radian
primary unit used in measuring angles. it is an alternative to degrees;
hypotheticals
[1] write the equation to be used
[2] input into the equation any changed factors for any variable of interest
[3] do the math of the equation and combine [meaning add || subtract || multiply || divide] all numbers that were not part of the original mathe
1 radian equals how many degrees?
how do you orientate yourself to the radians from degrees
1 radian approximately equals 57.29 degrees.
we know that pi = 180 degrees like 2 pi = 360 degrees. Pi has a value of 3.14. So 3.14 radians =180 degrees. Pi/2(1.57 radians) = 90 degrees; pi/4(0.785 radians) = 45 degrees;
uniform circular motion
motion of an object moving in a circle at constant speed.
radial acceleration
acceleration directed along the radius that points to the center for an object traveling along a circular path. radial and centripetal acceleration are synonyms.
angular position
concept is analogous to linear position variable x; the measurement of an angle [in radians] at a particular instant in time
frequency
the number of revolutions(2pi) per second
period
Time required for 1 complete revolution
angular displacement
(1)the difference in an objects final and initial angle measured in radians;
(2) the angle measures in radians through which a body rotates in a circular path;
variable for angular speed
average angular speed
instantaneous angular speed
when is angular speed positive and negative?
angular speed is positive when theta is increasing (moving counterclockwise) and negative when theta is decreasing (moving clockwise)
when are the average angular speed and instantaneous angular speed equal?
when angular speed is constant
What is true about a rigid object rotating about a fixed axis?
Every portion of the object has the same angular speed and acceleration; however the tangential speed and acceleration take different values depending on the distance from the given point of rotation
variable of angular acceleration
average angular acceleration
instantaneous angular acceleration
arc length
the length or distance around part of a circle
the length or distance around a portion of a circle.
the value will be some multiple of the r;
tangential speed
[1] the linear (non-rotational) speed of an object following a circular path. when objects move they always want to move in a straight line. If you were to remove the force causing the object to follow the circular path, the tangential speed is the speed
proof for tangential speed
proof for angular position
what is the synonym for angular speed?
rotational speed
what is the difference between rotational and tangential speed?
tangential (linear) speed increases as that point moves outward away from the center of rotation; the angular speed is the same at every point (no matter how outward(or toward) that point is from the center of rotation).
Caveat tangential speed
the equation for tangential speed only works when the rotational speed is in rad/s.
tangential acceleration
a measure of how the tangential (linear) speed of an object changes over time.
REMINDER tangential acceleration
tangential acceleration will usually be zero or something we don't particularly worry about unless directly asked.
The following are conditions where TA will not be zero. If the velocity speeds(due to changes in angular speed) or if there is a change in t
Proof for tangential acceleration
*what's the difference between record payers(phonographs) and CD players?
Record players rotate at constant angular speed while CD players work by picking up the sound at a constant tangential speed. As a result the record rotates constantly while a CD will have to speed up or slow down it's rotation depending on where the lase
centripetal acceleration
[1] the acceleration experienced by an object traveling in a circular path at constant speed. This acceleration points towards the center of the circle[circular path].
[2] acceleration that causes an object to follow a circular path. the only way an accel
what's the connection between centripetal acceleration and arc length?
similar triangles
the triangle for centripetal acceleration(change in tangential velocity) has the same angular displacement as the triangle for arc length.
the triangle between velocity vectors and the triangle between arc length radii have the same angl
proof of centripetal acceleration
(triangles)
proof of centripetal acceleration
There is no such thing as centrifugal force
Ex :Ball on a String (picture)
There is no such thing as centrifugal force
Ex :Ball on a String (explanation)
If you have ever done this yourself, you know that you feel a force pulling outward on your hand. This is not what's happening. To keep the ball moving in a circle you pull inwardly on the string, which in turn exerts the force on the ball. The ball exert
what is centripetal force?
Force that causes an object to follow a circular path. when object move they natural want to move in a straight line or follow a linear path. So centripetal force is the force that forces an object traveling in straight path to move in a circular path. Wi
How exactly would the friction force between a car on a flat circular track be an example off centripetal force?
One must understand that a car moves because of the static friction force between the wheels and the ground. If there was no static friction force the car's wheels would spin in place or the ground would be moving underneath it. As a result the static fri
Why does a ball move in a circle?
Because of inertia(Newton's first law) the ball wants to move in a straight line; However by exerting a radial force the string prevents the motion along a straight line and forces the ball to curve and follow the circular path. This tension force in the
Is the gravity we experience on the face of the earth a centripetal force?
No. gravity is not acting centripetally for us because while gravity is acting it is not acting to keep us traveling in a circular path around the earth. Nor are we traveling with some velocity independent of being in contact with the earth's rotation.
Wi
Centrifugal force
literally means center fleeing force. From the point of view of the object there seems to be some force that wants the object to fly away outward from the center of the object. however this "feeling" is fictitious and is why centrifugal forces are conside
True or False
If a person jumps straight up on a rotating space station he will come straight down.
