Most tested P3.7

Work Energy and Power

This topic covers the fundamental principles of mechanical energy, including its forms (kinetic and potential), how it is transferred via work, and the rate of transfer (power). Mastering these concepts is crucial for solving problems involving motion, forces, and height changes in various physical systems.

Part of the ESAT Physics syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • Work done by a force represents a transfer of energy. It is calculated as the force multiplied by the distance moved in the force's direction.
  • The principle of conservation of energy states that in a closed system, energy cannot be created or destroyed, only converted from one form to another (e.g., gravitational potential energy converting to kinetic energy).
  • In real-world scenarios, energy is often 'lost' from the system to the surroundings, typically as heat due to resistive forces like friction or air resistance. This is termed 'wasted energy'.
  • Efficiency measures how effectively a system converts total input energy into useful output energy, with the remainder being wasted.
  • Power is the rate at which work is done or energy is transferred over time, measured in Watts (Joules per second).

Diagram

Energy conversion on a rampmax GPE, KE = 0GPE -> KEmax KE, GPE = 0
As the ball rolls down, gravitational potential energy is transferred to kinetic energy. Ignoring friction, KE gained = GPE lost.
› Why does this happen?

Why does lifting an object give it potential energy?

It's a direct result of doing work against a force. To lift an object, you must apply an upward force equal to its weight (Force = mass x g). When you lift it through a height (h), you are doing work against the gravitational field. The work you do is calculated as Force x distance, which becomes (m x g) x h. This energy isn't lost; you have transferred it to the object, where it is now stored because of its new position in the field. This stored energy is its gravitational potential energy (GPE). If you let go, gravity does work on the object, converting this GPE back into kinetic energy as it falls.

Where does 'wasted' energy from friction actually go?

Energy is never truly lost, it just gets transferred into a less useful form. When an object moves against friction or air resistance, it has to do work against this resistive force. This work doesn't increase the object's kinetic or potential energy. Instead, it increases the internal energy of the object and its surroundings. The surfaces in contact rub together, causing the atoms and molecules within them to vibrate more vigorously. We detect this increase in random particle vibration as a rise in temperature. So, the kinetic energy is converted into thermal energy (heat).

Why is kinetic energy proportional to velocity squared (v^2)?

This shows how much more significant an object's speed is to its energy compared to its mass. Doubling an object's mass only doubles its kinetic energy, but doubling its speed quadruples its kinetic energy. Why? Remember that work done on an object (Force x distance) gives it kinetic energy. Imagine accelerating an object with a constant force. To make it reach double the speed, you have to apply that force over four times the distance. Since four times the work has been done (same force x four times distance), the object gains four times the kinetic energy. This v-squared relationship has important real-world consequences, like braking distances. A car travelling at 60 mph needs four times the braking distance to stop compared to one at 30 mph, because it has four times the kinetic energy to get rid of.

Formulae

Work = F × d

To calculate the energy transferred by a constant force F acting over a distance d, where the distance is measured parallel to the force's direction.

GPE = m × g × h

To find the change in gravitational potential energy of a mass m when its vertical height changes by h. Assume g is 10 m/s2 unless stated otherwise.

KE = 1/2 × m × v2

To calculate the kinetic energy of a mass m moving at speed v. Remember to square the velocity.

P = E / t

To calculate power P, given the energy E transferred over a period of time t.

Efficiency (%) = (Useful Output / Total Input) × 100

To determine how effectively a system converts input energy or power into a useful form. Outputs and inputs must be in the same units (e.g., both energy or both power).

Definitions

Work Done
The energy transferred to or from an object when a force causes it to move a certain distance. It is a scalar quantity, measured in Joules (J).
Kinetic Energy (KE)
The energy an object possesses due to its motion. It depends on the object's mass and the square of its speed.
Gravitational Potential Energy (GPE)
The energy an object possesses due to its position in a gravitational field. It is calculated relative to a defined zero-height level.
Power
The rate at which energy is transferred or work is done. The standard unit is the Watt (W), where 1 W = 1 J/s.

Worked example

A 4 kg block starts from rest and slides down a rough ramp. The ramp causes the block to descend a vertical height of 5 m. At the bottom of the ramp, the block's speed is measured to be 8 m/s. Assuming g = 10 N/kg, what is the work done by friction on the block?

  1. 1

    First, identify the initial energy of the system.

    At the top, the block is at rest (KE = 0), so all its energy is gravitational potential energy (GPE).

  2. 2

    Calculate the initial GPE:

    GPE = m × g × h = 4 kg × 10 N/kg × 5 m = 200 J
  3. 3

    Next, identify the energy at the bottom.

    The height is now zero (GPE = 0), so the energy is in the form of kinetic energy (KE).

  4. 4

    Calculate the final KE:

    KE = 1/2 × m × v2 = 1/2 × 4 kg × (8 m/s)2 = 2 × 64 J = 128 J
  5. 5

    Apply the principle of conservation of energy.

    The initial energy must equal the final energy plus any energy lost (work done against friction).

  6. 6

    Set up the energy balance equation:

    Initial GPE = Final KE + Work done by friction
  7. 7

    Solve for the work done by friction:

    200 J = 128 J + Workfriction

    Therefore, Workfriction = 200 J - 128 J = 72 J.

Answer: 72 J

Common mistakes

  • ×Forgetting the '1/2' in the kinetic energy formula (KE = 1/2 × m × v2) or forgetting to square the velocity (v2). This is a very frequent source of 'off by a factor' errors.
  • ×Failing to account for all energy transformations. In problems with friction or air resistance, simply equating GPE lost to KE gained is incorrect; you must include the work done against the resistive force.
  • ×Using inconsistent units. Always convert mass to kg, distance to m, and speed to m/s before substituting into formulae. Forgetting this is a common reason for calculation errors.
  • ×Mixing up vertical height (h) for GPE and the distance travelled along a slope (d) for work done. GPE depends only on the vertical change in position.

No-calculator tips

  • ✓When calculating KE = 1/2 × m × v2, deal with the 1/2 first if possible. If mass is an even number, halve it before multiplying by v2. If v2 is even, halve that instead. This keeps numbers smaller.
  • ✓For efficiency calculations, simplify the fraction (Useful / Total) before multiplying by 100. For example, if useful work is 45 J and total input is 60 J, simplify 45/60 to 9/12, then to 3/4, which is easily identified as 75%.
  • ✓Many numbers in ESAT are chosen to be easy to work with. Before diving into complex multiplication, look for opportunities to cancel or rearrange terms. In energy conservation problems, sometimes terms like 'm' or 'g' might cancel out if they appear in every term of the equation.

Read this topic in the official UAT-UK ESAT guide →

All Physics topics