06_Rotational Motion & Gravitation
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 their mean distances from the Sun.
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.
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
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.
period
Time required for 1 complete revolution
angular position
concept is analogous to linear position variable x; the measurement of an angle [in radians] at a particular instant in time
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
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"
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 satellite. The astronaut has a feeling of weightlessness similar to this because the astronaut and satellite both have an acceleration equal to gravity. In a way they both are falling towards earth but they never reach the earth because the tangential velocity pulls them away. the astronaut satellite is similar to the scenario of a person in an elevator that is free-falling. the person inside the freefalling elevator is moving at an acceleration equal to gravity just as the elevator is. Both the freefalling person and elevator experience a feeling of weightlessness. Because the elevator is inanimate we can't ask it how it feels but if it could speak it would say it feels weightless or as if it were in zero gravity.
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 a yo-yo being rotated, a person rotating from a position above or below the equator, or a teather ball attached to a pole. (3) don't confuse work/energy with rotational speed. (4) for banking problems there are frictional forces present acting which may have a partial affect on centripetal motion (5) For banking problems the object wants to naturally slide down so static friction for will act upwards (6) if an object is described as barely making it over the top the velocity is slowing down and Normal force is zero or slightly greater than zero.
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 pushing him down to the wall( or ground) but the normal force is correcting for his tangential velocity and forcing him to follow the circular path of rotation. (2) In artificial gravity the person feels Earth normal force. Since the station walls(or from the person's perspective the station's floor) is rotating at a certain speed so the person standing on the wall moves with a certain tangential speed . The person want's to fly off on a straight line tangent with the velocity given by the rotating station. However, the person is not allowed to fly off and follows the circular path because the wall(floor) is following a circular path.
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
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;
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 other to his helmet. The length of the strap is adjusted to keep his head vertical. (a) Which shoulder does his head tend to lean toward? (b) What force or forces produce the centripetal acceleration when there is no strap? (c) What force or forces do so when there is a strap?
(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
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) Both its velocity and acceleration are changing. (d) Its velocity and acceleration remain constant.
(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 acceleration as the object rotates.
CQ14 The orbital planes of all the planets must pass through the center of the Sun. Why?
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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?
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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;
[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 ratio of the tangential acceleration at (A) to that at (B) is (i) one half (ii) one quarter (iii) 2 (iv) 4 (v) the tangential acceleration is zero at both points 2. The ratio of the centripetal acceleration at (A) to that at (B) is (i) one half (ii) one quarter (iii) 2 (iv) 4 (v)the tangential acceleration is zero at both points 3 The angular speed is greatest at (A), (B) or it's equal at both.
1.(v) 2. (i) 3.(ii)
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 measure in earth years.
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 speed of the planet in orbit.
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 acts in the forward direction, the object will speed up and if the tangential force acts backward, the object will slow down.
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 of Earth. Recall that GPE associated with an object is nothing more than the negative of the work done by the force of gravity in moving the object. If an object falls under the force of gravity from a great distance (effectively infinity), the change in GPE is negative, which corresponds to a positive amount of gravitational work done on the system. This positive work is equal to the(also positive) change in kinetic energy. see example on page 210 SFV
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 string changes the velocity vectors direction. If the force where to vanish the object would immediately leave its circular path and move along a straight line tangent to the circle at the point where the force vanished.
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 slowed down. Observation(2) there must be some negative tangential acceleration in addition to the centripetal acceleration since the object slowed down.
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 points radially horizontal to keep object on circular path and upwards to prevent riders from crashing through their seats. Normal force would be smallest at the top
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 further away so that the swept out areas are equal.
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
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. Without centripetal force an object would move in a straight line. Centripetal force is not some new force that acts upon an object. its a descriptor of a force that already exist and is acting on the system towards the center (axis of rotation). It forces an object to follow a circular path instead of flying of tangentially Ex the gravitational force acting on a satellite rotating and the tension force in the string of a rotating yo-yo are described as centripetal forces because they are acting on the object keeping it in circular motion and are center seeking in nature.