True... As long as the space station continues to rotate at a constant speed, he will come straight down. This is so because before he jumped he was moving at the same speed relative to the space station and so when he comes down he and the space station
How is artificial gravity created in a rotating space station?
(1) artificial gravity is created by the normal force of the space station on the person. The normal force prevent the person from flying off tangential. Normal force points towards the center of the station. To the person this force seems centrifugal pus
Solving the TILTED AXIS
(1)Draw the freebody diagram with normal axis.(think PLUS sign)
(2) Imagine rotating the axis so as to line up with the tilted axis frame of reference
(3) redraw angles on new tilted axis.
REMINDER FBD for Newton's 2nd Law
Reaction Forces are not included in an FBD.
why is normal force zero for a roller coaster that just makes it to the top?
Because the object is just about to free fall down to the ground. when an object free falls there is no normal force acting only mg.
observation(1) is that if the roller coaster just makes it over the top the velocity of the roller coaster must have slowe
IN what direction would normal force be pointing for an object traveling at a constant speed at the bottom of the rollercoaster ride? At the top? Half way Point? explain each. which value of normal force would be smaller and explain why?
Both would be upwards. At the bottom Normal force prevents riders from falling through cart. At the top normal force prevents riders from falling out. at the midway point normal force points in the at and angle upwards and towards the center. Normal force
Newton's 2nd law at the top of a roller coaster
(1)Normal force is zero if the rollercoaster just makes it over
(2) At the top, centripetal acceleration has a negative value like mg because it's pointing down
REMINDER: Banking Problem
There is a static friction force as well as all the other forces. It acts upwards to prevent the car from sliding down.
Universal Gravitation Constant
Newton's law of Gravitation
what is an alternative equation for the velocity of an object in circular motion
Gauss's law
the gravitational force exerted by a uniform sphere on a particle outside the sphere is the same as the force exerted by a uniform sphere were it concentrated at it's center;
this law also applied to electric fields.
geosynchronous satellite
a satellite that stays above the same point on the equator of the Earth
What is apparent weightlessness?
weightlessness is when the force of Gravitational attraction is zero, where acceleration due to gravity is equal to zero. Apparent weightlessness is where gravity act so weight equals mg but the apparent weight(normal force) is zero because the object is
why does a person orbiting the earth in a satellite feel weightless?
The weightlessness experienced by a person in a satellite is synonymous to the weightlessness experienced by a person in a free-falling elevator. To understand why this is so we must first accept the fact that the satellite[ & person] is falling towards t
what do object that are orbiting experience when they are rotating due to gravity?
the feely apparent weightlessness.
*What is the difference between artificial gravity and an object rotating due to gravity? Due both feel weightless? Why or why Not?
the difference between artificial gravity and a person in a satellite rotating due to gravity is the following:
For the person in the satellite both the satellite and the person are moving with a tangential speed.
For the person in the rotating space stat
the new GPE
The convention about positive and negative GPE
As before, gravitational potential energy is a property of a system, in this case the object of mass m and Earth. The equation for new GPE is valid for the special case where the zero level for potential energy is at an infinite distance from the center o
Conditions for determining Escape Velocity launch speed
Initial: KE equation for Escape and GPE at earth radius
Final: final velocity is 0 and Final GPE is zero because you are infinitely far away or 1/0
Condition for determining Launch velocity and orbital velocity for rotation
Initial: KE equation and GPE at earth's radius:
Final: KE where velocity is calculated via equation for centripetal acceleration. gravitational Force at orbital height; GPE for height above earth's surface.
Escape Speed
value for Earth
the speed necessary for an object leaving the earth to leave earth and not be pulled back due to gravity.
Value for earth is 11.2 km/s or 25000 mph
Escape Speed proof
geocentric model
the model developed by Claudius Ptolemy that describes the Earth as the center of the universe
heliocentric model
the model showing the earth and other planets revolve in circular orbits around the sun.
KEPLER's first Law
All planets move in elliptical orbits about the sun where the Sun is in one focus and nothing is in the other;
Because the orbit is an ellipse the distance from the Sun to the planet continuously changes; which will affect centripetal acceleration and spe
focal point
(focus)
one of two points internal to an ellipse necessary to create the elliptical path of orbit.
KEPLER's 2nd Law
Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal periods of time.
Equal areas for Equal time intervals. The object in rotation will move faster when its closer to the sun and slower when it's fur
KEPLER's Third Law
(1) The square of the orbital period of any planet is proportional to the cube of the average distance from the planet to the sun
(2) The ratio of the squares of the periods of any two planets revolving about the Sun is equal to the ratio of the cubes of
KEPLER's Third Law Equation
(two versions)
KEPLER's Third Law proof (1planet)
KEPLER's Third Law proof (2 planet)
when can we use AU and the simplification of Kepler's third law?
