Centripetal Force — Why Going Twice as Fast Changes Everything
A practical investigation into mass, radius and rotational speed
There are some equations in physics that students can learn perfectly well without really appreciating what they mean.
Centripetal force provides a particularly good example.
The familiar relationship is:
F = mv^2/r
where:
F is the centripetal force,
m is the mass of the moving object,
v is its speed,
r is the radius of its circular path.
It looks straightforward enough.
But hidden inside that equation is a result that is surprisingly unintuitive:
the force depends on the square of the speed.
That means going twice as fast does not require twice the centripetal force.
It requires four times as much.
Go three times as fast and the required force becomes nine times as large.
That is something students can calculate on paper.
It is much more memorable when they can actually investigate it.
Circular Motion Is Constant Acceleration
One of the first conceptual difficulties is the word acceleration.
Ask a student:
"Can an object travelling at a constant speed be accelerating?"
A common answer is no.
That seems perfectly reasonable if acceleration has become mentally associated with a car getting faster.
But acceleration means a change in velocity, and velocity includes direction as well as speed.
Imagine an object travelling around a circular path at a perfectly constant speed.
Its speed may not change at all.
Its direction is changing continuously.
Therefore its velocity is changing continuously.
Therefore it is accelerating.
That acceleration is directed towards the centre of the circle and is called centripetal acceleration.
The corresponding resultant force must also act towards the centre.
Hence the name:
centripetal = centre-seeking.
The Force Is Not Pulling the Object Around the Circle
There is another useful idea to establish before beginning the experiment.
The instantaneous velocity of the moving object is tangential to its circular path.
The centripetal force acts approximately at right angles to that velocity, towards the centre.
The force is therefore continually changing the object's direction.
Remove that inward force and the object does not continue travelling around the circle.
It travels away approximately along the tangent.
That is an excellent idea to demonstrate physically if the apparatus allows it.
It also helps address the persistent misconception that an outward force must be keeping the object in circular motion.
For the laboratory analysis, what we need is an inward resultant force.
Three Variables — Three Experiments
I particularly like this investigation because the same apparatus can reveal three different mathematical relationships.
Rather than changing everything at once, I would divide the work into three investigations:
change the mass while keeping speed and radius constant;
change the radius while keeping mass and speed constant;
change the speed while keeping mass and radius constant.
That last experiment is the most interesting.
But I would not start with it.
The first two establish the method and give students relatively intuitive relationships before we encounter the surprise.
Experiment 1 — What Happens If We Increase the Mass?
Keep the radius and rotational speed constant.
Then increase the rotating mass.
From:
F = mv^2/r
if v and r remain constant:
F proportional to m
Double the mass and the required centripetal force should double.
Triple the mass and it should triple.
This is a simple linear relationship.
Students can collect several values and plot:
centripetal force against mass
If the experiment behaves well, the graph should be approximately a straight line through the origin.
This is already more useful than simply checking one calculated answer.
We are testing the form of a physical relationship.
A prediction worth making first
Before taking measurements, I would ask:
"If I double the rotating mass, what do you expect to happen to the force?"
Most students will probably predict that the force doubles.
Good.
Write that prediction down.
We will shortly encounter a variable for which intuition is much less reliable.
Experiment 2 — What Happens If We Increase the Radius?
Now keep mass and speed constant and change the radius.
The equation predicts:
F proportional to 1/r
Increasing the radius therefore reduces the required centripetal force, provided the linear speed really remains constant.
This needs careful thought because rotational experiments can introduce an important complication.
If we keep angular speed constant rather than linear speed, increasing the radius also increases the object's linear speed.
Since:
v = 2 pi r / T
or:
v = omega r
changing r while keeping the rotation rate constant does not keep v constant.
That can completely change what students observe.
This is an excellent experimental-design discussion.
It shows why physics is not merely about substituting numbers into equations.
We must understand what we are actually controlling.
Experiment 3 — Now Change the Speed
This is where the experiment becomes especially interesting.
Keep mass and radius constant.
Change only the linear speed.
Then:
F proportional to v^2
Before revealing that relationship, I would ask students for predictions.
Suppose our original speed is v and our original force is F.
What happens if we double the speed?
A very tempting prediction is:
2v gives 2F.
But that is not what the physics predicts.
Because speed is squared:
2v gives 4F.
Similarly:
3v gives 9F.
and:
4v gives 16F.
The required force rises extremely rapidly.
That is the result I want students to experience rather than merely memorise.
Make the Relationship Visible With Graphs
This experiment also provides an excellent opportunity to teach students something about graph transformations.
Plot:
F against v
and the result should be a curve.
That already tells us that force is not directly proportional to speed.
Now calculate v^2 for every measurement and plot:
F against v^2
The graph should become approximately linear.
That is powerful.
We have transformed experimental data to test a proposed mathematical model.
Instead of simply saying:
"The equation says F is proportional to v^2,"
we have asked the experiment whether that relationship appears to be true.
That is much closer to the way experimental physics actually works.
