Download Lesson 22 notes – Circular Motion - science

Lesson 11 notes – Circular Motion
Objectives
Be able to explain that a force perpendicular to the velocity of an object will
make the object describe a circular path.
Be able to explain what is meant by centripetal acceleration.
Be able to select and apply the equations for speed and centripetal
acceleration: v = 2r/T and a = v2/r.
Outcomes
Will know that circular motion occurs because of an unbalanced force which
makes an object accelerate towards the centre of the circular path.
Be able to explain that if an object has an unbalanced force on it, there must
be acceleration and that it is called the centripetal acceleration.
Be able to apply the equations for circular speed and centripetal acceleration
to solve problems correctly.
Be able to rearrange the equations for circular speed and centripetal
acceleration.
Be able to explain what the idea of centrifugal force is and why it is imaginary.
Be able to derive the equations for circular speed and centripetal acceleration.
Centripetal Force
A circle follows a curve all the way round and
we can describe it quantitatively as well as
qualitatively. All objects that follow a curved
path must have force acting towards the centre
of that curve. We call this force the centripetal
force. (Greek: Centre seeking).
Some examples of circular motion and the associated centripetal forces are:
Planetary orbits (almost!)
Electron orbits
Centrifuge
Gramophone needle
Car cornering
Car cornering on banked track
Aircraft banking
gravitation
electrostatic force on electron
contact force (reaction) at the walls
the walls of the groove in the record
friction between road and tyres
component of gravity
horizontal component of lift on the wings
Centripetal acceleration
Newton’s 1st Law says that an object will change direction if it feels an
unbalanced force at right angles to the direction it is travelling. This is the
reason things travel in circular paths. If there is always a force at 90 degrees
to the direction of motion then an object will travel in a circle. Since velocity is
speed in a given direction if an object is travelling at a constant speed but is
constantly changing direction it must be accelerating. This is what is
happening in circular motion. The acceleration is called Centripetal
Acceleration.
Circular velocity
The instantaneous linear velocity at a point in the circle is usually given the
letter v and measured in metres per second (m s-1).
Speed is defined as the distance / time.
For a circle, 1 complete circumference is 2r and T is the Time period for one
rotation (T)
So
v = 2r / T
Centripetal acceleration
If an object is moving in a circle at a constant speed, its direction of motion is
constantly changing. This means that its linear velocity is changing and so it
has a linear acceleration. The existence of an acceleration means that there
must also be an unbalanced force acting on the rotating object.
Consider an object of mass m moving with
constant speed (v) in a circle of radius r with
centre O.
v
v
Q
It moves from P to Q in a time t.
The change in velocity v is parallel to PO and v
= v sin
When  becomes small (that is when Q is very
close to P) sin is close to  in radians.
So v = v 
Dividing both sides by t gives:
v / t = v t
a = v2/r
a is the Centripetal Acceleration.

O
P
Extension
Centrifugal Force
This is a difficult concept. If we have something following a circular path and
there is a force pulling it to the centre there must be an equal and opposite
force pushing out of the circle right? Wrong.
The wrong idea:
The correct idea:
Centrifugal
force
Centripetal
Force
Centripetal Force
There is no equal and opposite force pushing the object out of its circular
path.
For there to be circular motion the forces must be unbalanced. There is a
centripetal force that is proportional to a centripetal acceleration.
If there is no more centripetal force the object does not fly out of the circle
away from the centre of the circle it just carries along in a straight line out of
the circle.
Think of the following examples:
Sitting in the back seat of a car as it corners: If the car turns to the left, you
feel as if you are being thrown to the right. In fact, your bum is in contact with
the seat, and gets pulled round to the left (providing there is sufficient friction).
The upper half of your body tries to carry on in a straight line. Viewed from a
point above the car, your upper half will be seen to be trying to follow a
tangential path while the car turns to the left.
Watching a marble roll on the surface of a table in a train as the train corners:
again, if the train turns to the left, the marble will appear to drift off to the right.
It is following a straight-line path, tangential to the curve. There is no friction to
pull it to the left, so no centripetal force.
An interesting example is a helium-filled balloon inside a cornering car. The
balloon leans in towards the centre of the circle. The air in the car tries to
continue in a straight line, so it is slewing to the right inside the car. The
balloon is lighter than the air, so it gets pushed towards the lower pressure at
the centre of the circle.