Grade by Grade
C. Energy and Momentum · Work, Energy and Simple Harmonic Motion

Friction and Thermal Energy: Energy Accounting on Slopes and Tracks

C2.3 C2.2 C2.1About 75 minutes
Learning goals
  • I can explain why total energy is conserved even though mechanical energy is not when friction acts.
  • I can calculate the thermal energy produced by kinetic friction, including on inclines.
  • I can solve energy problems with friction and with other forces that do work.
  • I can find an unknown friction force or coefficient from energy measurements.

Words to know

thermal energy
The internal energy of the particles of a substance; friction converts mechanical energy into thermal energy.
law of conservation of energy
Energy cannot be created or destroyed; it can only be transformed from one form to another or transferred.
non-conservative force
A force such as kinetic friction or air resistance that changes mechanical energy into other forms.
energy budget
A list of all the energies at the start, all the work done, and all the energies at the end of a process.

Where did the energy go?

Toboggan down a real hill and you never reach the speed the frictionless formula predicts. Coast to the bottom and you slide across the flat field until you stop. Your kinetic energy seems to disappear. It does not: it becomes thermal energy in the snow, the toboggan and the air, and a little sound. In this lesson you will keep track of that energy and use it to solve problems with friction on slopes and tracks.

Remember from Grade 11: SPH3U wrote Ek₁ + Eg₁ = Ek₂ + Eg₂ + E_th for problems with friction, where E_th is the thermal energy produced. Grade 12 extends this to inclines (where friction depends on the angle) and to forces that add energy.

Total energy is always conserved

The law of conservation of energy says energy cannot be created or destroyed, only transformed or transferred. Mechanical energy is conserved only without non-conservative forces; total energy is always conserved.

When kinetic friction acts over a sliding distance Δd, it converts mechanical energy into thermal energy:

E_th = F_f Δd = μ_k F_N Δd

The energy equation becomes

Ek₁ + Eg₁ = Ek₂ + Eg₂ + E_th

A more general form includes any other force that does work, such as your push or a motor (W_other):

Ek₁ + Eg₁ + W_other = Ek₂ + Eg₂ + E_th

On an incline of angle θ, remember that F_N = mg cos θ (when no other force pushes into or pulls away from the surface), so friction is μ_k mg cos θ, not μ_k mg.

Eg 4022 Jtop of slopeEk 2917 JEth 1105 Jbottom of slopeEth 4022 Jafter the runoutTotal energy stays 4022 J; mechanical energy does not.
The toboggan in Example 1. Gravitational energy at the top becomes kinetic energy plus thermal energy at the bottom, and all thermal energy after the runout. The total stays the same at every stage.
Worked example

Example 1: A real toboggan run

A toboggan and rider (30 kg in total) start from rest at the top of a 40 m slope inclined at 20°. μ_k = 0.10. (a) Find the speed at the bottom. (b) How far do they slide on level snow (μ_k = 0.10) before stopping?

  1. Height: h = 40 sin 20° = 13.68 m. Eg₁ = (30)(9.8)(13.68) = 4022 J.
  2. Friction on the slope: F_f = μ_k mg cos 20° = 0.10(30)(9.8)(0.9397) = 27.63 N. E_th = (27.63)(40) = 1105 J.
  3. (a) Ek₂ = Eg₁ − E_th = 4022 − 1105 = 2917 J. v = √(2 × 2917 ÷ 30) = 13.9 m/s.
  4. (b) On the level, F_f = μ_k mg = 0.10(30)(9.8) = 29.4 N. All 2917 J becomes thermal energy: 29.4Δd = 2917, Δd ≈ 99 m.

(a) about 14 m/s; (b) about 99 m. Without friction the bottom speed would be √(2gh) = 16.4 m/s.

20°40 m, μ_k = 0.10level snow, μ_k = 0.10: stops after about 99 mrunoutstart at rest (h = 13.7 m)
The toboggan run in Example 1: a 40 m slope at 20° followed by a level runout. Friction acts on both parts, so the runout is where the remaining kinetic energy turns into thermal energy.
Worked example

Example 2: Finding friction from energy

A 500 kg coaster car starts from rest at the top of a 30 m hill. After 120 m of track it reaches the lowest point (h = 0) moving at 22 m/s. Find the thermal energy produced and the average friction force.

  1. Eg₁ = (500)(9.8)(30) = 147 000 J; Ek₁ = 0.
  2. Ek₂ = ½(500)(22)² = 121 000 J; Eg₂ = 0.
  3. E_th = 147 000 − 121 000 = 26 000 J (2.6 × 10⁴ J).
  4. Average friction: F_f = E_th ÷ Δd = 26 000 ÷ 120 = 217 N.

About 2.6 × 10⁴ J of thermal energy; average friction (and drag) about 2.2 × 10² N. Engineers use exactly this kind of calculation to decide how tall each hill can be.

Worked example

Example 3: Pulling a crate up a rough ramp

A 40 kg crate starts from rest and is pulled 6.0 m up a 25° ramp by a rope parallel to the ramp with tension 300 N. μ_k = 0.30. Find its speed at the top.

  1. W_other = work by the rope = (300)(6.0) = 1800 J.
  2. ΔEg = mgΔd sin 25° = (40)(9.8)(6.0)(0.4226) = 994 J.
  3. E_th = μ_k mg cos 25° Δd = 0.30(40)(9.8)(0.9063)(6.0) = 639.5 J.
  4. Ek₂ = 0 + 1800 − 994 − 639.5 = 166.5 J. v = √(2 × 166.5 ÷ 40) = 2.9 m/s.

