Movement: skeleton and muscles · Quantitative
Biomechanics: forces in the body
Holding a 2 kg dumbbell, your biceps pulls with about 160 newtons. Why is it working eight times harder than the weight it is holding?
Start here
The muscle is losing, badly, and it is built that way on purpose.
Hold a bag of sugar on your flat hand. It weighs 10 N. Your biceps, at that instant, is pulling with about 80 N — and it is attached only about 4 cm from your elbow, while the bag sits about 32 cm away.
Something about that arrangement is doing the damage. What?
The two distances. A force that acts a long way from a joint has far more turning effect than the same force acting close to it — and your muscles are all attached close in. That is the trade this whole lesson is about, and it can be worked out to the newton.
Every force acting on a bone is trying to turn it about a joint. How much turning it does depends on two things: how big the force is, and how far from the joint it acts.
At the bench · the forearm rig
Three measurements. The fourth number is yours.
Meter not fitted yet
Commit first. You move the load from 32 cm out to 16 cm — half the distance. What happens to the force the muscle needs?
2.0 kg
4.0 cm
Hand distance
Weight of the load
20 N
Load, from the elbow
32 cm
Muscle, from the elbow
4.0 cm
Force in the muscle
not measured — you work it out
Work it out first. Then fit the meter and see whether the rig agrees with you.
Turning effect = force × distance from the joint
The triangle
Cover the one you want
T = F × d
F = T ÷ d
d = T ÷ F
T is on its own at the top, with the other two side by side underneath. Cover it and you are left with F × d — multiply.
F sits underneath, with T above it. Cover it and you are left with T over d — divide.
d sits underneath, with T above it. Cover it and you are left with T over F — divide.
Two things side by side means multiply. One thing over another means divide.
T — turning effect, in N m
F — force, in N
d — distance from the joint, in m
Nothing moving: F₁ × d₁ = F₂ × d₂
Worked example · one step at a time
A 2 kg dumbbell, 32 cm out. The biceps attaches at 4 cm.
Step 0 of 4
Formula
F × d(muscle) = W × d(load)
Nothing is moving, so the turning effect of the muscle equals the turning effect of the load.
Insert
F × 0.04 m = 20 N × 0.32 m
The 2 kg dumbbell weighs 2 × 10 = 20 N. Both distances go into metres.
Fine-tune
F × 0.04 = 6.4, so F = 6.4 ÷ 0.04
Right-hand side first, then rearrange so F is on its own.
Answer
F = 160 N
Eight times the weight of the dumbbell — because 0.32 is eight times 0.04.
Your turn · the same four steps
Your rig: 2.0 kg at 32 cm, muscle at 4.0 cm.
Commit to each line, then open the worked version and compare it with yours.
Step 1 · Formula
Nothing is moving, so the two turning effects are equal. Which line says that?
Step 2 · Insert
Put your rig's numbers in, with the distances in metres.
Steps 3 and 4 · Work it out, then answer
Divide, round to the nearest newton, and choose the unit.
Your rig, done four ways
Formula
F × d(muscle) = W × d(load)
Nothing moves, so the two turning effects are equal.
Insert
F × 0.04 m = 20 N × 0.32 m
Weight is mass × 10, and both distances go into metres.
Fine-tune
F × 0.04 = 6.40, so F = 6.40 ÷ 0.04
Right-hand side first, then rearrange so F is on its own.
Answer
F = 160 N
That is 8.0 times the weight of the load, because the load acts 8.0 times further from the joint.
Key fact
A force turns a bone about a joint by force × distance from the joint. Muscles attach close in, so they must pull many times harder than the load they hold.
Measured, not guessed · three force meters
Which muscle group pulls hardest?
Not ranked yet
Three groups of muscles, each measured three times on a force meter by the same person. Put them in order before you look.
Hand grip
312 N · 298 N · 305 N
305 N
mean of three
Biceps, pulling up
196 N · 210 N · 203 N
203 N
mean of three
Leg press, both legs
1450 N · 1390 N · 1425 N
1422 N
mean of three
The bigger the muscle, the bigger the force — and the three readings for each are never identical, so each one is reported as a mean. A single pull tells you almost nothing.
Seven words
Say your answer out loud before you turn each card over. If you cannot say it, you do not know it yet.
Think again
“Your arm is a lever, and levers make things easier — so the muscle pulls less than the weight.”
You have just calculated the opposite. Commit to what the body is getting out of the deal before you read on.
Speed and distance. The biceps shortens by about 4 cm and your hand travels about 32 cm — eight times as far, in the same time, so eight times as fast. A muscle can only shorten by a fraction of its own length and it cannot do it quickly. Attaching it close to the joint converts a small, slow, powerful pull into a long, fast movement.
You pay for it in force, every single time. That is not a design fault; it is the price of a hand that can throw something. A body built the other way round would be immensely strong and would move like a crane.
Mastery ladder
Not started yet.
Rungs 3 and 4 you mark yourself.
Rung 1 · Calculate
A load of 30 N is held 30 cm from the elbow. The muscle attaches 5 cm from the elbow. What force must the muscle pull with?
Rung 2 · The one that catches people
Why does the biceps have to pull so much harder than the weight it is holding?
Rung 3 · Explain
The arrangement in your arm costs a lot of force. Explain what the body gets in return, using the two distances.
Rung 4 · Take it somewhere new
Someone carries a 5 kg bag. Held against the chest, its weight acts about 10 cm from the shoulder joint; held out at arm’s length, about 60 cm. Explain, with a calculation, why the second one is so much harder — and say what that means for how you should lift a box.
Key note
Turning effect = force × distance from the joint. When nothing is moving, the two turning effects are equal, so a muscle attached 4 cm from the elbow must pull eight times harder than a load held 32 cm out. Force is bought with distance, and speed is bought with force.
Going further
Your Achilles tendon is the exception that proves the rule. It attaches behind the ankle joint, well back from it, which is a long way as body attachments go — and that is exactly why you can push off the ground hard enough to run and jump. Elite sprinters tend to have a slightly shorter heel bone than average, giving the tendon a smaller distance to work with, and a bigger force is needed for every stride. What they get in return is speed: a small shortening of the calf muscle throws the foot down faster. The trade is the same one your elbow makes, tuned differently.
Before this lesson
Taught in full in
- Physics: Forces
Where the turning effect of a force is developed on its own, away from the body.
At GCSE this becomes
- Moments, levers and gears, and the mechanics of the musculoskeletal system.
Where to next
- Next: A balanced diet
Nutrition and digestion
- Previous: Antagonistic muscle pairs
Ask Mr Badmus AI
Stuck on which distance goes where?
Weight in newtons is taken as mass in kilograms × 10 N/kg throughout.
Lesson content © MrBadmusAI.