Download Test Hints – gravity

AP Physics – Little AP Test Review Helper – Gravity/Torque
We studied the universal law of gravity and centripetal force together.
Gravity
 Force that acts between two bodies that have mass. This force is given by the equation for
gravity, which is called the universal law of gravity.
FG  
Gm1m2
r2
N m
G  6.67 x 1011
2
G is the universal gravity constant. It is given by:
2
kg
Both bodies attract one another with the same magnitude force. A ball falls to the earth because of
the force of gravity, so the ball falls. The earth also falls towards the ball, but the motion is so small
that you don’t notice it.
This is an inverse square function. Force of gravity decreases by the square of the distance. r is the
distance between the center of mass for each of the objects.
The acceleration of gravity on a planet can be found by setting the equation of weight, w  mg ,
equal to the universal law of gravity.
m1 g 
Gm1m2
r2
g
Gm2
where m2 is the mass of the planet.
r2
Of course, you remember Kepler’s 3 law of planetary motion:
Planets move in ellipses with the sun at one focus.
Planets sweep equal areas in equal times (speeding up in close to the sun.)
R3/T2 is the same for all planets (later explained when Newton set the Gravity = Centripetal Force)
Circular Motion:
T
1
f
Period:
The time for one revolution.
Frequency:
How often a repeating event happens. For circular motion it would be measured in
revolutions per second.
Instantaneous velocity:
Speed:
f is frequency.
The actual velocity (speed and direction) at any instant of time for a body that is
traveling along a circular path. Its direction is tangent to its path. Also known
as tangential velocity.
Magnitude of instantaneous velocity. Also known as linear speed or tangential speed.
The speed is equal to the circumference of the circle divided by the time to make one
revolution. v 
x
t

2 r
T
1
Centripetal Acceleration: Equation provided for AP Test.
v2
ac 
r
Centripetal means “center seeking”.
Direction is always towards center of circle.
Centripetal Force:
Force that changes direction of instantaneous velocity for a body that is traveling
in a circle.
Equation is not given, you have to derive it using centripetal acceleration and second law:
v2
ac 
r
 F  FNet  ma
 v2 
F  m 
 r 
mv 2
F
r
v
ac
ac
ac
v
v
Objects want to travel in straight line path – first law, but centripetal force changes the direction
of the velocity, causing them to follow a circular path. Since object is pushed or pulled toward
center, there is a feeling of being forced away from the center to the third law. This sensation is
called the centrifugal force. Centrifugal force is a fictitious force.
Centripetal force is always caused by some real force; tension in a string, friction, gravity,
magnetic force, &tc.
Problem gouge:
1. Think centripetal force if the object is going in a circle, accelerating towards the center, or changing
direction.
2. In circular motion, the centripetal force is the sum of the forces acting on the object, so you can set it
equal to the sum of all forces. FC  F If one of the forces is a tension and the other force is the
weight of the object, you get; FC  t  mg . You can then substitute for the centripetal force;
FC  t  mg
mv 2
 t  mg
r
You can then solve for whatever you want.
2
3. To find the minimum speed for an object to make a vertical circle, set the centripetal force equal to the
weight of the object:
mv2
 mg
r
FC  mg
v  rg
4. Finding the orbital velocity of a satellite. Set the centripetal force equal to the gravitational force.
m2v 2
FC 
r
v2 
Gm1
r
F G
v
m1m2
r2
Gm1
r
mm
m2v 2
G 1 2
r
r2
so
Gm
r
v
so
The mass is that of the body being orbited, like the sun for a planet or the earth for the moon, &tc.
5. Finding the orbital period of a satellite. Solve the velocity equation for time:
v
x
t
t
x
v
x  2 r
so t 
2 r
v
Gm
r
The velocity of a satellite we already know - v 
2 r
t
Gm
r
Torque 
t
2
2
2  r 2


Gm
r
2
2  r 3


Gm
 2 2 r 3
t
Gm
plug it in:
r3
t  2
Gm
  rF sin
The applied force that causes rotation:
So there is excess torque in either the clockwise or counterclockwise direction. This will cause the
object to rotate.
1. As always, ask what the object is doing. Is it rotating or is it standing still?
2. Set the direction of motion as positive. It will either rotate clockwise or counterclockwise. If you
pick the wrong direction your final answer will be negative, telling you that you did thing in
reverse. But, the answer will be correct nonetheless. If it is not moving pick one direction to be
positive, it really doesn’t matter. But the other must be negative, so that the torque cancels.
3. Identify the sum of torque equation.
   clockwsise   counterclockwsise
or
   counterclockwsise   clockwsise
3
4. Substitute the relevant force equations and solve (examples assume clockwise was positive direction)
Rotating you get some + / Not Rotating
    rF sin clockwsise    rF sin counterclockwsise
0    rF sin clockwsise    rF sin counterclockwsise
Angular momentum: Depends on mass (like regular momentum) and it also depends on mass
distribution. As an ice skater brings their arms closer to the body they begin to spin faster, since the
mass has a shorter distance to travel.
Angular momentum is conserved. The radius gets smaller, but angular velocity increases (vice versa as
the skater moves arms outward). A galaxy, solar system, star, or planet forms from a larger cloud of
dust. As the cloud is pulled together by gravity its radius shrinks. So the angular velocity must
increase. These objects all begin to spin faster and faster. That is why we have day and night.
Rotational analogues
And so on…


t
Angular acceleration (rad/s 2 )
4