Download Uniform Circular Motion

Uniform Circular Motion
Existence and properties of centripetal acceleration:
There are two ways to prove the existence of centripetal acceleration.
The first begins with acceleration. This is defined as “the rate of change of velocity.”1 therefore, it is
described mathematically by the model:
𝒂=
∆𝒗
∆𝑡
𝒗 is velocity, a vector quantity. Defined mathematically as:
𝒗=
∆𝒔
∆𝑡
Therefore, they require magnitude and direction. In circular motion, there is a constant change in
direction meaning that the velocity is constantly changing. As the velocity is changing, there must be
acceleration. Due to the speed being constant acceleration must be perpendicular to the direction of
the velocity, therefore, only effecting direction.1 As can be seen by the diagram below the
acceleration must be directed towards the axis of the circular motion to change the direction of the
velocity in a way that causes circular motion.
Fig 1
The second way is using Newton’s first and second law of motion. His first law states that a body; “A
body remains at rest and a moving body continues to move with a constant velocity unless and
external force acts on it.”1 If there is a change in direction there is also a change in velocity.
Therefore, due to Newton’s first law of motion, an external force must be acting on the body for its
velocity to change. Newton’s second law describes the force acting on a body, it can be defined by
the expression:
𝐹 = 𝑚𝒂
Showing that for there to be a force acting there must also be acceleration acting in the same
direction.1 For the body to constantly change direction, rotate around an axis, and keep a constant
speed the force must also be perpendicular to the direction of the velocity at all times.1 The forces
direction must be towards the axis, essentially pulling the body inwards. As the velocity is always
changing the centripetal forces direction is also changing, therefore, the body moves circularly.
Centripetal acceleration is very different from the types of acceleration usually encountered as rather
than affecting the magnitude of velocity is effecting the direction. If were to remain in a fixed
direction a parabolic arc would be seen like that of projectile motion.1
Causes of uniform circular motion:
As shown by Newton’s second law of motion, F=ma, for there to be acceleration there must be a
force acting on the body and this force will be in the same direction as the acceleration, as
acceleration is the only other vector quantity in the equation. As seen in the equation below:
𝑣2
𝑎𝑛𝑑 𝐹 = 𝑚𝑎
𝑟
𝑚𝑣 2
∴𝐹=
𝑟
∵ 𝑎𝑐 =
Due to the direction of the acceleration changing, to cause the velocity to also change direction, the
force must also constantly be changing direction to be perpendicular to the velocity.
Examples of uniform circular motion:
1) Rotation of body on a fixed radius string:
This is the most basic and commonly though of example for circular motion. Due to the body
being on the end of a string, the string is pulling the body in towards the axis to complete the
circular path. Therefore, the tension force on the string is equal to the centripetal force acting on
the body.1 As seen in the diagram below:
Fc=Ft
v
Fig 2
∴ 𝐹𝑡 =
𝑚𝑣 2
𝑟
If the centripetal force required rotating the body is more than the maximum tension force of the
string it will snap.1 The body will fly of in the direction of the velocity at that moment because
there is no external force (Newton’s first law) to change the direction.
2) Car cornering on a flat road:
When turning, vehicles are subject to centripetal force and acceleration because they are
engaging in a circular motion. As only connection between the car and the road is the tires,
therefore, the friction force between the tire and the road is the centripetal force acting on the car
to complete the turn.1 As seen in the diagram below:
v
Ff
Fig 3
ac and Fc
∴ 𝐹𝑓 = 𝐹𝑐
Similarly to the body on the end of a string, if the centripetal force required to turn the car is greater
than the friction force between the road and the tires the car will slide off the road as predicted by
Newton’s first law of motion.1 The direction in which the car slides will be tangential to the path it
was following as that is the direction of the velocity at any time. (the direction of the velocity at that
time)
3) Satellites and planets:
Satellites and planets mostly orbit in an approximately circular path meaning that they are under
the effects of centripetal force, however, they are also under the effect of gravitational force in
the same direction.1 Since the centripetal force is acting inwards along some medium and
gravitational force is the only thing that can provide this, centripetal force is equal to
gravitational force.1 This is shown in the diagram below:
Fig 4:
Path of inertia is the direction of the velocity at
that point as if the gravitational/ centripetal force
were to fail this is the path the planet of satellite
would take.
The diagram shows that the centripetal force is
inwards towards the axis of the planets rotation,
as the axis is a larger planer or star this force is
also gravitational force so they’re equal.
∴ 𝐹𝐺 = 𝐹𝑐
4) Car cornering on optimally banked road:
When cornering on a banked road the centripetal force is equal to the addition of the frictional
force, between the cars tire and the road, and the horizontal component of the cars normal force
(𝐹𝑁 sin 𝜃).1 (Seen in Fig 5) However, on an optimally banked road the angle is to such a degree
that the only force which the centripetal force is equal to is 𝐹𝑁 sin 𝜃. The optimal banking angle
depends on the speed of the cars and the radius of the bend.
Fig 5
Using this diagram a model for the optimal banking angle can be derived from fig. 6 (a vector
diagram):
𝐹𝑁 sin 𝜃 =
𝑚𝑣 2
𝑟
FN
𝐹𝑁 cos 𝜃 = 𝑚𝑔
𝜃
Fig 6
𝐹𝑁 sin 𝜃 𝑚𝑣 2
∴ tan 𝜃 =
=
𝐹𝑁 cos 𝜃 𝑟𝑚𝑔
𝑣2
∴ tan 𝜃 =
𝑟𝑔
𝑣2
∴ 𝜃 = tan−1 ( )
𝑟𝑔
References
1) Olesnicky, A 2009, ‘Motion in Two Dimensions’, Physics SACE 1: Part A, ed. DRPE Orell,
Lee, Younger, Collins, Greg Eather , Addelaode Tuition Centre, 21 Fourth Street, Bowden
SA 5007.
2) Fig 1: Centripetal Acceleration and Velocity Diagram, n.d., illustration, M & EMs, accessed
22 May 2013, <http://www.jonahgreenthal.com/physics/lessons/mech/2.2/>.
3) Fig 4: Centripetal force from planet orbit, n.d., illustration, Mansfield CT, accessed 22 May
2013, <http://www.mansfieldct.org/Schools/MMS/staff/hand/lawsCentripetalForce.htm>.
4) Fig 5: Banked Turn- No Friction, n.d., illustration, accessed 22 May 2013, <
http://batesvilleinschools.com/physics/phynet/mechanics/circular%20motion/banked_no_frict
ion.htm>.