In the previous chapter we studied forces as the cause of changes in motion. Now we study the effects of forces over a distance - the concepts of work, energy, and power. These ideas provide an alternative and often simpler way to solve mechanics problems. Instead of analysing forces and accelerations, we can track the transfer of energy from one form to another.
Work is done when a force moves a body through a distance. Energy is the capacity of a body to do work, and it exists in many forms - kinetic, potential, heat, chemical, and others. Power measures the rate at which work is done. The most powerful idea in this chapter is the conservation of mechanical energy, which states that the total mechanical energy of a system is constant when only conservative forces act.
These concepts connect mechanics to all other branches of physics. They explain why a roller coaster speeds up as it descends, how a pendulum keeps swinging, and how collisions transfer energy between bodies. This chapter lays the groundwork for thermodynamics and rotational motion in later chapters.
In physics, work is done when a force applied on a body displaces it. Work is defined as the product of the magnitude of the displacement and the component of the force along the displacement. If a constant force F acts at an angle theta to the displacement s, the work done is:
W = F * s * cos theta
Work is a scalar quantity and its SI unit is the joule (J), where 1 J = 1 N m. If the force and displacement are in the same direction, theta = 0 and W = Fs, the maximum work. If they are perpendicular (theta = 90 degrees), the work done is zero. If the angle is between 90 and 180 degrees, the work done is negative; for example, work done against friction or by gravity when a body moves upward.
Work depends on the angle theta, not just on force and displacement. Pushing a wall without moving it does no work, because the displacement is zero. A porter carrying a load walks horizontally, and the vertical supporting force does no work because the displacement is horizontal and perpendicular to the force.
In general, the force on a body may change as the body moves. The work done by a variable force along a path is obtained by summing up the work done in infinitesimal displacements. This is expressed as an integral:
W = integral of F(s) ds
Geometrically, the work done by a variable force equals the area under the force-displacement graph between the two positions. This graphical interpretation is very useful, especially when the force is not described by a simple algebraic function, as in the stretching of a spring.
For a spring, the restoring force is F = -k x, where k is the spring constant and x is the displacement. The work done in stretching the spring from x1 to x2 is:
W = (1/2) k (x2^2 - x1^2)
This work is stored in the spring as potential energy.
The kinetic energy of a body is the energy it possesses by virtue of its motion. A body of mass m moving with speed v has kinetic energy:
K = (1/2) m v^2
Kinetic energy is a scalar and has the SI unit joule. If a body of mass m accelerates from velocity u to v under a constant force over a displacement s, then using v^2 = u^2 + 2as, the work done is:
W = F * s = m * a * s = (1/2) m v^2 - (1/2) m u^2
This is the work-energy theorem: the work done by the net force on a body equals the change in its kinetic energy.
Delta K = K_f - K_i = W
The work-energy theorem applies to variable forces as well. It is a powerful tool because it relates work (a path-dependent quantity computed from forces) directly to the change in kinetic energy, without needing to know the acceleration at every instant.
Potential energy is the energy stored in a body by virtue of its position or configuration. Gravitational potential energy of a body of mass m at a height h above a reference level is:
U = m g h
A spring stretched or compressed by x stores elastic potential energy:
U = (1/2) k x^2
Forces such as gravity and the spring force are called conservative forces because the work done by them is independent of the path and depends only on the initial and final positions. The work done by a conservative force is negative of the change in potential energy. Non-conservative forces like friction depend on the path and dissipate energy.
The principle of conservation of mechanical energy states that if only conservative forces act on a system, the total mechanical energy (kinetic plus potential) is constant:
K + U = constant
As a body falls freely, its potential energy decreases and its kinetic energy increases by the same amount, keeping the total constant.
When non-conservative forces like friction are present, the total mechanical energy is not conserved. The work done by non-conservative forces equals the change in total mechanical energy:
W_friction = Delta(K + U)
For example, a block sliding on a rough floor slows down because friction does negative work, converting kinetic energy into heat. The total energy of the universe is always conserved - energy is neither created nor destroyed, but it can be converted from one form to another. This is the law of conservation of energy, one of the most fundamental laws of physics.
Power is the rate at which work is done or energy is transferred. If work W is done in time t, the average power is:
P = W / t
The instantaneous power is:
P = dW/dt
For a body moving with velocity v under a force F, the instantaneous power is:
P = F * v
The SI unit of power is the watt (W), where 1 W = 1 J/s. A larger practical unit is the horsepower (1 hp = 746 W). In the electric power industry, energy is often measured in kilowatt-hours (kWh), where 1 kWh = 3.6 x 10^6 J. Power helps us compare how quickly different machines do the same work.
When two bodies collide, momentum is always conserved if no external force acts. Collisions are classified as elastic or inelastic. In an elastic collision, both momentum and kinetic energy are conserved. In an inelastic collision, momentum is conserved but kinetic energy is not; some kinetic energy is converted into heat, sound, or deformation energy.
For a perfectly inelastic collision, the two bodies stick together and move with a common velocity. The coefficient of restitution e is 1 for a perfectly elastic collision, 0 for a perfectly inelastic collision, and between 0 and 1 for real collisions.
In a head-on elastic collision of two equal masses, the bodies simply exchange their velocities. This result is important in billiards and particle physics.
| Quantity | Formula | SI Unit |
|---|---|---|
| Work | W = F s cos theta | J |
| Kinetic energy | K = (1/2) m v^2 | J |
| Gravitational PE | U = m g h | J |
| Elastic PE | U = (1/2) k x^2 | J |
| Power | P = W / t = F v | W |
| Impulse | J = F delta t | N s |
| Collision Type | Momentum Conserved | Kinetic Energy Conserved |
|---|---|---|
| Elastic | Yes | Yes |
| Inelastic | Yes | No |
| Perfectly inelastic | Yes | No (bodies stick) |
In this chapter we introduced the scalar concepts of work, energy, and power. Work is defined as Fscos theta, kinetic energy as (1/2)mv^2, and potential energy as mgh for gravity and (1/2)kx^2 for a spring. The work-energy theorem connects the net work done on a body to the change in its kinetic energy. For conservative systems, the total mechanical energy K + U remains constant, giving us the powerful principle of conservation of mechanical energy. Power measures the rate of doing work, P = W/t = Fv. We also distinguished elastic collisions, in which kinetic energy is conserved, from inelastic collisions, in which it is not. These ideas will be used in the study of rotational motion, where energy and angular momentum play analogous roles.