

   When a force acting on an object remains constant in both magnitude and direction along a straight displacement, the work done is simply given by:

   $W = F \Delta x \cos\theta$

   However, in most real-world scenarios, forces vary as a function of position. Common examples include the restoring force of an ideal spring, gravitational forces over astronomical distances, or aerodynamic drag forces that change with speed and position.

   Because the value of the force continuously changes, we cannot multiply a single force value by the total displacement. Instead, we divide the total path into infinitesimal displacements $dx$. Over each minute displacement, the force $F_x(x)$ can be considered approximately constant, yielding an infinitesimal amount of work:

   $dW = F_x(x)\,dx$

   Area Under the Force-versus-Displacement Curve

   A fundamental method for evaluating the work done by a variable force is using a force-versus-position graph ($F_x$ plotted against $x$).

   If we approximate the total displacement from $x_i$ to $x_f$ as a sequence of small rectangular strips of width $\Delta x_k$ and height $F_{x,k}$, the total work is approximated by the sum of their areas:

   $$W \approx \sum_{k} F_{x,k} \,\Delta x_k$$

   In the limit as the width of the intervals approaches zero ($\Delta x \to 0$), this sum becomes exact and equals the area bounded by the force curve and the position axis:

   • Positive Work: Regions where $F_x > 0$ for positive displacement $\Delta x > 0$ lie above the position axis and represent positive work.
   • Negative Work: Regions where $F_x < 0$ lie below the position axis and represent negative work done on the object.
   • Net Work: The net work equals the algebraic sum of the areas (areas above the axis minus areas below the axis).

   Analytical Integration and Spring Forces

   Mathematically, the summation of infinitesimal work elements leads to the definite integral:

   $$W = \int_{x_i}^{x_f} F_x(x)\,dx$$

   A classic application is an ideal spring adhering to Hooke's Law, where the restoring force exerted by the spring is proportional to its displacement from equilibrium:

   $$F_s(x) = -kx$$

   Here, $k$ is the spring constant and $x$ is the displacement from the unstretched position ($x = 0$).

   Work Done by the Spring

   When the spring moves from an initial position $x_i$ to a final position $x_f$, the work done by the spring force is:

   $$W_s = \int_{x_i}^{x_f} (-kx)\,dx = -\left[\frac{1}{2}kx^2\right]_{x_i}^{x_f} = \frac{1}{2}kx_i^2 - \frac{1}{2}kx_f^2$$

   Work Done by an Applied External Force

   To stretch or compress the spring quasi-statically (without accelerating it), an external agent must apply an opposing force $F_{\text{ext}} = +kx$. The work done by this external force is:

   $$W_{\text{ext}} = \int_{x_i}^{x_f} kx\,dx = \frac{1}{2}kx_f^2 - \frac{1}{2}kx_i^2$$

   Connecting Variable Work to Kinetic Energy

   The Work-Energy Theorem states that the net work done by all forces acting on a particle equals the change in its kinetic energy:

   $$W_{\text{net}} = \Delta K = K_f - K_i$$

   This principle holds true for variable forces as well. Using Newton's second law $F_{\text{net}} = m\,a = m\frac{dv}{dt}$ and applying the chain rule $\frac{dv}{dt} = \frac{dv}{dx}\frac{dx}{dt} = v\frac{dv}{dx}$:

   $$W_{\text{net}} = \int_{x_i}^{x_f} F_{\text{net}}\,dx = \int_{x_i}^{x_f} m v \frac{dv}{dx}\,dx = \int_{v_i}^{v_f} m v\,dv$$

   Evaluating this integral yields:

   $$W_{\text{net}} = \left[ \frac{1}{2}mv^2 \right]_{v_i}^{v_f} = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$$

   Summary of Key Takeaways:

   • Work done by a position-dependent force is evaluated via $W = \int_{x_i}^{x_f} F_x(x)\,dx$.
   • Graphically, work corresponds to the net signed area between the $F_x(x)$ curve and the displacement axis.
   • Regardless of whether forces are constant or variable, the net work always equals the change in kinetic energy of the object.