D. Energy and Society · Work, Energy and PowerConservation of Energy with Friction: Work, Heat and Thermal Energy
D2.3 D3.1 D3.4 D2.1 D3.3About 75 minutes
Learning goals- I can explain where mechanical energy goes when friction or air resistance acts.
- I can use Ek₁ + Eg₁ + W_applied = Ek₂ + Eg₂ + E_th to solve problems with friction.
- I can calculate the thermal energy produced by friction, E_th = F_f Δd.
- I can distinguish thermal energy, temperature and heat.
Words to know
thermal energy (E_th)
The total kinetic and potential energy of the particles of a substance. Friction transforms mechanical energy into thermal energy.
temperature
A measure of the average kinetic energy of the particles of a substance, in °C or K.
heat (Q)
The transfer of thermal energy from a warmer object to a cooler one because of their temperature difference.
non-conservative force
A force such as friction or air resistance whose work turns mechanical energy into other forms (mostly thermal energy).
regenerative braking
A braking system in electric and hybrid vehicles that uses the motor as a generator, turning some of the vehicle's kinetic energy into electrical energy stored in the battery.
The toboggan that came up short
Using the method of the last lesson, a toboggan starting from rest at the top of a 15 m hill would reach the bottom at √(2 × 9.8 × 15) ≈ 17 m/s. Real toboggans are much slower. Is energy conservation wrong? No. Friction with the snow and air resistance have been taking mechanical energy away the whole way down, and turning it into thermal energy: the snow, the toboggan's base and the air get a tiny bit warmer. The total energy is still conserved; we just need an extra term to track it.
Thermal energy, temperature and heat
These three words are often mixed up.
- Thermal energy is the total energy of the random motion (and positions) of all the particles in an object. A bathtub of warm water has more thermal energy than a cup of hot tea, because it has so many more particles.
- Temperature measures the average kinetic energy of the particles. The tea has a higher temperature.
- Heat is thermal energy being transferred from a hotter object to a colder one. Objects don't "contain" heat; they contain thermal energy, and heat flows between them.
When friction acts, it does negative work on the sliding object, and that energy shows up as thermal energy in both surfaces. The Thermal and Nuclear Energy unit shows how to calculate the temperature change this causes.
The full energy equation
For kinetic friction acting over a sliding distance Δd, the mechanical energy transformed into thermal energy is
E_th = F_f Δd
Add it to the conservation equation. If a person or motor also does work on the object (W_applied, for example pedalling or pulling), add that as energy coming in:
Ek₁ + Eg₁ + W_applied = Ek₂ + Eg₂ + E_th
Read it as: energy at the start + energy put in = energy at the end + energy turned into thermal energy. With no friction and no applied work, it becomes the equation from the last lesson.
Mechanical energy is not conserved when friction acts, but total energy always is.
Worked exampleExample 1: How much friction on the toboggan?
A toboggan and rider (total mass 60 kg) start from rest at the top of a hill 15 m high. The slope is 50 m long. At the bottom they are moving at 12 m/s. (a) How much mechanical energy was transformed into thermal energy? (b) What was the average friction force (including air resistance)?
- Reference level: bottom of the hill. Start: Ek₁ = 0, Eg₁ = (60)(9.8)(15) = 8820 J.
- End: Ek₂ = ½(60)(12)² = 4320 J, Eg₂ = 0. No applied work.
- (a) Ek₁ + Eg₁ = Ek₂ + Eg₂ + E_th: 0 + 8820 = 4320 + 0 + E_th, so E_th = 4500 J (4.5 × 10³ J).
- (b) E_th = F_f Δd, so F_f = 4500 J ÷ 50 m = 90 N.
4.5 × 10³ J became thermal energy, from an average friction force of 90 N.
Worked exampleExample 2: Finding a speed with friction
A 2.0 kg block starts from rest and slides down a ramp 3.0 m long that drops 1.5 m in height. The friction force along the ramp is 4.0 N. How fast is the block moving at the bottom?
- Eg₁ = (2.0)(9.8)(1.5) = 29.4 J; Ek₁ = 0.
- E_th = F_f Δd = (4.0 N)(3.0 m) = 12 J. (Use the length of the ramp, the distance the block slides.)
- Ek₂ = 29.4 − 12 = 17.4 J.
- v = √(2 × 17.4 ÷ 2.0) = √17.4 ≈ 4.2 m/s
About 4.2 m/s (without friction it would have been √(2 × 9.8 × 1.5) ≈ 5.4 m/s).
