Unit notes

Work transfers energy through a force acting over displacement. The system boundary determines whether an interaction is treated as external work or as a change in internal potential energy. Mechanical energy includes kinetic and relevant potential energies.

Near Earth, gravitational potential energy change can be written mgΔy for an object–Earth system with approximately constant g. A spring stores elastic potential energy ½kx² relative to its unstretched state in the ideal Hooke’s-law model. A potential-energy zero can be chosen conveniently; differences determine physical predictions.

If Earth is outside the chosen system, gravity can be treated as an external force doing work. If Earth is inside, use gravitational potential energy instead. Do not count both gravitational work and the same gravitational potential-energy change as separate inputs in one account.

Total energy is conserved, but mechanical energy K + U need not be. Friction can convert mechanical energy into thermal energy of contacting bodies. A motor or person can transfer energy across the system boundary. Identify these transfers before setting initial mechanical energy equal to final mechanical energy.

A broader system can include thermal energy and other stores. The phrase “energy is lost” should specify that energy leaves a chosen system or changes form; it is not destroyed. An isolated system can conserve total energy while its mechanical energy decreases.

For conservative interactions, potential-energy change depends on endpoints rather than path. On a frictionless track with the same starting and ending heights, an object’s final speed can be independent of track shape. With friction, path length and normal force can affect the transfer into thermal energy.

At a one-dimensional turning point, kinetic energy is momentarily zero, so total mechanical energy equals potential energy if no other relevant energy store is changing. The accessible region satisfies K ≥ 0. An energy diagram gives possible positions but does not alone give the time spent reaching them.

Average power is energy transferred per time. For a force acting on an object, instantaneous power is the dot product of force and velocity. Two devices can perform the same work with different power if they take different times.

Efficiency compares useful output energy or power with input. Define what counts as useful for the device. A motor delivering a fixed power may exert less forward force as speed rises because P = Fv for aligned force and velocity.

A force approach is useful for acceleration, tension, and time-dependent motion. An energy approach often relates speeds and positions without needing elapsed time or the detailed force at every point. Neither method is universally better. Choose the approach matching the requested quantity and the information available.

For a block–Earth system, gravity can be represented by changing gravitational potential energy. For the block alone, gravity does external work. Both descriptions can be correct if used consistently. Counting gravity’s work and also adding the same potential-energy decrease as a separate input double-counts the transfer. Draw the system boundary before writing the energy balance.

For a force component along the displacement, work is the signed area under the force-versus-position graph. A positive area adds kinetic energy through that force; a negative area removes it. A changing force can be handled by geometric areas when its graph has simple segments. The height of the graph is force, not work.

For an ideal spring, force magnitude grows with displacement from equilibrium and elastic energy is kx²2. The square means compression and extension by the same magnitude store the same energy in the ideal model. The spring force points toward equilibrium, so its work depends on whether the spring moves toward or away from its relaxed length.

If kinetic friction acts, mechanical energy of the chosen system may decrease while internal thermal energy increases. Total energy is still conserved when the system and transfers are accounted for. “Energy is lost” should identify the form or destination rather than imply disappearance.

Average power is energy transferred divided by elapsed time. Instantaneous power from a force is the dot product of force and velocity, or Fv when they are parallel. Two motors can do the same work at different power if they take different times. Efficiency compares useful output energy or power with input, and should use the same type of quantity in numerator and denominator.

For a small displacement, dW=F·dr. Only the force component along that displacement transfers energy as work. Sum these contributions along the actual path. In one dimension, W=∫F_x(x)dx; a changing force must be integrated. If force is given as a function of time, use dr=v dt before integrating work.

The net work changes translational kinetic energy, ΔK=W_net. Work by one particular force need not equal ΔK if other forces also do work. A constraint force can be nonzero yet do zero work when perpendicular to motion. In rolling, deciding whether static friction does work requires identifying the point of contact and reference frame.

A conservative force has F_x=−dU/dx. A steep negative potential slope means a large positive force. An equilibrium occurs at zero slope, not necessarily zero potential. A minimum is stable because small displacements generate restoring forces; a maximum is unstable. Adding a constant to U does not change the force.

