Topic 3: Work, energy, and power

Energy is the currency of physics: a system that does work pays for it by changing one of its energy stores. In this unit you learn to track that energy as it moves between motion, height, and stretched springs, and to use conservation as a shortcut that sidesteps the messy details of forces and acceleration. You also learn when energy quietly leaks away as heat, and how fast work gets done.

Work done by a constant force

Work is the transfer of energy that happens when a force acts on an object as the object moves. For a constant force, the work done equals the force component along the direction of motion multiplied by the displacement: W = F*d*cos(theta), where theta is the angle between the force vector and the displacement vector. The unit of work is the joule (J), equal to one newton-meter (N*m). The cos(theta) factor matters: only the part of the force lined up with the motion does work. A force pushing exactly along the motion (theta = 0) does the most work, W = F*d. A force perpendicular to the motion (theta = 90 degrees) does zero work, because cos(90) = 0; this is why the normal force on a sliding box and the tension on a ball swung in a horizontal circle do no work. When the force opposes the motion (theta = 180 degrees), cos(180) = -1 and the work is negative, meaning energy is being removed from the object.

Reading work off a force-distance graph

When a force is not constant, the formula W = F*d no longer applies directly, but a graph of force versus position still tells the whole story. The work done equals the area between the force curve and the horizontal axis. For a constant force this area is simply a rectangle, recovering W = F*d. For a force that grows linearly with position, such as a stretched spring, the area is a triangle. Areas above the axis count as positive work and areas below count as negative work. Treating work as an area lets you handle any varying force without calculus, which is exactly the level expected on the AP Physics 1 exam.

The work-energy theorem

The work-energy theorem links force and motion through energy: the net work done on an object equals its change in kinetic energy, W_net = (delta)KE. Net work means the sum of the work done by every force acting on the object, equivalently the work done by the net force. If the net work is positive, the object speeds up; if negative, it slows down; if zero, its speed is unchanged. This theorem is powerful because it connects what forces do over a distance to the resulting change in speed, without ever needing to find the acceleration or the time. Whenever a problem gives you forces and distances and asks about a final speed, the work-energy theorem is usually the fastest route.

Kinetic energy

Kinetic energy is the energy an object has because of its motion. It is given by KE = (1/2)*m*v^2, where m is the mass and v is the speed, and it is measured in joules. Two features of this expression are worth memorizing. First, kinetic energy depends on the square of the speed, so doubling the speed quadruples the kinetic energy; this is why stopping distances grow so dramatically with speed. Second, kinetic energy is a scalar and is always positive or zero, regardless of the direction of travel. Because it depends only on speed, a moving object has the same kinetic energy whether it travels north or south at a given pace.

Gravitational potential energy

Gravitational potential energy is energy stored in the configuration of an object and the Earth due to height, and near the surface of the Earth it is PE_grav = m*g*h. Here m is the mass, g is the gravitational field strength (about 9.8 N/kg), and h is the height above a reference level you are free to choose. Only changes in gravitational potential energy are physically meaningful, so you can set the zero of height wherever it is most convenient, often the lowest point in the problem. Lifting an object raises this store; letting it fall lowers it, converting potential energy into kinetic energy along the way.

Elastic (spring) potential energy

When a spring is stretched or compressed it stores elastic potential energy, given by PE_spring = (1/2)*k*x^2, where k is the spring constant in N/m and x is the displacement from the spring's natural, unstretched length. Like kinetic energy, this expression depends on the square of x, so the energy stored climbs steeply as the spring is deformed further. The spring constant k measures stiffness: a larger k means a stiffer spring that stores more energy for the same stretch. This connects to Hooke's law, F = k*x, which gives the restoring force the spring exerts; the energy formula is exactly the triangular area under the Hooke's law force-distance graph.

Conservation of mechanical energy

Mechanical energy is the sum of kinetic and potential energy. When only conservative forces such as gravity and ideal spring forces act, mechanical energy is conserved: KE_i + PE_i = KE_f + PE_f. Energy is not created or destroyed, only converted between stores. Worked example: a 2.0 kg ball is released from rest at a height of 5.0 m. How fast is it moving just before it hits the ground? Take the ground as the zero of height. Initially the ball has only potential energy, PE_i = m*g*h = 2.0*9.8*5.0 = 98 J, and KE_i = 0. At the bottom all of that has become kinetic energy, so KE_f = 98 J. Solving (1/2)*m*v^2 = 98 J gives v^2 = 2*98/2.0 = 98, so v = 9.9 m/s. Notice the mass canceled in spirit; for free fall the final speed does not actually depend on mass.

Conservative versus non-conservative forces and friction

A conservative force, such as gravity or an ideal spring force, does work that depends only on the start and end points, not on the path taken, and any energy it stores can be fully recovered. A non-conservative force, most importantly friction and air resistance, does work that depends on the path and removes mechanical energy from the system, usually converting it into thermal energy (heat). When friction is present, mechanical energy is no longer conserved, but total energy still is. The accounting becomes KE_i + PE_i = KE_f + PE_f + (energy lost to friction), where the energy lost equals the friction force times the distance over which it acts. Always ask whether friction is present before assuming mechanical energy is conserved.

Power

Power is the rate at which work is done or energy is transferred: P = W/t, where W is the work and t is the time. The unit of power is the watt (W), equal to one joule per second (J/s). Two engines can do the same total work, but the more powerful one does it in less time. Power can also be written in terms of force and velocity for an object moving at constant speed: P = F*v, the product of the applied force and the speed. This form explains why a car needs far more engine power to maintain high speed against air resistance: the resisting force grows with speed and the speed itself is larger, so their product rises sharply.