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 exerts an equal and opposite force and this is the force your hand feels. If centrifugal force existed the ball would fly outwardly. But it doesn't the ball flies off tangentially in the direction of the velocity it had at the moment it was released.
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.
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. With that said gravity would be acting centripetally for a satellite in orbit around the earth. The satellite is traveling with some velocity(independent of the earth) and the gravitational force is keeping the object traveling with it's velocity in circular orbit and preventing it from flying off.
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
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 friction force propels the car forward opposite to the direction of tire rotation. Since the car is turning in a circular path it is the static friction force that provides the centripetal acceleration.
REMINDER FBD for Newton's 2nd Law
Reaction Forces are not included in an FBD.
*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 laser is above the track. Said another way the angular speed of the CD must vary according to the radial position of the laser.
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 at the equator tends to fly off on a tangent and a bulge is created at the equator.
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.
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 the Earth. Just like an elevator's force due to gravity points towards the center of the earth, so too does the satellite's gravitational force point. In the elevator, the person feels apparent weightlessness because he and the elevator experience the same acceleration. The person in the satellite feels the same weightlessness because he & the satellite also experience the same acceleration towards the earth.
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 will have travelled the same rotational distance so relative to the space station he will come straight down.
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 acceleration can cause an object to go in a circle is if it acts in a direction perpendicular to the objects motion[velocity]. if acceleration acts in the same direction the object speeds up' if it acts in the opposite direction it slows down; but if it acts in the perpendicular direction the object travels in a circle. [3] the only way an object can move in a circular path is if some force acts on it that prevents it from going in a straight line. any force that does this is called a centripetal force. centripetal acceleration is the acceleration that causes an object to go in a circle. unlike causes it to expereince an acceleration perpendicular to the direction the object moves...when given the option most objects want to move in a straight line.. but in order for something to move in a circle their most be something that prevents it from doing so.
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 that object would move with. [2] the linear (non-rotational) speed of an object forced to follow a circular path. [3]the speed at which an object moving in a circular path would move if the force acting on the particle towards the center were to be eliminated.
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 mathematical equation. [4] factor out the numbers that were not part of the original equation.
tangential acceleration
a measure of how the tangential (linear) speed of an object changes over time.
geosynchronous satellite
a satellite that stays above the same point on the equator of the Earth
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.
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)
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 to weigh more at night than during the day?
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 Force
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 considered fictitious forces. Your not actually being pushed away from the center though it feels like it. what your actually feeling is a course correction of your path from straight line to circular. Said another way, the inertia from your velocity wants you to keep going straight but the centripetal force makes you follow the circular path. Centrifugal force is a fake and fictional force just like the force you feel when your are in an accelerating car. When a car accelerates you get pushed back into your seat. Nothing is actually pushing you back into your seat. Rather the seat is pushing you forward and you fall backwards.
uniform circular motion
motion of an object moving in a circle at constant speed.
focal point (focus)
one of two points internal to an ellipse necessary to create the elliptical path of orbit.
radian
primary unit used in measuring angles. it is an alternative to degrees;
what is the synonym for angular speed?
rotational speed
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 angle.
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).
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 the radius during rotation.
*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 station, the person is moving with a tangential speed but the space station is not moving at all. It's walls are rotating at the same speed as the person but the space station itself is not moving. so when the person in the space station put his feet on the floor, he experiences a contact normal force. When the person in the satellite puts his feet on the floor, no contact force is created because the satellite floor is moving at the same speed as the man. this is equivalent too being in an elevator that is freefalling. your feet don't remain on the floor because you are falling at the same speed as the elevator. (like in Inception) the person and the satellite are both moving where in artificial gravity technically the person is moving but the station is not(it is rotating
Caveat tangential speed
the equation for tangential speed only works when the rotational speed is in rad/s.
what do object that are orbiting experience when they are rotating due to gravity?
the feely apparent weightlessness.
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.
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 radius and tangential velocity can be observed from the planet and the mass of the planet can be determined
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;
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.
frequency
the number of revolutions(2pi) per second
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
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 value lower than what it actually is.
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 accelerating in free fall where the acceleration equals g.
when are the average angular speed and instantaneous angular speed equal?
when angular speed is constant