Chapter 5 Learning Objectives Circular motion and rotation Uniform circular motion Students should understand the uniform circular motion of a particle so they can Relate the radius of the circle and the speed or rate of revolution of the particle to the magnitude of the centripetal accel ID: 221236
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Slide1
Dynamics of Uniform Circular Motion
Chapter 5Slide2
Learning Objectives- Circular motion and rotation
Uniform circular motion
Students should understand the uniform circular motion of a particle, so they can:
Relate the radius of the circle and the speed or rate of revolution of the particle to the magnitude of the centripetal acceleration.
Describe the direction of the particle’s velocity and acceleration at any instant during the motion.
Determine the components of the velocity and acceleration vectors at any instant, and sketch or identify graphs of these quantities.
Analyze situations in which an object moves with specified acceleration under the influence of one or more forces so they can determine the magnitude and direction of the net force, or of one of the forces that makes up the net force, in situations such as the following:
Motion in a horizontal circle (e.g., mass on a rotating merry-go-round, or car rounding a banked curve).
Motion in a vertical circle (e.g., mass swinging on the end of a string, cart rolling down a curved track, rider on a Ferris wheel).Slide3
Table Of Contents
5.1 Uniform Circular Motion
5.2 Centripetal Acceleration
5.3 Centripetal Force
5.4 Banked Curves
5.5 Satellites in Circular Orbits5.6 Apparent Weightlessness and Artificial Gravity5.7 Vertical Circular MotionSlide4
Chapter 5:Dynamics of Uniform Circular Motion
Section 1:
Uniform Circular MotionSlide5
Other Effects of Forces
Up until now, we’ve focused on forces that speed up or slow down an object.
We will now look at the third effect of a force
Turning
We need some other equations as the object will be accelerating without necessarily changing speed.Slide6
Uniform circular motion is the motion of an object
traveling at a constant speed on a circular path.
DEFINITION OF
UNIFORM CIRCULAR MOTIONSlide7
Let
T
be the time it takes for the object to
travel once around the circle.Slide8
Example 1: A Tire-Balancing Machine
The wheel of a car has a radius of 0.29m and it being rotated
at 830 revolutions per minute on a tire-balancing machine.
Determine the speed at which the outer edge of the wheel is
moving.Slide9
Newton’s Laws1
st
When objects move along a straight line the sideways/perpendicular forces must be balanced.
2
nd
When the forces directed perpendicular to velocity become unbalanced the object will curve.3rd The force that pulls inward on the object, causing it to curve off line provides the action force that is centripetal in nature. The object will in return create a reaction force that is centrifugal in nature.Slide10
Chapter 5:Dynamics of Uniform Circular Motion
Section 2:
Centripetal AccelerationSlide11
In uniform circular motion, the speed is constant, but the
direction of the velocity vector is
not constant.Slide12Slide13
The direction of the centripetal acceleration is towards the
center of the circle; in the same direction as the change in
velocity.Slide14
Conceptual Example 2: Which Way Will the Object Go?
An object is in uniform circular
motion. At point
O
it is released
from its circular path. Does the object move along the straightpath between O and A or along the circular arc between pointsO and P ?Straight pathSlide15
Example 3: The Effect of Radius on Centripetal Acceleration
The bobsled track contains turns
with radii of 33 m and 24 m.
Find the centripetal acceleration
at each turn for a speed of
34 m/s. Express answers as multiples of Slide16
Chapter 5:Dynamics of Uniform Circular Motion
Section 3:
Centripetal ForceSlide17
Recall Newton’s Second Law
When a net external force acts on an object
of mass
m
, the acceleration that results is
directly proportional to the net force and hasa magnitude that is inversely proportional tothe mass. The direction of the acceleration isthe same as the direction of the net force.Slide18
Recall Newton’s Second Law
Thus, in uniform circular motion there must be a net force to produce the centripetal acceleration.
The centripetal force is the name given to the net force required to keep an object moving on a circular path.
The direction of the centripetal force always points toward the center of the circle and continually changes direction as the object moves.Slide19
Problem Solving Strategy – Horizontal Circles
Draw a free-body diagram of the curving object(s).
Choose a coordinate system with the following two axes.
a) One axis will point
inward
along the radius (inward is positive direction). b) One axis will point perpendicular to the circular path (up is positive direction).Sum the forces along each axis to get two equations for two unknowns. a) FRADIUS: +FIN F
OUT = m(v2)/ r b) F
: FUP FDOWN = 0
Do the math of two equations with two unknowns.
R
Slide20
Just in case…The third dimension in these problems would be a direction tangent to the circle and in the plane of the circle.
We choose to ignore this direction for objects moving at constant speed.
If an object moves along the circle with changing speed then the forces tangent to the circle have become unbalanced.