About 2.9 m/s. Of the 1800 J the rope supplied, 55% was stored as Eg, 36% became thermal energy and only 9% became kinetic energy.

Energy accounting checklist

  1. Choose the reference level and list the energies at the start and the end.
  2. Identify every force that does work: gravity is in Eg; friction gives E_th; anything else (a push, a motor, a rope) is W_other.
  3. Find friction carefully: F_N = mg cos θ on a plain incline; it changes if another force pushes into or pulls off the surface.
  4. Write Ek₁ + Eg₁ + W_other = Ek₂ + Eg₂ + E_th and solve.
  5. Check: E_th must be positive. A negative value means a sign or height error.
Just for fun

Rub your hands together quickly for ten seconds and they warm up. Congratulations: you have just converted kinetic energy into thermal energy using friction, which is the same thing a hockey player's skates do to the ice, only much less dramatically.

Hands-on activity

Track the lost energy of a sliding book

You need: a hardcover book, a board or a stiff tray at least 60 cm long, a stack of books to prop it, a tape measure, a kitchen scale, and a phone with a stopwatch or slow-motion video.

  1. Weigh the book. Prop the board so the book just slides down on its own when given a tiny push (raise it slowly until it does). Measure the height h of the top end and the length of the board.
  2. Release the book from rest at the top and time its slide down the board. Repeat 3 times and average.
  3. From the distance and time, find the average speed, and double it to get the speed at the bottom (this works if the acceleration is constant and the book starts at rest).
  4. Calculate Eg at the top and Ek at the bottom. The difference is E_th.
  5. Divide E_th by the board length to estimate the average friction force, and by mg cos θ to estimate μ_k.

What to look for: E_th is usually a large fraction of Eg (often more than half) for a book on wood. Your μ_k should be somewhere between about 0.2 and 0.5 for a book cover on wood or cardboard. Trials that differ a lot usually mean the book was nudged at the start.

Your turn

1

A box slides 3.0 m across a floor against a 5.0 N friction force. How much thermal energy is produced?

2

A 50 kg skier starts from rest at a height of 20 m and reaches the bottom at 15 m/s. How much thermal energy was produced?

3

A cyclist and bike (90 kg) coast from 12 m/s to rest on level ground in 60 m. Find the average resistive force.

4

A 5.0 kg block slides from rest down a ramp 4.0 m long and 2.0 m high. Friction is 8.0 N. Find its speed at the bottom.

5

Is energy conserved when friction acts? Is mechanical energy conserved? Explain.

6

A ball dropped from 2.0 m bounces back to 1.4 m. What percentage of its mechanical energy was lost in the bounce?

More practice: warm-up

1

What form of energy does kinetic friction produce?

2

Write the formula for the thermal energy produced by kinetic friction.

3

On an incline of angle θ (no other forces into the surface), what is the normal force?

4

Write the energy equation with friction (no other forces doing work).

5

Why do brakes get hot?

More practice: on your own

1

A 0.16 kg puck slows from 15 m/s to 12 m/s while sliding 30 m. Find the thermal energy produced, the friction force and μ_k.

2

A 1200 kg car rolls from rest down a hill 20 m high and reaches 16 m/s at the bottom. What percentage of its Eg became thermal energy?

3

A 25 kg crate is pushed at constant speed 8.0 m up a 15° ramp with μ_k = 0.20, pushing parallel to the ramp. How much work does the pusher do?

4

A 60 kg skier passes the top of a 100 m long slope (30 m high) at 2.0 m/s and reaches the bottom at 18 m/s. Find the average friction force.

5

A student says, "Friction destroys energy." Correct the statement.

Challenge

1

A block is released from rest and slides 5.0 m down a 30° incline (μ_k = 0.25), then onto a level floor with the same μ_k. How far along the floor does it slide?

2

A 600 kg coaster car crosses the top of a 45 m hill at 2.0 m/s. It travels 150 m of track to the top of a 30 m hill, with an average friction force of 300 N. Find its speed at the top of the second hill.

3

A 0.25 kg ball falls from rest through 10 m and hits the ground at 12 m/s. Find the average air-resistance force and compare it with the ball's weight.

Check your understanding

Quiz

1.

Which statement is true when kinetic friction acts on a sliding object?

2.

A 10 kg box slides 4.0 m across a level floor with μ_k = 0.25. How much thermal energy is produced?

3.

On a 30° incline, the friction force on a sliding 2.0 kg block with μ_k = 0.50 is:

4.

A 2.0 kg ball falls from 5.0 m and lands at 9.0 m/s. How much energy became thermal energy?

5.

Which energy equation correctly includes a person's push (W_other) and friction?

6.

If a calculated value of E_th comes out negative, it most likely means:

For parents and teachers

Big idea (C2.3): with friction, mechanical energy is not conserved, but total energy is: the missing energy becomes thermal energy, E_th = F_fΔd. Students set up an energy budget from start to finish, including work done by people or motors.

How to help: ask your teen to explain where a sled's energy goes when it stops at the bottom of a hill. In the car, notice how braking hard makes the brake area warm after a long hill (do not touch: brakes can be very hot).

Common difficulties:

  • Using F_N = mg on an incline instead of mg cos θ.
  • Putting E_th on the wrong side of the equation.
  • Forgetting friction acts over the whole sliding distance (the slope length, not the height).
  • Mixing the slope and level parts of a problem. Treat each part separately.