For fixed mechanical energy E, motion is allowed only where E≥U. Turning points satisfy E=U, making kinetic energy zero. Speed follows √[2(E−U)m]. The same energy line can admit separate allowed regions that the particle cannot cross without extra energy. Distinguish a turning point from an equilibrium.

Instantaneous power is dW/dt=F·v. Average power is total work divided by elapsed time; it is not generally the product of average force and average speed. A motor delivering constant power exerts a larger force at smaller nonzero speed, so acceleration is not constant.

Choose a system and include internal energy when friction or deformation matters. For large changes of altitude use gravitational U=−GMm/r with zero at infinity. The approximation mgh describes a small height change near a surface. Conservation of total energy remains valid when mechanical energy decreases, provided the transferred energy is accounted for.

Kinetic energy depends on speed squared and is nonnegative. Work is a signed energy transfer through force acting over displacement. A force parallel to motion does positive work; one opposite motion does negative work; an exactly perpendicular force does no work at that instant. A large force can therefore do zero work if there is no displacement or if it is perpendicular to the motion.

For a selected particle, net work equals change in kinetic energy. For a larger system, interactions within the system can be represented by potential-energy changes. If Earth is included with the object, gravity can be treated through gravitational potential energy. If it is outside, gravitational work is an external transfer. Use one consistent accounting method rather than counting both the same work and potential change.

Mechanical energy need not remain constant when friction or other processes convert it into thermal energy. Total energy accounting can still be valid. State the system boundary and the transfer or conversion instead of saying energy “disappears.”

Area under force versus position gives work for the component along the displacement. This is different from area under force versus time, which gives impulse. For a spring, force changes with displacement, so using final force times distance overestimates the work magnitude by a factor of two in the simple linear case.

Potential-energy differences are physically significant; a zero level is chosen for convenience. Near Earth’s surface, mgΔy is useful when g is approximately constant. Elastic potential energy is ½kx² measured from the spring’s relaxed length. A negative change in potential energy can supply an increase in kinetic energy when no other transfer intervenes.

A potential-energy graph helps identify allowed positions for a given total mechanical energy: kinetic energy is E−U and cannot be negative. At a turning point, kinetic energy can be zero. The graph’s slope indicates the direction of a conservative force: motion tends toward lower potential energy when released from rest.

Power is the rate of energy transfer. A machine can do the same work in less time and have greater average power. For a force and velocity, instantaneous power depends on the component of force along velocity. Energy units are joules; power units are joules per second or watts.

Efficiency compares useful output energy or power with the corresponding input. Choose the same time interval and system boundary for numerator and denominator. Lower efficiency does not violate conservation: other outputs, often thermal, account for the difference.

Energy methods often avoid detailed time dependence, making them efficient for speed after a displacement. They generally do not by themselves give elapsed time or every individual force. Choose Newton’s laws or kinematics when those quantities are the target, and combine methods when needed.

An energy approach connects initial and final states without necessarily finding elapsed time. Define a system so gravitational or spring potential energy is appropriate, then account for energy crossing the boundary through work or other transfers. A closed mechanical-energy model is valid only when dissipative transfers are negligible or explicitly included.

For gravity near Earth, ΔU=mgΔy depends on height change, not path length. A reference height changes individual potential values but not their difference. A negative chosen potential value is allowed; it reflects a reference, not negative mass or impossible energy.

Friction can transform mechanical energy into thermal energy. If the rough surface is outside the chosen system, account for its work; if included, track internal thermal energy. These are compatible bookkeeping choices when used consistently. Do not count the same transfer twice.

Work by a constant force is FΔr cos θ. Only the component along displacement contributes. For a varying force, the area under force-versus-position gives work in one dimension. The area under force-versus-time instead gives impulse, so axis labels are essential.

Power is the rate of energy transfer. For force and velocity, instantaneous power is the dot product F·v. The same amount of work can be done with different power if the time differs. A force can act without doing work when displacement is perpendicular to it.

A spring’s energy depends on displacement from its equilibrium length, not its absolute position. The energy change uses the difference between squared displacements. Doubling extension quadruples stored energy for an ideal Hooke’s-law spring, while doubling the force only doubles the extension within the model.