CHAPTER 5

Uniform circular motion is the motion of an object traveling at a constant speed on a circular path. If T (period) is the time it takes for an object to travel once around a circle, of radius, r, then the velocity of the object is given by: Where v t is the tangential velocity of the object Uniform Circular Motion

Acceleration in Uniform Circular Motion An object that moves at constant speed in a circular path is accelerating. This acceleration is known as Centripetal Acceleration (a c ) and it is directed towards the center of the circular path. © 2012 OpenStax College

Acceleration in Uniform Circular Motion This object traveling in a circular path also experiences a force that is also directed towards the center of the circular path This force is known as Centripetal Force (F c ) Therefore change in velocity of this object is also directed toward the center of the circular path © 2012 OpenStax College

Rotation and Centripetal Acceleration Note!!! If an object is rotating about a fixed axis, even at a constant speed, every point in that object is undergoing a centripetal acceleration as well.

Centripetal Acceleration The magnitude of the centripetal acceleration is given by the formula Where : v t = tangential speed r = radius of the circle. The direction of the force is toward the center of the circle. © 2012 OpenStax College

The force causing constant circular motion is given by: This force is not a new force. It just tells us the amount of force that must be provided, by a tension, gravity, etc., in order for an object to move in a circle. Centripetal Force © 2012 OpenStax College

A pail of water is swing in a circular path of radius r and a speed at the top of v t. Find the force exerted by the pail on the water. Find the minimum speed with which the bucket can be swung and still have the water remain in the pail at the top. What is the normal force on the water at the bottom of the swing if the pail is moving with this same speed? Circular Motion Example 1 vtvt FCFC

Banked Curves: © 2012 OpenStax College

Banked Curves

Putting the equations together: ***This equation can be used to calculate the speed of an object traveling on a banked curve

Centripetal Acceleration This formula agrees with our results.

Vertical Circular Motion

Example problem #1

Trebuchet project: 2-part project Both test grades Part 1 (research and design) due 12/14 Part 2 (launch and review) due _________ ,2 to 3 people in a team Opportunities for extra credit points Socrative: rm # phy2103

Example problem #2

5.2 Centripetal Acceleration Example 3: The Effect of Radius on Centripetal Acceleration The bobsled track contains turns with radii of 33 m and 24 m. Find the centripetal acceleration at each turn for a speed of 34 m/s. Express answers as multiples of

5.2 Centripetal Acceleration

5.4 Banked Curves Example 8: The Daytona 500 The turns at the Daytona International Speedway have a maximum radius of 316 m and are steeply banked at 31 degrees. Suppose these turns were frictionless. At what speed would the cars have to travel around them in order to remain on the track?

Circular Motion Example 3 What is the minimum speed the cars must travel to go around a loop of diameter d?

If you did not get the chance to complete your lab yesterday, you may work on it at this time (make sure you have completed these notes) Review Tomorrow  Free Response Test on Thursday

Newton’s Universal Law of Gravitation Gravitational force is the mutual force of attraction between particles of matter There is a gravitational force between any 2 objects The larger the object…the more pull it has For example, there is a gravitational force between 2 pencils Since they have the same mass, they have the same gravitational pull

If the objects are larger, they will have more gravitational force Our gravitational force is extremely small (due to our small mass) relative to the earth’s gravitational force (which has a much larger mass) As a result, our gravitational force is cancelled out due to earth’s larger gravitational force

Gravitational force is the force that keeps planets in orbit and keeps them from coasting off in a straight line Gravitational force is an attractive force It depends on the distance between two objects and The magnitude of the masses

Increase the distance-decrease the gravity Gravitational force is directly proportional to the product of the two masses involved Gravitational force is inversely proportional to the square of the distance of separation

Increase the distance-decrease the gravity

Inverse square of distance – complete the chart Original distance New distance reduceinverseSquare Change in F g 1020¼ x 20104x

Example problem #1

Satellites in Circular Orbits There is only one speed that a satellite can have if the satellite is to remain in an orbit with a fixed radius.

Finding the speed of an Earth Satellite write these equations at the bottom of your notes!! Centripetal acceleration: Centripetal force: The force is the force of Gravity The mass of the satellite divides out, so it doesn’t matter.

Finding the speed of an Earth Satellite (Continued) Solving for v:

Newton’s Cannon Newton compared the motion of a falling object to the motion of the Moon. An object in orbit is actually falling toward the Earth, at just the same rate that the Earth curves away from it. Source: Brian Brondel, Newton Cannon.svg, Wikimedia Commons,

Gravitational Field Strength of a Point Mass (or Spherically Symmetric Mass) write these equations at the bottom of your notes!! The gravity field around a point mass, or a spherically symmetric mass, depends only on the mass and the distance away from the center.

Satellites in Circular Orbits

Example #2: Orbital Speed of the Hubble Space Telescope Determine the speed of the Hubble Space Telescope orbiting at a height of 598 km above the earth’s surface.

Satellites in Circular Orbits

Kepler’s contributions to astronomy Used Tycho’s data to formulate three laws of planetary motion. The planets move in elliptical orbits with the Sun at one center of the ellipse. The planets trace out equal areas in equal times. The square of a planet’s period is proportional to the cube of its distance from the sun. T 2 ~ r 3

Kepler’s First and Second Law Here is an animated visualization of Kepler’s first two laws. html