Work & Energy

A sprinter at full speed carries something a sleeping cat does not. Physics calls it kinetic energy, and the way you give it to something is by doing work on it.

Kinetic energy#

Kinetic energy can grow, shrink or stay the same. With no force, velocity and hence K stay constant. And since K depends only on speed, a force can only change K if it has a component along the motion. A falling stone speeds up (gravity points along its velocity); a stone thrown upward slows down (gravity points against it); a ball whirled on a string keeps its speed (the tension is perpendicular to the velocity).

Make this precise. Differentiate K:

where $F_t$ is the tangential component of the resultant force and θ is the angle between the force and the velocity.

Work#

The quantity $\vec F\cdot d\vec r = F\,dr\cos\theta$ is the work done by the force during the small displacement $d\vec r$. Over a finite path, add up the pieces:

The work–energy theorem#

If $\vec F$ is the resultant force, integrating (8.2) gives

If several forces act, $\vec F = \vec F_1 + \vec F_2 + \dots$, and the total work is the sum of the separate works $W_1 + W_2 + \dots$. So the theorem says: the works done by all the forces add up to the change in kinetic energy.

Most work calculations in mechanics fall into three families: forces that never change, springs, and forces that push sideways. Learn these and the integral almost never needs doing by hand.

A constant force#

If $\vec F$ is constant in size and direction, it comes out of the integral: $W = \vec F\cdot\int d\vec r = \vec F\cdot\vec r$, where $\vec r$ is the total displacement.

Gravity near the Earth

The weight $mg$ is constant near the surface. If a particle moves from A to B, with AB making angle θ with the vertical, then $W = mg(AB)\cos\theta = mgh$, where $h$ is the height descended. Climbing a height h gives $-mgh$. Remarkably, the curve followed between A and B doesn’t matter, and a round trip gives zero.

The spring force#

A block on a smooth table is attached to a spring; x = 0 is the natural length. The spring’s force on the block is $-kx$, which changes as the block moves, so we must integrate. Over a small step dx the work is $-kx\,dx$, and

The pattern: the spring does negative work while you deform it and positive work as it relaxes. A round trip nets zero.

A force perpendicular to the velocity#

If $\vec F \perp \vec v$ at every instant, then $\vec F\cdot d\vec r = \vec F\cdot\vec v\,dt = 0$ throughout. The tension on a ball whirled in a circle, the normal force on a body sliding over a fixed surface and the magnetic force on a moving charge all do zero work.

Work as an area#

For a force along a line, $\int F\,dx$ is the area under the F–x graph, just like distance under a v–t graph.

Walking up the stairs and running up them take the same work. Running is harder because you deliver that work faster: more power.

Power#

The rate at which a force does work is its power. In a small time dt the work is $dW = \vec F\cdot d\vec r$, so

That is instantaneous power. Over an interval, average power is the total work divided by the time taken.

Kinetic energy that disappears and later comes back must have been stored somewhere. Potential energy is the name for that store, and it only exists for certain kinds of force.

Work inside a system#

Two oppositely charged beads on a smooth table attract each other. No outside force acts, yet both speed up and the system’s kinetic energy grows. How? Each bead does work on the other. The two forces are equal and opposite, but so are the displacements (each moves toward the other), so both works are positive and they add up.

Energy that comes back#

Now let the beads attract each other but start out flying apart. They slow down (K falls), stop, turn round, and come back together. When they return to their starting separation, they have exactly their original speeds again. The kinetic energy wasn’t lost, only stored, and the amount stored depended only on the configuration (the separation). We call that configuration-dependent energy potential energy.

Conservative and non-conservative forces#

Compare two round trips:

• Drag a block a distance $l$ across a rough table and back. Friction opposes you both ways: it does $-\mu mgl$ each way, $-2\mu mgl$ in total. That energy is gone for good.
• Let a block oscillate on a spring. Without friction it passes the centre at the same speed every time; with friction, a little slower each pass until it stops.

Defining potential energy#

For a conservative internal force, define the change in potential energy as the negative of the work it does:

Three points follow:

• Only changes in U are defined. You may pick any configuration as U = 0, just as you choose where to put the origin.
• No potential energy can be defined for friction: its work depends on the path, so there is no single “stored” amount.
• In a rigid body the distances between particles never change, and the internal forces do no net work (their works cancel in pairs). So U only changes when separations change.

These two lines are all you need for the next lesson.

Of all the ideas in physics, this may be the most useful: energy changes form, but the total never changes. Use it and many hard motion problems need just one line.

Conservation of mechanical energy#

Combine the last two lessons. If the internal forces are conservative and the external forces do no work, then $K_f - K_i = -(U_f - U_i)$, which rearranges to

If non-conservative forces (like friction) or external forces do work, the mechanical energy changes by exactly that work:

Gravitational potential energy#

Take “block + Earth” as the system. The Earth is so massive that it hardly moves, so we can use a frame attached to it. When the block rises by h, gravity does $-mgh$ on it, so the system’s potential energy rises by $mgh$. By custom we call this the potential energy “of the block”:

Elastic potential energy of a spring#

Stretch a massless spring slowly by x. The wall’s force on it does no work (its end doesn’t move); the block’s pull does $\int_0^x kx\,dx = \tfrac12kx^2$. The spring’s kinetic energy stays zero, so all of that work is stored as potential energy. Compressing by x stores the same amount. Choosing U = 0 at natural length:

All the forms of energy#

Mechanical energy is not the whole story. A body at rest still has internal (thermal) energy from the random motion and interaction of its molecules. Light and sound carry energy; chemical reactions release it. And Einstein showed that mass itself is a form of energy: