Maths › Further Mechanics 1 › Work, energy and power
Work, energy and power
Energy methods restate Newton's laws in terms of work rather than force. The work done by the forces acting equals the change in kinetic energy, and where no resistance acts the total mechanical energy stays the same.
Builds on Forces and Newton's laws and Friction and inclined planes.
IN THIS TOPIC
- Calculate work done by a force at an angle, and by gravity and friction.
- Apply the work-energy principle to motion on a slope.
- Use conservation of mechanical energy where no resistance acts.
- Use P = Fv to link engine power, driving force and acceleration.
COMMON MISCONCEPTION
The work done by a force of 20 N moving an object 5 m is always 100 J.
What counts as work
Work is the component of force along the displacement, times the displacement, so it is Fd cos θ with θ the angle between them. A force at right angles to the motion does no work at all, which is the reason the normal reaction never appears in an energy equation. Fd on its own, with the angle forgotten, would also be wrong for a force partly opposing the motion, where the work is negative.
The booklet has no energy formulae at all, so both of these have to be memorised. Both are measured in joules. Heights for gravitational potential energy are measured from whatever level you choose, so choose one and use it in every total on the page.
Energy accounting
The work-energy principle says the total work done by all the forces equals the change in kinetic energy. On a rough slope that reads as energy released by gravity, minus energy lost to friction, equals kinetic energy gained. Written that way it settles problems that would otherwise need several equations of motion.
When no resistance acts, nothing is lost and the sum of kinetic and gravitational potential energy stays the same throughout. That is conservation of mechanical energy, and it is worth saying explicitly that the surface is smooth before you use it.
WORKED EXAMPLE
Down a rough slope
A particle of mass 5 kg slides 4 m from rest down a slope at 30° to the horizontal, with coefficient of friction 0.2. Find its speed at the bottom, taking g = 9.8 m/s².
Work by gravity = 5(9.8)(4)sin30° = 98 J.
Normal reaction = 5(9.8)cos30° = 42.4 N, so friction = 8.49 N and the work against it is 8.49 × 4 = 33.9 J.
Kinetic energy gained = 98 − 33.95 = 64.05 J, so ½(5)v² = 64.05 and v = 5.06 m/s.
Power
Power is the rate of doing work, measured in watts. For a vehicle moving at speed v with driving force F:
It is not printed anywhere in the booklet. Learn it. At a fixed power the driving force falls as the vehicle speeds up, so the acceleration tails off. Maximum speed arrives when the driving force has dropped to equal the resistance. Setting P/v equal to the resistance solves for it in one line.
GUIDED PRACTICE
A car accelerating
A car of mass 1200 kg works at 20 kW. At the moment its speed is 15 m/s the resistance is 500 N. Find the acceleration, and the maximum speed if the resistance stays at 500 N.
Show the working
Driving force = 20000/15 = 1333 N.
Resultant = 1333 − 500 = 833 N, so a = 833/1200 = 0.694 m/s².
At maximum speed the acceleration is zero, so the driving force equals 500 N: 20000/v = 500 gives v = 40 m/s.
In practice resistance grows with speed, so the real maximum would be lower.
ASSESSMENT FOCUS
- Include cos θ in the work done whenever the force is not along the motion.
- Keep the normal reaction out of energy equations. It never does work.
- Set out the energy equation as a sentence of terms before substituting numbers.
- For power questions, decide whether the vehicle is accelerating or at maximum speed before writing anything.
- Watch the units. Power in kilowatts must become watts before it meets a force in newtons.
CHECK YOURSELF
A force of 30 N acts at 60° to the direction of motion while an object moves 8 m. Find the work done.
Show a hint
Only the component along the motion counts.
Show the answer
Work = 30 × 8 × cos60° = 30 × 8 × 0.5 = 120 J, half what the force would do if it acted along the motion.
Work is Fd cos θ, and the work-energy principle says the total work done by all forces equals the change in kinetic energy.
With no resistance, kinetic plus gravitational potential energy stays constant.
Power is P = Fv, so the driving force falls as speed rises, and maximum speed is where it has dropped to equal the resistance.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the work, energy and power questions page.
CHECK YOUR PROGRESS
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- Calculate work done by a force at an angle, and by gravity and friction.
- Apply the work-energy principle to motion on a slope.
- Use conservation of mechanical energy where no resistance acts.
- Use P = Fv to link engine power, driving force and acceleration.
Open the full revision checklist to see every objective in the course in one place.