Dynamics Unit- Auxiliary Forces Sub-Unit
Regents Physics
1 | P a g e
2.2.1 Hooke’s Law
Hooke’s Law
The restoring force exerted by an object (FS) is proportional to the amount it is stretched or
compressed (x) and its ‘spring constant’ (k).
Equation
Example #1
• A spring with a spring constant of 10 newtons per meter is stretched 0.5 meter.
What force must be applied to the spring?
Example #2
• What is the spring constant of an elastic object if it stretches 2.0 centimeters
when pulled with a force of 30 newtons?
PRACTICE
1) What distance will a spring with a spring constant of 1200 newtons per meter stretch
when a force of 50 newtons is applied to it?
2) What does the slope of the graph below represent? How do you know that the graph
represents data taken for an object that does obey Hooke’s Law?
FS
2.2.2 Universal Gravitation
What is Gravity?
An attractive force between two objects with mass.
• Strength of attraction depends on:
o __________________________________________________________
o __________________________________________________________
o __________________________________________________________
Equation
Example #1
• What is the force of gravitational attraction between two asteroids separated by
3000 meters if they have masses of 4.0 x 105 kilograms and 6.0 x 105 kilograms?
2.2.1 – Calculate the amount of spring force; spring constant; or amount that a spring’s length will be changed. Interpret the slope of a graph of force vs. stretch. Determine if an object obeys Hooke’s Law.
- How much force is needed to stretch a spring with a spring constant of 1000
newtons per meter a distance of 0.04 meter?
[40N]
- What is the spring constant of a spring that is compressed a distance of 0.06
meter when pushed with a force of 50 newtons?
[833N/m]
- What distance will a spring with a spring constant of 600 newtons per meter be
stretched when a force of 3.0 x 102 newtons is applied?
[0.5m]
F
x
x
F
F
x What does the
slope of each of the graphs represent?
Sketch a graph that represents an object that
does NOT obey Hooke’s Law.
[k; 1/k]
[any non-direct line is correct]
Dynamics Unit- Auxiliary Forces Sub-Unit
Regents Physics
3 | P a g e Gravitational Proportions
• Force of gravity is _______________________ proportional to mass.
• Force of gravity is _______________________ proportional to the square of
distance.
Gravitational Fields
• Objects with ________________________ produce gravitational fields.
• Field lines point inward from __________________________________.
Fg
r
- What is the effect on a gravitational system if…
o the distance between masses is doubled
o the distance between masses is tripled
o the distance between masses is halved
Fg
m
- What is the effect on a gravitational system if…
o one mass is doubled
o both masses are doubled
PRACTICE
1) Two masses are attracted by a force of 20 newtons.
a. What would the force between them be if both masses were tripled?
b. What would the force between them be if the distance separated them
were doubled?
2) An astronaut with a mass of 50 kilograms is standing on Earth’s surface.
a. Calculate his weight while on the Earth’s surface.
b. The astronaut moves to an altitude that is one Earth radius above the
surface of the Earth. Calculate his weight at this altitude. Is gravitational force increased or
decreased?
1. ¼ the distance
2. ½ one mass and ½ distance
3. 3x one mass
4. 2x distance
5. 3x both masses; 4x distance
What is the effect on a gravitational system if…
1. 3x both masses
2. 4x distance
3. 2x one mass and 3x the other
4. ½ distance and 2x one mass
Dynamics Unit- Auxiliary Forces Sub-Unit
Regents Physics
5 | P a g e
2.2.3 Circular Motion
Uniform Circular Motion - Velocity
• Objects in uniform circular motion have a constant speed and a constantly
CHANGING velocity – changing in direction but not magnitude.
• If velocity is always tangential then why do things move in circles?
Centripetal Force
• Inertia causes objects to travel ____________________________
• Paths can be bent by ____________________________________
• _______________________________________ bends an object’s path into a
circle – pulling toward the _______________________________
Review Questions
2.2.2 – Calculate the force of gravity between two masses. Determine the effect of changing variables on the force of gravity.
- Determine the gravitational force of attraction experienced by two 5.0
kilogram masses separared by a distance of 2.5 meters.
[2.7 E -10 N]
- Determine the gravitational force between the Earth and the Moon.