AU stands for Astronomical Units. AN AU is the measured distance from the earth to the Sun which would substitute in for the semimajor axis. This unit can be used in placed of the radius in Kepler's third law equation and when the Period of Time is measur
semi-major axis:
half the longest distance across the elliptical orbit. it is considered to be the average distance from the planet to the sun. basically it's the "radius of the longest distance of an ellipse
[PICTURE]
QQ 7.6 A race track is constructed such that two arcs of radius 80 m at (A) and 40 m at (b) are joined by two stretches of straight track. In a particular trial run, a driver travels at a constant speed of 50 m/s for one complete lap.
1. The rat
[ANSWERS]
QQ 7.6 A race track is constructed such that two arcs of radius 80 m at (A) and 40 m at (b) are joined by two stretches of straight track. In a particular trial run, a driver travels at a constant speed of 50 m/s for one complete lap.
1. The rat
1.(v) 2. (i) 3.(ii)
QQ7.7 An object moves in a circular path with constant speed v. Which of the following statements is true concerning the object? (a) Its velocity is constant, but its acceleration is changing. (b) Its acceleration is constant, but its velocity is changing
(c) It's velocity is changing due to the object following the curve and the direction changing. Yes the speed is not changing but the velocity changes due to change in direction. acceleration is changing because of the direction of centripetal acceleratio
CQ1. In a race like the Indianapolis 500, a driver circles the track counterclockwise and feels his head pulled toward one shoulder. To relieve his neck muscles from having to hold his head erect, the driver fastens a strap to one wall of the car and the
(a) the head leans towards his outside shoulder (towards the outside of the curve)
(b) tension in the neck muscles produce the centripetal acceleration
(c) the tension in the strap provides the centripetal acceleration allowing the neck muscle to relax
CQ3. An object executes circular motion with a constant speed whenever a net force of constant magnitude acts perpendicular to its velocity. What happens to the speed if the force is not perpendicular to the velocity?
An object can move in a circle even if the total force on it is not perpendicular to it's velocity, but then its speed will change. Resolve the total force into an inward radial component and perpendicular tangential component. If the tangential force act
CQ4. Explain why Earth is not spherical in shape, but bulges at the equator.
The earth is not spherical, it bulges at the equator. Earth is not a solid object and spins fast at the equator since the equator is furthest from the axis of rotation. Therefore the centripetal force and linear speed is greatest there. Each small portion
CQ5 If a car's wheels are replaced with wheels of greater diameter, will the reading of the speedometer change? Explain.
the speedometer measure angular speed and calculates speed based on the radius of the tire. With the bigger wheels angular speed will be the same but however the linear speed will be higher because of the increased radius. the speedometer will read a valu
CQ6. At night, you are farther away from the Sun than during the day. What's more, the force exerted by the Sun on you is downward into Earth at night, and upward into the sky during the day. If you had a sensitive enough bathroom scale, would you appear
during the daytime you will way less because the attraction between you and the sun works against the attraction force between you and the earth. At night the attraction between you and the sun works in the same direction as you and the earth. Normal Forc
CQ7. Correct the following statement: "The race car rounds the turn at a constant velocity of 90 miles per hour.
constant "SPEED"
velocity is changing because the direction is constantly changing though its magnitude(speed) doesn't
CQ8. Why does an astronaut in a spacecraft orbiting Earth experience a feeling of weightlessness?
(1) When a person is free falling, there is no normal force acting and as a result the person feels weightless.
(2) The acceleration of the person and that of the satellite are both equal to g. Therefore the person feels weightless with respect to satelli
CQ9 Explain why it's easier to determine the mass of a planet when it has a moon
the gravitational force exerted on the moon by the planet produces a centripetal acceleration and as a result will orbit the planet in a circular path. For the 2nd law Force the centripetal force equals to the gravitational force of attraction. Both the r
CQ10. Because of Earth's rotation about its axis, you weigh slightly less at the equator than at the poles. Why?
Normally the net force acting on you is zero so N= mg. At the equator net force equals to some centripetal acceleration(ac) so N = Mg- Mac
CQ12. Describe the path of a moving object in the event that the object's acceleration is constant in magnitude at all times and (a) perpendicular to its velocity; (b) parallel to its velocity.
The path will be circular. This is hard to explain but because the acceleration is always perpendicular to the velocity at each new instant. In the old problem the acceleration is perpendicular to the initial velocity not all the new velocities.
CQ14 The orbital planes of all the planets must pass through the center of the Sun. Why?
...
CQ18.A satellite in orbit is not truly traveling through a vacuum�it's moving through very thin air. Does the resulting air friction cause the satellite to slow down?
...
Solving Centripetal Problems
(1) You can solve the problems radially via tilted axis, Fx, Fy. or you can break the centripetal acceleration into angles via trig
(2) all that matters when solving these problems is the circular motion. the radius from rotation needs to be resolved like