Download Chapter 5

Chapter 5-Dynamics of Uniform Circular Motion
5.1: Uniform Circular Motion
Definition of Uniform Circular Motion
Uniform circular motion is the motion of an object traveling at a constant (uniform) speed on a circular path.
The period 𝑇 is the time required to make one complete revolution.
When discussing constant linear motion we used:
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑣=
𝑡𝑖𝑚𝑒
So now that we are dealing with circular motion we need to equate this to a circle. So this becomes
𝑣=
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 2𝜋𝑟
=
=
𝑡𝑖𝑚𝑒
𝑝𝑒𝑟𝑖𝑜𝑑
𝑇
Note:
 Although this is considered a constant circular motion we need to remember that the velocity vector still
has a changing direction and only the magnitude remains constant.
 The change in the velocity vector means acceleration is occurring.
5.2: Centripetal Acceleration
Centripetal Acceleration is a vector quantity that tells us how much an object is accelerating radially toward the
center of the circle.
Centripetal Acceleration
Magnitude: The centripetal acceleration of an object moving with a speed 𝑣 on a circular path of radius 𝑟 has a
magnitude 𝑎𝑐 given by
𝑣2
𝑎𝑐 =
𝑟
Direction: The centripetal acceleration vector always points toward the center of the circle and continually
changes directions as the object moves.
Note:
 An object that is experiencing uniform circular motion can never be in equilibrium.
 If it helps to remember what centripetal is; the word centripetal means “center seeking”.
5.1 & 5.2
Practice Problems: 2, 3, 6
Homework: 1, 5, 7, 10
5.3: Centripetal Force
Centripetal Force
Magnitude: The centripetal force is the name given to the net force required to keep an object of mass 𝑚, moving
at a speed 𝑣, on a circular path of radius 𝑟, and it has a magnitude of
𝐹𝑐 =
𝑚𝑣 2
𝑟
Direction: The centripetal force always points toward the center of the circle and continually changes direction as
the object moves.
Note:
 There must always be a force to keep the object moving with a circular motion, if it is not obvious look deeper
because it does exist.
5.3
Practice Problems: 15, 18
Homework: 11-14, 16
5.4: Banked Curves
When a car travels without skidding around a curve it is the static friction that holds the car in place. For times
when the static friction is not large enough to hold a car in place we use other methods to obtain the needed
centripetal force. The most common solution is to make a banked curve. This is most noticeable in car racing
where the corners are banked sometimes around 30o.
On a given bank we can determine a few of the forces that are present (assuming a frictionless surface).
We see for the y-direction we have:
𝐹𝑛 cos 𝜃 − 𝑚𝑔 = 𝑚𝑎𝑦
Where 𝑎𝑦 = 0 (We don’t want the car flying off in the air or sinking down into the ramp.)
For the x-direction we have:
𝐹𝑛 sin 𝜃 = 𝑚𝑎𝑥
2
Where 𝑎𝑥 = 𝑎𝑐 = 𝑣 ⁄𝑟
Since 𝐹𝑛 is the same we actually will take a ratio of the x and y directions to create a new equation that relates 𝑣
and 𝜃.
2
𝐹𝑛 sin 𝜃 𝑚𝑣 ⁄𝑟
=
𝐹𝑛 cos 𝜃
𝑚𝑔
tan 𝜃 =
𝑣2
𝑟𝑔
Note:
 This all is done without friction; with friction remember to use the same starting spot and moving from there.
5.4
Practice Problem: 23
Homework: 20-22
5.5: Satellites in Circular Orbits
For satellites in circular orbits it is the force of the earth’s gravity that provides the needed centripetal force.
There is only one speed that a satellite can have if the satellite is to remain in an orbit with a fixed radius.
We can see this by using the standard equation for gravitational forces.
𝑚𝑀𝑒 𝑚𝑣 2
𝐹𝑐 = 𝐺 2 =
𝑟
𝑟
By solving for 𝑣 we get
𝐺𝑀𝑒
𝑣=√
𝑟
You will notice that the mass of the satellite is irrelevant to the speed it must maintain. This equation works for
any object revolving about another object; you only need to replace earth’s values with the value of the other
object that is at the center of the revolution.
From earlier equations we can also determine the period needed for such a revolution. Using
𝐺𝑀𝑒
2𝜋𝑟
𝑣=√
=
𝑟
𝑇
And solving for 𝑇 we get
𝑇=
2𝜋𝑟 3⁄2
√𝐺𝑀𝑒
This equation is very important when determining the placement of satellites as well as astronomical calculations.
This allows us very easily to make quick calculations to approximate the period of revolving objects or to work
backwards and then determining the mass of an object being revolved around.
Note.
 All geosynchronous satellites have the same radius and speeds.
5.6: Apparent Weightlessness and Artificial Gravity
An object orbiting the earth still experiences the force of gravity but because it is above the effects of air
resistance it is basically in free fall (only experiences acceleration due to gravity). This causes any object not
attached to experience weightlessness. (Imagine jumping out of a plane with no air resistance, it would be like
you were floating, that is until you reached the earth’s surface…. Ouch.)
The effect of weightlessness causes many problems for astronauts while in space, many that are not known. If we
were to have extended trips into space in the future we would probably have to use some time of artificial gravity.
Artificial gravity is easily created by a centrifuge. A centrifuge is an object that spins; this spinning motion can be
used to create the accelerations needed to simulate the earth’s gravitational pull.
5.5 & 5.6
Practice Problems: 27, 31
Homework: 28-30, 35