Download Unit A: Kinematics Exam

Name:_________________
Summer
Physics 20
UNIT C: Circular Motion, Work and Energy
-1-
10
5.1 Defining Circular Motion
KEY TERMS
Axle: shaft on which a wheel rotates
Axis of rotation: imaginary line that passes through the centre of rotation perpendicular
to the circular motion
Uniform circular motion: motion in a circular path at a constant speed
Centripetal acceleration: acceleration acting toward the centre of a circle; centre
seeking acceleration
Centripetal force: force acting toward the centre of a circle causing an object to move in
a circular path
Speed and Velocity in Circular Motion
When we spin an object in a circular motion we see the following take place:
- If we were to let go of the string, the object would then resume its velocity at
a tangent from the circle
- The speed of the wheel remains constant but
- When time goes on the direction of velocity changes
- Since there is a change in velocity we know there is an acceleration
If we were to calculate the acceleration towards the center of the object we would use:
 
 v f  vi
a
t f  ti
Notice that when we do this we are able to find the direction of the acceleration.
This acceleration toward the centre of the circle is called the centripetal acceleration.
This centripetal is Latin for centre seeking. Since we have acceleration we must also
have a force that is causing that acceleration, we call this force: centripetal force. Often
we mistake, centripetal force with an outward force that we feel when the string pulls,
but this is simply the reaction force of the centripetal force.
-2-
5.2 Circular Motion and Newton’s Laws
KEY TERMS
Cycle: one complete back-and-forth motion or oscillation
Revolution: one complete cycle for an object moving in a circular path
Period: the time required for an object to make one complete oscillation (cycle)
Frequency: the number of cycles per second measured in hertz (Hz)
Rpm: revolutions per minute
Period and Frequency of Circular Motion
We consider a cycle or a revolution to be one complete rotation of a wheel.
The time it takes for that rotation we call the period, we measure the period in s/cycle.
The frequency tells us the number of cycles in a set amount of time. The units for
frequency are cycle/s or hertz (Hz). It has the following relationship:
1
1
T  or f 
T
f
The frequency can also be written in terms of rpm. Rpm’s are 60 times bigger than
hertz:
Hz x 60 = rpm
rpm  60 = Hz
Circular Motion
d
we can solve for circular speed by the same idea:
t
Circumference 2r
v

time
T
v
Similarly we can solve for the centripetal acceleration: ac 
turns into:
t
v 2 4 2 r
ac 
 2
r
T
Using the idea that F=ma we can determine the centripetal force:
mv2 4 2 mr
Fc  mac 

r
T2
From kinetics we learnt that: v 
-3-
Application of Horizontal Circular Motion
When we have a car driving at a constant velocity on a curved road we have centripetal
force keeping the car on the road. In the case for cars around corners we use the force
of friction as the centripetal force.
Fc  F f
Application of Vertical Circular Motion
When talking about vertical circular motion we often refer to scenarios dealing with
roller coasters or a bucket swinging
Things to remember:
- Gravity will always act downwards at the same force
- Normal force will always be perpendicular from its surface
- Tension will always be towards the center
- Centripetal force will always be towards the center of the circle
-4-
5.3 Satellites and Celestial Bodies in Circular Motion
KEY TERMS
Ellipse: elongated circle; consists of two foci, a major, and a minor axis
Eccentricity: degree to which an ellipse is elongated; number between 0 and 1, with 0
being a perfect circle and 1 being a parabola
Orbital period: time required for a planet to make one full orbit; may be measured in
earth days
Orbital radius: distance between the centre of the ellipse and the planet; average
orbital radius corresponds to the semi-major axis
Artificial satellite: artificially created object intended to orbit Earth or other celestial
body to perform a variety of tasks
Orbital perturbation: irregularity or disturbance in the predicted orbit of a planet
Kepler’s Laws
1. All planets in the solar system have elliptical orbits with the sun at one
focus.
Eccentricity = 0
Eccentricity = 1
Eccentricity = 0.25 Eccentricity = 0.5
foci
Minor axis
Semi-major axis
Major axis
Semi-minor axis
2. A line drawn from the Sun to a planet sweeps out equal areas in equal
times.
Area 1 swept in
120 days
Area 2 swept in
120 days
Area 1 = Area 2
-5-
3. The ratio of planet’s orbital period squared to its orbital radius cubed is a
constant. All objects orbiting the same focus (e.g., planets, the Sun) have
2
2
T
T
the same constant. a3  b3
ra
rb
Kepler’s constant for planets orbiting the sun are:
1a   1a2/AU3
T
K  E3 
1AU 3
rE
2
2

2
TE
3.15567 x107 s
K 3 
3
rE
1.50 x1011



2
 2.95x10-19s2/m3
Speed of Satellites
We can determine the velocity of the object that is orbiting a celestial object by
combining two known formulas:


Fg  Fc
m1v 2 Gm1m2

r
r2
Gm2
v
r
-6-
6.1 Work and Energy
KEY TERMS
Energy: the ability to do work
Work: a measure of the amount of energy transferred when a force acts over a given
displacement. It is calculated as a the product of the magnitude of applied force and the
displacement of the object in the direction of that force
Gravitational potential energy: the energy of an object due to its position relative to
the surface of Earth
Reference point: an arbitrarily chosen point from which distance is measured
Elastic potential energy: the energy resulting from an object being altered from its
standard shape, without permanent deformation
Kinetic energy: the energy due to the motion of an object
Work
When we have a force acting over a distance, a transfer of energy has occurred we call
this transfer of energy, work.
W  Fd
W  F||d
W  ( F cos  )d
The force and the distance travelled must be in the same direction in order for work to
be done.
Gravitational Potential Energy
We can give objects energy simply by lifting them up. By applying a force to the object
over a distance, in this case height, we are having work done on the object.
W  Fd
W  Fg h
W  mgh  E p
We call this type of work “a change in potential energy”: E p
When dealing with objects relative to the ground we can simplify this formula to:
E p  mgh : “Potential energy = mass times gravity times height”
Hooke’s Law
Robert Hooke, was able to determine the relationship between springs, displacement
and a constant. When graphing the applied Force to the spring vs. the resultant position,
we find that there becomes a linear relationship between the two. The slope of these
two variables was found to be:
F F f  Fi
k

or F  kx
x x f  xi
-7-
“k” is know as a spring/elastic constant, this k value is unique for every spring. The value
of this constant tells us how hard we must push/pull on a spring in order for it to
expand/ contract. The “x” is the distance away from equilibrium x=0.
Elastic Potential Energy
We can determine how much energy a spring or elastic has by calculating its elastic
potential energy. We use the following formula:
1
E p  kx2
2
We are using the same “k” spring constant, and “x” distance from equilibrium that was
used in Hooke’s Law.
Kinetic Energy
When we have an object in motion, we then have to consider kinetic energy: E k. The
kinetic energy of an object is calculated using the equation:
1
Ek  mv 2
2
-8-
6.2 Mechanical Energy
KEY TERMS
Mechanics: the study of kinematics, statics, and dynamics
Mechanical energy: the sum of the potential and kinetic energies
The Work-Energy Theorem
-9-
6.3 Mechanical Energy in Isolated and Non-isolated Systems
KEY TERMS
Isolated system: a group of objects assumed to be isolated from all other objects in the
universe
Non-isolated system: a system in which there is an energy exchange with the
surroundings
- 10 -
6.3 Work and Power
KEY TERMS
Power: the rate of doing work
Efficiency: ratio of the energy output to the energy input of any system
- 11 -