A Numerical Example
Suppose we have:
m = 0.50 kg
r = 1.0 m
v = 2.0 m/s
Then:
F = mv^2/r
F = 0.50 x 2.0^2 / 1.0
F = 2.0 N
Now double the speed:
v = 4.0 m/s
F = 0.50 x 4.0^2 / 1.0
F = 8.0 N
The speed has doubled.
The force has increased from 2 N to 8 N.
Now consider:
v = 6.0 m/s
F = 0.50 x 6.0^2 / 1.0
F = 18 N
That rapidly increasing force is why speed matters so enormously in circular motion.
Try It Interactively
A useful way of developing intuition is to change just one quantity at a time. Start with a mass of 2 kg, speed of 4 m/s and radius of 2 m, then double the speed while leaving everything else unchanged.
The important question is not simply "What is the new force?"
It is:
"Can you predict the new force before changing the control?"
That turns the equation into a physical model.
Why This Matters Outside the Laboratory
The squared speed relationship is not merely an examination curiosity.
It appears whenever objects move around curved paths.
Cars travelling around bends
A vehicle travelling around a bend requires an inward resultant force.
At modest speeds that force may be easily provided by the interaction between tyres and road.
Increase the speed and the required force rises as v^2.
This immediately explains why taking the same bend substantially faster is not merely slightly more demanding.
Roller coasters
Circular and curved sections of roller-coaster track can produce large accelerations because relatively high speeds combine with relatively small radii.
Satellites and planets
Orbital motion is another form of curved motion.
Gravity provides the inward force needed to continually change the direction of the velocity.
An orbiting spacecraft is not travelling because there is no gravity.
Quite the opposite.
Gravity is fundamental to maintaining the orbit.
Laboratory centrifuges
Centrifuges exploit rapid rotational motion to separate materials.
Increasing rotational speed can have a dramatic effect because of the squared relationship.
A Particularly Good Student Challenge
Once students have collected their measurements, I would give them an unknown data set.
Do not tell them which variable was changed.
Give them values of force and another quantity and ask:
"Does this look like F proportional to x, F proportional to x^2, or F proportional to 1/x?"
Now they must investigate.
They could:
inspect ratios;
calculate x^2;
calculate 1/x;
plot alternative graphs;
decide which produces the best straight line.
That turns a centripetal-force practical into a much broader lesson about mathematical modelling and experimental evidence.
What Would I Measure With Modern Equipment?
This is one of those experiments where modern sensors can transform the lesson.
Instead of merely watching a rotating mass and measuring a hanging weight, we can potentially record force continuously while also determining rotational period or speed.
The interesting part is then the graph.
Students can see the force changing rather than simply receiving one number at the end of the experiment.
With suitable data-logging equipment, I would want to display the measurements live.
That gives us opportunities to stop the experiment and ask:
"Why has the force just increased?"
"What would happen if we increased the speed by another 20%?"
"What graph should we get?"
Students become participants in the investigation rather than spectators waiting for an answer.
Experimental Problems Are Part of the Science
Real experiments rarely produce perfect textbook graphs.
There may be:
friction;
uncertainty in radius;
fluctuations in rotational speed;
sensor calibration errors;
vibration;
difficulty measuring the exact centre of rotation;
uncertainty in timing.
That is not a reason to avoid the experiment.
It is one of the reasons to do it.
Students can add error bars, repeat measurements and identify anomalous results.
They can also ask whether a discrepancy means the theory is wrong or simply that the experiment has limitations.
That distinction lies at the heart of good experimental science.
One Important Safety Point
Rotating apparatus deserves respect.
A small mass moving quickly possesses substantial kinetic energy, and the forces on attachments increase rapidly with speed.
The apparatus should therefore be properly secured, rotating components checked before use, speeds kept within the equipment manufacturer's limits, and observers kept clear of the plane of rotation.
Ironically, the very relationship we are investigating explains why this becomes increasingly important as the apparatus gets faster.
The Bigger Lesson — Equations Should Make Predictions
For me, this is the real value of an experiment like this.
Students sometimes encounter equations as instructions:
"Find the numbers, substitute them and calculate the answer."
But an equation is much more interesting than that.
It is a model of how nature behaves.
The equation:
F = mv^2/r
makes three distinct predictions.
Increase mass and force increases proportionally.
Increase radius, while maintaining the appropriate other conditions, and the relationship changes inversely.
Increase speed and force rises with the square of speed.
We can test those predictions.
That is what turns an equation into physics.
Conclusion — Twice as Fast Is Not Twice as Demanding
The most memorable moment in this experiment may come before any measurement is made.
Ask:
"If I make this object travel twice as fast around exactly the same circle, how much more force will I need?"
The intuitive answer is often:
"Twice as much."
Then perform the experiment.
The answer should be approximately:
four times as much.
That difference between intuition and evidence is precisely why practical physics is so valuable.
A student can memorise F = mv^2/r for an examination.
But watching the force rise dramatically as the apparatus speeds up gives that little superscript 2 a physical meaning.
And once you have actually seen what it does, it becomes considerably harder to forget.
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