7.1 Work

Work and energy are fundamental concepts in physics, shaping our understanding of how objects interact and move. These principles explain everything from pushing a box across a room to the complex motions of planets in space.

Work involves force and displacement, while energy represents the capacity to do work. Together, they form a powerful framework for analyzing physical systems, from simple machines to complex natural phenomena.

Work and Energy

Work by constant forces

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• Work is the product of force and displacement in the direction of the force calculated using $W = F \cdot d \cdot \cos\theta$ where $W$ is work (J), $F$ is force (N), $d$ is displacement (m), and $\theta$ is the angle between force and displacement vectors
• When force is constant and in the same direction as displacement, work simplifies to $W = F \cdot d$ (pushing a box across a floor)
• Work is a scalar quantity measured in joules (J) which is equivalent to applying 1 N of force over a distance of 1 m
• Positive work occurs when force and displacement are in the same direction ($0^\circ \leq \theta < 90^\circ$) such as pushing a cart forward
• Negative work occurs when force and displacement are in opposite directions ($90^\circ < \theta \leq 180^\circ$) like pulling a wagon uphill
• No work is done when force is perpendicular to displacement ($\theta = 90^\circ$) as seen when carrying a heavy object horizontally at a constant speed
• Work is a form of energy transfer, measured in the same units as energy (joules)

Work from variable forces

• For variable forces, work is calculated by integrating force over displacement using $W = \int_{x_1}^{x_2} F(x) \, dx$ where $F(x)$ is force as a function of position $x$
• Hooke's law describes the force exerted by a spring as [F = -kx](https://www.fiveableKeyTerm:F_=_-kx) where $k$ is the spring constant (N/m) and $x$ is displacement from equilibrium (m)
• Work done by a spring is $W = \frac{1}{2}kx^2$ where $x$ is the total displacement from equilibrium which equals the area under the force-displacement curve for a spring
• The work done by a spring is independent of the path taken and only depends on initial and final positions (compressing a spring 0.1 m requires the same work whether done quickly or slowly)
• The work done on a spring changes its potential energy

Work and force-displacement curves

• Work done by a force can be visualized as the area under the force-displacement curve (plotting force on the y-axis and displacement on the x-axis)
• For constant forces, the area is a rectangle with width $d$ and height $F$ so $W = F \cdot d$ is equivalent to the area of this rectangle
• For variable forces, the area under the curve can be calculated using integration with $W = \int_{x_1}^{x_2} F(x) \, dx$ representing the area under the force-displacement curve between $x_1$ and $x_2$
• The sign of work (positive or negative) depends on the direction of force relative to displacement
  • If force is in the same direction as displacement, the area under the curve is positive (pushing a lawnmower forward)
  • If force is opposite to displacement, the area under the curve is negative (pulling back on a bowstring)

Energy and Power

• Work is related to changes in kinetic energy through the work-energy theorem
• Power is the rate at which work is done or energy is transferred, measured in watts (W)
• The principle of conservation of energy states that the total energy of an isolated system remains constant
• Efficiency is the ratio of useful work output to total energy input, often expressed as a percentage