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Work and Power
The Physics Definition of Work
Work is defined as the scalar product of force and displacement, measuring the energy transferred by a force acting over a distance.
$$W = \vec{F} \cdot \vec{s} = Fs \cos \theta$$
Work is the component of force in the direction of motion multiplied by displacement.
$W$=Work done(Joules (J))
$F$=Magnitude of applied force(Newtons (N))
$s$=Magnitude of displacement(Meters (m))
$\theta$=Angle between force and displacement vectors(degrees)
$\theta = 0^\circ$
→$W = Fs$ (Maximum positive work)
$\theta = 90^\circ$
→$W = 0$ (No work done)
Force Component: Only the component $F \cos \theta$ along the displacement vector contributes to work.
Unit Definition: One Joule is the work done by a 1 N force moving an object 1 m in the force's direction.
Nature of Work: Directional Dependency
The sign of work depends on the relative orientation of force and motion, determining if energy is added to or removed from a system.
$$W = 0 \text{ if } \theta = 90^\circ$$
Positive Work: Occurs when $0^\circ \le \theta < 90^\circ$ (force aids motion).
Negative Work: Occurs when $90^\circ < \theta \le 180^\circ$ (force opposes motion, e.g., friction).
Zero Work: Occurs when force is perpendicular to displacement (e.g., Centripetal Force or normal force on a horizontal plane).
Zero Work Scenarios
Zero work occurs when there is no displacement component in the direction of force.
$$W = 0 \text{ when } \vec{F} \perp \vec{s}$$
Zero work occurs when force is perpendicular to displacement, OR when displacement is zero.
$\theta = 90^\circ$=Force perpendicular to motion(degrees)
$s = 0$=No displacement occurs(m)
Satellite in circular orbit
→Gravity does $W = 0$ (centripetal force)
Pendulum bob swings
→Tension does $W = 0$ (always perpendicular to motion)
Car on horizontal road
→Normal force does $W = 0$ (perpendicular to motion)
Pushing a wall: Force applied but $s = 0$, so $W = 0$.
Circular motion: Centripetal Force is always perpendicular to tangential velocity.
Work Done by a Variable Force
When force changes with position, total work is calculated as the integral of the force function over the displacement path.
$$W = \int_{x_1}^{x_2} F(x) dx$$
Work equals the area under the curve of a Force vs. Displacement graph.
$F(x)$=Force as a function of position(N)
$dx$=Infinitesimal displacement element(m)
$x_1, x_2$=Initial and final positions(m)
$F = kx$
→$W = \frac{1}{2}kx^2$ (Work done by a spring)
Graphical Method: Area above the x-axis represents positive work; area below represents negative work.
Summation Approach: Total work is the limit of the sum of $F_i \cos \theta_i \Delta d_i$ as $\Delta d$ approaches zero.
Power: The Rate of Energy Transfer
Power quantifies how quickly work is performed or energy is converted.
$$P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$$
Power is the time-derivative of work or the dot product of force and velocity.
$P$=Instantaneous power(Watts (W))
$dW$=Small amount of work done(J)
$dt$=Time interval(s)
$v$=Instantaneous velocity(m/s)
Constant $v$
→$P = Fv$ (Power required to overcome resistance)
Standard Units: The Watt (1 J/s). Larger units include horsepower (1 hp ≈ 746 W).
Average vs. Instantaneous: Average power uses total work/time; instantaneous power is the rate at a specific moment.
The Work-Energy Theorem
The Work-Energy Theorem connects force-based mechanics to energy-based analysis. It states that net work equals change in Kinetic Energy.
$$W_{net} = \Delta K = K_f - K_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$$
The work done by all forces (net work) equals the change in kinetic energy.
$W_{net}$=Sum of work done by all forces(J)
$\Delta K$=Change in kinetic energy(J)
$K = \frac{1}{2}mv^2$=Kinetic energy(J)
Positive net work
→$KE$ increases ($v_f > v_i$)
Negative net work
→$KE$ decreases ($v_f < v_i$)
Zero net work
→$KE$ unchanged (speed constant)
Sign Rule: If an object speeds up, $W_{net} > 0$. If it slows down, $W_{net} < 0$.
Friction Work: Always negative (reduces $KE$).
Gravity Work: Depends only on vertical height change: $W_g = mgh$, regardless of the path taken.
Dimensional Analysis
Dimensional analysis provides a way to verify equations and check answer consistency.
$$[W] = [ML^2T^{-2}], \quad [P] = [ML^2T^{-3}]$$
Work and power dimensions derived from their defining formulas.
$[W] = [F][s] = [MLT^{-2}][L]$=Work dimension from force × displacement($[ML^2T^{-2}]$)
$[P] = [W]/[t]$=Power dimension from work/time($[ML^2T^{-3}]$)
Checking formula validity
→Both sides must have same dimensions
Converting units
→1 J = $10^7$ erg (CGS to SI)
Quick Check: Work and Energy have same dimensions: $[ML^2T^{-2}]$.
Power vs. Work: Power has one extra $T^{-1}$ (time in denominator).
Proportionality Relationships
Proportionality analysis lets you determine how changes in one variable affect another without full calculations.
$$W \propto F, \quad W \propto s, \quad W \propto \cos \theta$$
From $W = Fs \cos \theta$: If $F$ doubles and $s, \theta$ constant, $W$ doubles.
From $P = W/t$: If same work in half time, $P$ doubles.
From $P = Fv$: If $v$ triples and $F$ constant, $P$ triples.