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Circular motion

Content objective:

What are we learning? Why are we learning this?

content, procedures, skills

Vocabulary objective

Tier II: High frequency words used across content areas. Key to understanding directions, understanding relationships, and for making inferences.

Tier III: Low frequency, domain specific terms

Building on what we already know

What vocabulary & concepts were learned in earlier grades?
Make connections to prior lessons from this year.
This is where we start building from.

For our unit on the Coriolis effect see Coriolis effect.

Uniform Circular Motion (UCM) is motion in a circle of constant radius at constant speed.

Keep track of the position and speed.

     r = radius (a position vector)

     v = velocity (also a vector)

Velocity is always tangent to the circle

The magnitude of the velocity (speed) always stay same as we go round the circle

But the direction is always changing.




The direction of the acceleration always points inward.


This inward pointing acceleration is called centripetal acceleration.

Acceleration is always perpendicular to the velocity.

centripetal acceleration perpendicular to velocity

Period and Frequency

period  =  T = time to make 1 revolution around the circle

frequency  =  f  =  # of revolutions per second.

             f = 1 / T

             T = 1 /  f

Look at this animation: How long does it take to make one revolution? That amount of time is T, the period.



Centripetal velocity and acceleration

What is the velocity of something in UCM?

v  =  velocity

R = radius

By definition, v  =  distance / time

By definition, distance around circle = circumference = 2πr

Therefore V =  circumference / time  =


How much (inward pointing) acceleration does this object experience?

Every physics textbook has a proof. Here we just state the result:


We also may write the acceleration in terms of T (period)


Newton’s Second Law for circular motion

For an object to be in uniform circular motion, there must be a net force acting on it.

We already know the acceleration, so can immediately write the force:

F = ma        and       a = v2/r

So     F  = m·v2/r

Here’s a computer program written in Scratch.

It calculates centripetal force, using radius (r), period (T) and mass (m)
(right click on it to expand.)



Direction of centripetal force

Think about a ball on a string. We see that the direction of the centripetal force must be inward. (If there was no inward force then the ball would fly outwards!)

Direction centripeltal force circular

There is no centrifugal force pointing outward; what happens is that the natural tendency of the object to move in a straight line must be overcome.

If the centripetal force vanishes then the object flies off tangent to the circle.

It’d fly off in a straight line (Newton’s 1st law of motion!)

centripetal force NO outward centrifugal force

Highway Curves, Banked and Unbanked

When a car goes around a curve, there must be a net force towards the center of the circle of which the curve is an arc. If the road is flat then that force is supplied by friction.

Car going round curve centripetal force

If the frictional force is insufficient then the car will tend to move more nearly in a straight line, as the skid marks show.

Centripetal motion car friction off track Skidding

As long as the tires do not slip, the friction is static. If the tires do start to slip, the friction is kinetic, which is bad in two ways:

1. The kinetic frictional force is smaller than the static.

2. The static frictional force can point towards the center of the circle, but the kinetic frictional force opposes the direction of motion, making it very difficult to regain control of the car and continue around the curve.

Honors: Banking a car

Banking the curve can help keep cars from skidding.

In fact, for every banked curve, there is one speed where the entire centripetal force is supplied by the horizontal component of the normal force, and no friction is required.

Banking centripetal force example car

This occurs when

Banking centripetal force example equation


What’s happening here? Explain using the concepts covered in this unit.


PhysicsFootnotes writes:

As he rises, the radius of the hole increases, and the faster he has to run to generate a normal reaction force which (combined with friction) has a sufficient vertical component to balance his weight. Just like the things you put the coins in, only in reverse.

How fast would he need to run, in order to do this? Set up a force-vector diagram. His centripetal acceleration has a value of v-squared/r. Increasing r means you have to increase v to compensate.



Examples of centripetal force video clips

A car going around a curve

What force does the car put on the road?
What force does the road put on the car?



You’re a passenger in a car, making a right-hand turn.
As the car turns right, you begin sliding to the left.
The car is turning to the right, due to the inward force
(tires against road, and road back against the tires)

So why are you forced left/outward?
In actuality, the car is turning, while you continue in a straight path!

G forces on a roller coaster


Full discussion here: http://www.physicsclassroom.com/mmedia/circmot/rcd.cfm


Physics of Batman The Dark Knight


PowerPoint Chap 10 Circular Motion Hewitt

Physics PowerPoints holtonsworld.com


As the small pebble stirs the peaceful lake; The centre mov’d, a circle straight succeeds, Another still, and still another spreads.
– Alexander Pope, Essay on Man (ep. IV, l.364)

People travel to wonder at the height of mountains, at the huge waves of the sea, at the long courses of rivers, at the vast compass of the ocean, at the circular motion of the stars; and they pass by themselves without wondering.
– Saint Augustine of Hippo (354 – 430 CE)


Learning Standards

2016 Massachusetts Science and Technology/Engineering Curriculum Framework
HS-PS2-1. Analyze data to support the claim that Newton’s second law of motion is a mathematical model describing change in motion (the acceleration) of objects when acted on by a net force.

HS-PS2-10 (MA). Use free-body force diagrams, algebraic expressions, and Newton’s laws of motion to predict changes to velocity and acceleration for an object moving in one dimension in various situations

Massachusetts Science and Technology/Engineering Curriculum Framework (2006)

1.8 Describe conceptually the forces involved in circular motion.

SAT Physics Subject Test Learning Objectives

Circular motion, such as uniform circular motion and centripetal force

Common Core Math

CCSS.MATH.CONTENT.7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
CCSS.MATH.CONTENT.8.EE.C.7 Solve linear equations in one variable
CCSS.MATH.CONTENT.HSA.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (including isolating a variable)
CCSS.MATH.CONTENT.HSA.CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

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