[1.99 E 20 N]
- A gravitational force F attracts two objects of mass M toward each other when
R meters apart.
o The force between the masses is changed to _________ if both
masses are changed to mass 2M.
o The force between the masses is changed to _________ if the
distance between the objects is chaned to 3R.
o If the system is changed as follows force F becomes…
M, 2M, R ______F
M, M, 2R ______F
2M, 2M, R/2 ______F
M, M, 4R _____F
3M, 2M, R/2 _____F
[4F, F/9, 2F, F/4, F, F/16, 24F]
the ‘d’ in this case is: ______________________________
the ‘t’ in this case is: ______________________________
t
d
v
avg
Velocity is
_____________________to the circle at all points
• As the Gravitron starts to spin, friction between your body and the ride starts you moving along.
• Once you are moving, your body wants to keep moving ____________________
but you can’t because the walls keep pushing you back toward the center of the ride.
• But you “feel” like you are being thrown outward! What is that called?
What is the sensation that you feel?
Centripetal Acceleration
• Centripetal force is a _______________________________________
o Causes __________________________________________
o In the ___________________________________________
Equation
• What is the centripetal acceleration of a toy ball on the end of a 1.44 meter long
string if it is moving at 12 meters per second?
Centripetal Force - Equations
Example #2
• What is the centripetal force acting on a 2000 kilogram airplane if it turns with a
radius of 1000 meters while moving at 300 meters per second?
Example #3
• Is it possible for a 1000 kilogram car to make a turn with a radius of 50 meters
while moving at 15 meters per second with rubber tires and on dry asphalt? • centrifugal (center fleeing) force
A ‘fictitious’ or ‘pseduo’ force that is experienced from INSIDE
a circular motion system
“Inertial” forces come from changes in motion that are perceived from the
INSIDE of a system – WHAT YOU FEEL
• centripetal (center seeking) force A true force that pushes or pulls an object toward the center of a circular path
“Real” forces are the actual pushes or pulls that affect motion as seen by an outside observer – WHAT ACTUALLY
Dynamics Unit- Auxiliary Forces Sub-Unit
Regents Physics
7 | P a g e PRACTICE
1) The Shooting Star carnival ride slings its cars around in a vertical circle. The speeds
and forces acting on the cars in this case are NOT constant as they are in uniform circular motion.
a. At which point in the ride with the rider experience a sensation of near
weightlessness?
b. At which point in the ride with riders experience the greatest amount of
force?
2) A rider and bicycle with a combined mass of 60 kilograms makes a turn with a radius
of 15 meters. The bike has rubber tires and makes the turn on dry concrete.
a. Calculate the amount of friction force acting on the bicycle tires.
b. Determine the maximum turning speed of the bicycle as it moves through
the turn.
3) Sketch the relationship between the variables for the following graphs.
4) Show that the mass of an object is not important in determining the minimum turning
radius, maximum turning speed, or minimum coefficient of friction in a system in which a vehicle makes a turn.
a. Write the equations for Fc and Ff and set them equal to each other.
b. Make a substitution for the normal force (how is it found?)
c. Show that mass cancels out in the equations!
A
B D
C
ac
v
ac
r
Fc
ac
Fc
- What is the minimum coefficient of friction needed for a 500 kilogram motorcycle to make a 20 meter radius turn if it is moving at 12 meters per second?
[0.73]
- A moon with a mass of 3.5 x 1020 kilograms orbits planet Hoth at a distance of
2.5 x 108 meters. Hoth has a mass of 6.6 x 1024 kilograms.
o Determine the gravitational force that planet Hoth and its moon
exert on one another.
[2.5 x 1018 N]
o Assuming that the gravitational force in this system is what provides
the centripetal force that makes the moon orbit Hoth, determine the acceleration of Hoth’s moon due to the gravitational field of Hoth.
[0.007 N/kg]
Review Questions
2.2.3 – Draw vectors to represent centripetal force; centripetal acceleration; and/or velocity in circular motion systems. Calculate speed, force, and acceleration in circtular motion systems. Determine an unknown parametr in a circular motion system in which centripetal force is generated by a particular type of force (Fg, Ff, etc) .
- Draw vectors to represent velocity, centripetal force, and centripetal
acceleration in each of the three systems shown below.
[v is tangent ac and Fc to center]
- A 3.5 kilogram object is swung in a circular path on the end of a 0.4 meter long
string. The object makes one trip around the circle every 0.2 seconds.
o Calculate the speed of the object (hint: v = d/t)
[12.6 m/s]
o Determine the centripetal acceleration of the object.
[397 m/s2]
o Calculate the centipetal force acting on the object.