Worked exampleExample 3: Pedalling up a hill
A cyclist and bike (80 kg in total) are moving at 6.0 m/s at the bottom of a hill. The cyclist does 5000 J of work pedalling up to the top, 4.0 m higher. Friction and air resistance turn 1500 J into thermal energy. How fast is the cyclist moving at the top?
- Start: Ek₁ = ½(80)(6.0)² = 1440 J; Eg₁ = 0; W_applied = 5000 J.
- End: Eg₂ = (80)(9.8)(4.0) = 3136 J; E_th = 1500 J.
- 1440 + 0 + 5000 = Ek₂ + 3136 + 1500, so Ek₂ = 1804 J.
- v₂ = √(2 × 1804 ÷ 80) = √45.1 ≈ 6.7 m/s
About 6.7 m/s at the top.
A ball dropped from 1.00 m bounces to 0.73 m, then 0.51 m, and so on. Since Eg = mgh, each bounce height shows how much mechanical energy is left: about 70% after each bounce. The rest becomes thermal energy and sound.Braking, and getting some energy back
When a car brakes to a stop, its friction brakes turn all of its kinetic energy into thermal energy in the brake discs and pads, which then spreads into the air. A 1500 kg car stopping from 25 m/s (90 km/h) turns about ½(1500)(25)² ≈ 4.7 × 10⁵ J into thermal energy every time.
Regenerative braking in electric and hybrid vehicles uses the motor as a generator to slow the car. Part of the kinetic energy is transformed into electrical energy and stored in the battery instead of becoming thermal energy. This is one way engineers improve the efficiency of transportation, a link to the efficiency lessons coming up.
Just for funFriction gets blamed for "wasting" energy, but without it you couldn't walk, tie your laces, or keep a toque on your head. Even skates need a little sideways grip to push off the ice.
Hands-on activityWhere does the bounce go?
You need: a ball that bounces well (a tennis ball or a rubber ball), a metre stick or tape measure taped upright against a wall, and a phone that records video (slow motion helps).
- Hold the ball so its bottom is at 1.00 m and drop it (don't throw it) beside the metre stick. Film it.
- Play the video back frame by frame and read the height of the bottom of the ball at the top of each of the first three bounces.
- For each bounce, calculate the percentage of the previous height it reached. Since Eg = mgh, this is also the percentage of mechanical energy kept.
- Rub your palms together quickly for 10 s. What do you feel? Connect this to the bouncing ball.
- Write a short conclusion: what happened to the "missing" energy, and was the law of conservation of energy broken?
What to look for: tennis balls often keep somewhere around half to three-quarters of their height on each bounce on a hard floor; your numbers will vary with the ball and floor. Rubbing your hands warms them: the work you do against friction becomes thermal energy, just like the ball's lost mechanical energy. Energy was not destroyed; it became thermal energy and sound.
Your turn
1A 40 kg child slides from rest down a 3.0 m high slide and reaches the bottom at 6.0 m/s. How much thermal energy was produced?
Eg₁ = (40)(9.8)(3.0) = 1176 J. Ek₂ = ½(40)(36) = 720 J. E_th = 1176 − 720 = 456 J ≈ 4.6 × 10² J.
2A hockey puck (0.17 kg) slides across the ice at 10 m/s and stops after 40 m. Find the thermal energy produced and the friction force.
E_th = Ek₁ = ½(0.17)(100) = 8.5 J. F_f = 8.5 J ÷ 40 m ≈ 0.21 N.
3A 5.0 kg box slides 2.0 m across a floor against 15 N of friction. How much thermal energy is produced?
E_th = F_f Δd = (15)(2.0) = 30 J.
4Explain the difference between temperature and thermal energy, using a lake and a mug of hot chocolate.
The hot chocolate has the higher temperature (faster average particle motion). The lake has far more thermal energy, because it has vastly more particles, even though each one moves more slowly on average.
5A 1000 kg car moving at 20 m/s brakes to a stop using friction brakes. How much thermal energy is produced in the brakes (ignoring air resistance)?
E_th = ½(1000)(20)² = 2.0 × 10⁵ J.
6A student says, "When a ball bounces lower, the law of conservation of energy is broken." Explain why the student is wrong.
The ball's mechanical energy decreases, but that energy becomes thermal energy and sound. The total energy is conserved; only the mechanical part shrinks.
More practice: warm-up
1What form of energy does friction mostly produce?
