Circular Motion: Gravitation Chapter 5

5-1 Kinematics of Uniform Circular Motion  Uniform circular motion is when an object moves in a circle at constant speed.  Examples  The magnitude of the velocity remains constant, but the direction continually changes so the object accelerates  Acceleration is the change in velocity which includes both speed and direction

Acceleration is defined as: Where Δv is the change in velocity during a short time interval Δt

5-1 Kinematics of Uniform Circular Motion Looking at the change in velocity in the limit that the time interval becomes infinitesimally small, we see that (5-1)

Centripetal Acceleration  Centripetal means center-pointing  Or radial acceleration because it is directed along the radius. aRaRaRaR

Rearrange:

 But, Δl/Δt is the linear speed, v, so

Summarize  An object moving in a circle of radius r at a constant speed v has an acceleration a R directed towards the center of the circle and the magnitude will be a R =v 2 /r

 Acceleration is directed towards the center, but which way is the velocity directed?  In which direction would the object go if the string broke?  Tangential to the circle.

Frequency and Period  Frequency is the number of revolutions per second  Period is the time for one revolution  Remember one revolution would be equal to the circumference of a circle  2πr so the speed around a circle can be written as

5-2 Dynamics of Uniform Circular Motion  According to Newton’s second law ΣF=ma  This is true of linear motion as well as circular motion  So  ΣF R =ma R = mv 2 r  So the net force must also be directed towards the center

 Centripetal force isn’t a new kind of force it just indicates the direction.  This force must be applied by another object on the object in circular motion.  String on a ball  Earth’s gravitation pull on a satellite

 There is no outward force acting on the circular objects.  Or centrifugal force.  It feels like it because when you swing a ball on a string you feel the “outward” pull, but that is the equal and opposite force the ball pulls on the string according to Newton’s second law.  If there was an outward force then when the string broke the ball would fly outward and not linearly.

Vertical Circle

5-3 Highway Curves  Banked and unbanked

5-5 Centrifugation  Used to separate materials  Particles have a tendency to resist change in motion (Inertia) so they go towards the bottom of test tube  Causes rapid sedimentation

5-6 Newton’s Law of Gravitation  Newton’s wondered about the forces that kept the circular orbit of the Moon around the Earth  Also knowing that falling object accelerate they must have a force acting on them also  When an object has a force exerted on it, that force must be exerted by another object  Newton concluded that that other object is Earth  This is even the case with the moon  In fact, all object have a force of attraction between them

 Newton figured that the force decrease as the distance between object increased  Where r is the distance from center of object to center of object

 He also determined that

Conclusion  Every particle in the universe attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distances between them. This force acts along the line joining the two particles.

Where G=6.67 x Nm 2 /kg 2

5-7 Gravity Near the Earth’s Surface

5-9 Kepler’s Laws  Before Newton there was German Johannes Kepler  He worked out a detailed description of the motion of the planets around the sun.  Now known as Kepler’s Laws

Summary  1 st law - The path of each planet about the Sun is an ellipse with the Sun at one focus  2 nd law – Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal area in equal time periods.  3 rd law – The ratio of the squares of the period of any two planets revolving about the Sun is equal to the ratio of the cubes of their mean distances from the Sun

 Kepler analyzed data to arrive at his results  50 years later Newton derived Kepler’s law mathematically from the Universal gravitation and the laws of motion.