You can sum the tangential forces to find the rate at which speed changes with time, a
TAN
. The linear kinematics equations can then be used to describe motion along or tangent to the circle. FTAN: FFORWARD FBACKWARD = m aTAN
R
tanSlide21
Example 5: The Effect of Speed on Centripetal Force
The model airplane has a mass of 0.90 kg and moves at
constant speed on a circle that is parallel to the ground.
The path of the airplane and the guideline lie in the same
horizontal plane because the weight of the plane is balanced
by the lift generated by its wings. Find the tension in the 17 mguideline for a speed of 19 m/s.Slide22
5.3.1. A boy is whirling a stone at the end of a string around his head. The string makes one complete revolution every second, and the tension in the string is
F
T
. The boy increases the speed of the stone, keeping the radius of the circle unchanged, so that the string makes two complete revolutions per second. What happens to the tension in the sting?
a) The tension increases to four times its original value.
b) The tension increases to twice its original value.c) The tension is unchanged.d) The tension is reduced to one half of its original value.e) The tension is reduced to one fourth of its original value.Slide23
Chapter 5:Dynamics of Uniform Circular Motion
Section 4:
Banked CurvesSlide24
On an unbanked curve, the static frictional force
provides the centripetal force.
Unbanked curveSlide25
On a frictionless banked curve, the centripetal force is the
horizontal component of the normal force. The vertical
component of the normal force balances the car’s weight.
Banked CurveSlide26Slide27
Example 8: The Daytona 500
The turns at the Daytona International Speedway have a
maximum radius of 316 m and are steely banked at 31
degrees. Suppose these turns were frictionless. As what
speed would the cars have to travel around them?Slide28
Chapter 5:Dynamics of Uniform Circular Motion
Section 5:
Satellites in Circular OrbitsSlide29
There is only one speed that a satellite can have if the
satellite is to remain in an orbit with a fixed radius.
Don’t worry, it’s only rocket scienceSlide30Slide31
Example 9: Orbital Speed of the Hubble Space Telescope
Determine the speed of the Hubble Space Telescope orbiting
at a height of 598 km above the earth’s surface.Slide32
Period to orbit the EarthSlide33
Geosynchronous OrbitSlide34
Chapter 5:Dynamics of Uniform Circular Motion
Section 6:
Apparent Weightlessness and Artificial GravitySlide35
Conceptual Example 12: Apparent Weightlessness and
Free Fall
In each case, what is the weight recorded by the scale?Slide36
Example 13: Artificial Gravity
At what speed must the surface of the space station move
so that the astronaut experiences a push on his feet equal to
his weight on earth? The radius is 1700 m.Slide37
Chapter 5:Dynamics of Uniform Circular Motion
Section 7:
Vertical Circular MotionSlide38
Circular MotionIn the previous lesson the radial and the perpendicular forces were emphasized while the tangential forces were ignored. Each class of forces serves a different function for objects moving along a circle.
Class of Force
Purpose of the Force
Radial Forces
Curves the object off a straight-line path.
Perpendicular Forces
Holds the object in the plane of the circle.
Tangential Forces
Changes the speed of the object along the circle.Slide39
Circular MotionMost of the horizontal, circular problems occurred at constant speed so that we could ignore the tangential forces. The vertical, circular problems have objects moving with and against gravity so that speed changes. Tangential forces become significant. The good news is that perpendicular forces can now be ignored unless hurricanes are present.Slide40
Problem Solving Strategy for Vertical Circles
Draw a free-body diagram for the curving objects.
Choose a coordinate system with the following two axes.
a) One axis will point inward along the radius.
b) One axis points tangent to the circle in the circular plane, along the direction of motion.
Sum the forces along each axis to get two equations for two unknowns. a) FRADIUS: +FIN FOUT = m(v2)/ r b) FTAN : F
FORWARD FBACKWARDS = maYou can generally expect the weight of the object to have components in both equations unless the object is exactly at the top, bottom or sides of the circle.
If the object changes height along the circle you may need to write a conservation of energy statement. This goes well with centripetal forces since there is an {mv
2} in both kinetic energy terms and in centripetal force terms.Do the math with 3(a) and 4 or perhaps 3(a) and 3(b).Slide41
Minimum/Maximum Speed Problems
Sometimes the problem addresses “the minimum speed” that an object can move through the top of the circle or “maximum speed” that an object can move along the top of the circle.
If the bucket of water turns too slowly you get wet.
If a car tops a hill too quickly it leaves the ground.
Allowing v
2/r to equal g can solve many of these questions. By solving for v you will find a critical speed. Slide42
Conceptual Example: A Trapeze Act
In a circus, a man hangs upside down from a trapeze, legs
bent over and arms downward, holding his partner. Is it harder
for the man to hold his partner when the partner hangs
straight down and is stationary of when the partner is swinging
through the straight-down position?Slide43