2Write the formula for thermal energy produced by friction.
3Is mechanical energy conserved when friction acts? Is total energy?
Mechanical energy: no. Total energy: yes, always.
Heat is the transfer of thermal energy from a warmer object to a cooler one.
5A ball starts with 50 J of mechanical energy and has 35 J left after one bounce. How much became thermal energy and sound?
More practice: on your own
1A 70 kg skier starts at 2.0 m/s at the top of a 20 m high slope that is 80 m long. The average friction force is 50 N. Find her speed at the bottom.
Ek₁ = ½(70)(4) = 140 J; Eg₁ = (70)(9.8)(20) = 13 720 J; E_th = (50)(80) = 4000 J. Ek₂ = 140 + 13 720 − 4000 = 9860 J. v = √(2 × 9860 ÷ 70) = √281.7 ≈ 17 m/s.
2A 0.50 kg ball is dropped from 2.0 m and rebounds to 1.4 m. How much mechanical energy was lost in the bounce? What percentage was kept?
Lost: (0.50)(9.8)(2.0 − 1.4) = 2.9 J (2.94 J). Kept: 1.4 ÷ 2.0 = 70%.
3A 25 kg crate is pushed up a ramp. The person does 900 J of work; the crate rises 2.5 m and ends at rest, as it started. How much thermal energy was produced by friction?
900 = Eg₂ + E_th = (25)(9.8)(2.5) + E_th = 612.5 + E_th, so E_th ≈ 288 J ≈ 2.9 × 10² J.
4A sled slides from rest down a frictionless 5.0 m high hill, then across a flat snowy field with 30 N of friction. The sled and rider have mass 50 kg. How far does it slide across the field before stopping?
At the bottom: Ek = mgh = (50)(9.8)(5.0) = 2450 J. All becomes thermal: 2450 = 30 Δd, so Δd ≈ 82 m.
5Explain how regenerative braking improves the efficiency of an electric car.
Instead of turning all of the car's kinetic energy into thermal energy in the brakes, the motor works as a generator, turning some of it into electrical energy stored in the battery. That stored energy can be used again to drive the car, so less energy is wasted.
Challenge
1A 0.060 kg ball dropped from 1.50 m rebounds to 1.20 m on each bounce, keeping the same percentage of its energy each time. (a) What percentage of its mechanical energy does it keep per bounce? (b) How high is the third bounce?
(a) 1.20 ÷ 1.50 = 80%. (b) 1.50 × 0.80³ = 1.50 × 0.512 ≈ 0.77 m.
2A 900 kg car rolls (engine off) from rest down a hill 12 m high. At the bottom it is moving at 13 m/s. The road down the hill is 200 m long. Find the average force of friction and air resistance.
Eg₁ = (900)(9.8)(12) = 105 840 J. Ek₂ = ½(900)(169) = 76 050 J. E_th = 29 790 J. F = 29 790 ÷ 200 ≈ 1.5 × 10² N (149 N).
3A meteoroid enters the atmosphere at very high speed and glows. Use energy ideas to explain the glow.
Air resistance does large negative work on the meteoroid, transforming its enormous kinetic energy into thermal energy in the meteoroid and the surrounding air. They get so hot that they give off radiant energy (light).
Quiz
1. When a sled slides to a stop on a flat field, its kinetic energy has mostly become:
2. A box slides 4.0 m against a friction force of 12 N. The thermal energy produced is:
3. Which statement is true when friction acts on a moving object?
5. A child (30 kg) slides from rest down a 2.0 m high slide and reaches the bottom at 5.0 m/s. The thermal energy produced is about:
6. Regenerative braking turns some of a car's kinetic energy into:
Big idea: Total energy is always conserved, even when mechanical energy is not (D2.3, D3.1). Friction transforms mechanical energy into thermal energy, which links to efficiency (D3.4).
How to help: Next time you are tobogganing, skating or cycling, ask: "Where is the energy going right now?" Rubbing hands together on a cold day is a quick demonstration of work becoming thermal energy.
Common difficulties:
- Putting E_th on the wrong side of the equation. It is energy that leaves the mechanical accounts, so it goes on the end side with Ek₂ and Eg₂.
- Using the height of a ramp instead of its length in E_th = F_f Δd (friction acts along the whole sliding distance).
- Mixing up heat, temperature and thermal energy. The Thermal and Nuclear Energy unit develops these carefully.
Safety: the bounce activity is safe; keep it away from windows and breakable things.