DPST1021 Chap.5 Work, Energy and Power
Work, Energy and Power
Lectures 8 to 10 shift the mechanics block from forces to energy. Instead of tracking acceleration moment by moment, you track how much energy a force adds to or removes from a body, and many problems that would need calculus with Newton's laws reduce to a single line of algebra.
The course syllabus lists mechanical work, the vector dot product, variable forces including Hooke's law, kinetic energy, the work-energy theorem, potential and internal energy, and power.
The chapter begins with kinetic energy and with work defined as the dot product of force and displacement, so only the part of a force along the motion counts.
The work-energy theorem then says that the net work on a body equals its change in kinetic energy.
Lecture 9 extends work to forces that vary: a spring obeying Hooke's law, gravity near the Earth's surface, and any force given as a function of position, where work becomes an integral or an area under a graph.
Lecture 10 introduces potential energy for conservative forces, the distinction from non-conservative forces such as friction, when the total of kinetic and potential energy stays fixed, and power as the rate of doing work.
Key relations come from the formula sheet, including the link between force and the slope of a potential energy curve. These lectures sit inside the Test 1 range, and energy methods return in later chapters on collisions, rotation and oscillations.
What this chapter covers
- 01
Energy of motion: kinetic energy, one half m v squared, a scalar in joules that ignores direction.
- 02
Work by a constant force: the dot product of force and displacement, positive when the force helps the motion and zero when it acts at right angles.
- 03
Work-energy theorem: the statement that the net work done by all forces on a body equals the change in its kinetic energy over the same interval.
- 04
Variable forces: work found by integrating force over position or measuring the area under a force-position graph, with the Hooke's-law spring as the standard case.
- 05
Potential energy: stored energy defined for conservative forces such as gravity and springs, related to force by F equals minus dU/dx.
- 06
Conservation and power: mechanical energy stays constant when only conservative forces do work, while power measures the rate at which work is done, in watts.
Worked example · free
A pendulum released from the side
- 1The ball drops a height h = L(1 − cos 60°) = 1.2 × 0.50 = 0.60 m between release and the lowest point.
- 1Gravity does W = mgh = 0.40 × 9.80 × 0.60 = 2.35 J, positive because the ball moves downward.
- 1The tension always acts along the string, perpendicular to the circular path, so it does zero work.
- 1By the work-energy theorem, ½mv² = 2.35 J, so v = √(2 × 9.80 × 0.60) = 3.4 m/s; the mass cancels.
Key terms
- Work
- The energy transferred by a force acting through a displacement, equal to the dot product of force and displacement and measured in joules.
- Kinetic energy
- The energy a body has because of its motion, equal to one half of its mass times the square of its speed.
- Conservative force
- A force whose work between two points does not depend on the path taken, so a potential energy can be defined for it.
- Potential energy
- The energy stored in a system because of its configuration, such as the height of a mass or the stretch of a spring.
- Mechanical energy
- The sum of kinetic and potential energy, constant whenever only conservative forces do work on the system.
- Power
- The rate at which work is done or energy is transferred, equal to force times velocity along the force.
Work, Energy and Power FAQ
Can work be negative?
Yes. Work is negative when the force has a component opposite to the displacement, so it takes kinetic energy away. Friction on a sliding block, gravity on a rising ball and a brake on a moving car all do negative work. A negative value is a direction of energy flow, not an error.
Does carrying a heavy bag across a level room do any work on it?
In the physics sense, almost none. Your upward supporting force is perpendicular to the horizontal displacement, so its dot product is zero once the bag moves at steady speed. Your muscles still use chemical energy to stay tense, but that energy does not end up as kinetic or potential energy of the bag.
Why does the choice of zero for gravitational potential energy not matter?
Only changes in potential energy enter the energy equation, and shifting the zero adds the same constant to the starting and finishing values. The difference is unchanged. Picking the lowest point of the motion as zero simply keeps the numbers positive and reduces arithmetic slips.
Is power the same as energy?
No. Energy is an amount, measured in joules; power is how quickly that amount is transferred, measured in joules per second, or watts. Two lifts raising the same load to the same floor do equal work, but the one that arrives in half the time delivers twice the power.
Exam move
Start each problem by deciding whether an energy method is the quicker route; it usually is when the question asks for a speed or a height rather than a time. Practise dot products in component form until finding work from two vectors takes seconds, and practise the angle form for slopes and ropes. For springs, sketch the force-extension line and read work as an area, then confirm with the half k x squared formula.
Write every energy problem as initial kinetic plus potential, plus non-conservative work, equals final kinetic plus potential, and state your zero of height first. Mix in power questions that ask for force times velocity at steady speed. Finally, read potential energy graphs: locate equilibria, mark turning points for a given total energy, and estimate the force from the slope.
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