Dynamics of Uniform Circular Motion

CHAPTER 5Dynamics of Uniform Circular MotionUniform 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 vt is the tangential velocity of the object Uniform Circular Motion

Acceleration in Uniform Circular MotionAn object that moves at constant speed in a circular path is accelerating.This acceleration is known as Centripetal Acceleration (ac) and it is directed towards the center of the circular path.

2012 OpenStax CollegeWhy do we say an object going in a circle is accelerating?

How do we know the acceleration is toward the center of the circle?

3Acceleration in Uniform Circular MotionThis object traveling in a circular path also experiences a force that is also directed towards the center of the circular pathThis force is known as Centripetal Force (Fc)Therefore change in velocity of this object is also directed toward the center of the circular path

2012 OpenStax CollegeWhy do we say an object going in a circle is accelerating?

How do we know the acceleration is toward the center of the circle?

4Rotation and Centripetal AccelerationNote!!!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 AccelerationThe magnitude of the centripetal acceleration is given by the formula

Where :vt = tangential speed r = radius of the circle.The direction of the force is toward the center of the circle.

2012 OpenStax College6The 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 vt. 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 1vtFCBanked 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 curveCentripetal AccelerationThis formula agrees with our results.

Vertical Circular Motion

Example problem #114Trebuchet project:2-part projectBoth test gradesPart 1 (research and design) due 12/14Part 2 (launch and review) due _________ 20151,2 to 3 people in a teamOpportunities for extra credit points

Socrative: rm # phy2103Example problem #25.2 Centripetal AccelerationExample 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 CurvesExample 8: The Daytona 500

The turns at the Daytona International Speedway have a maximum radius of 316 m and are steeply banked at 31degrees. 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?

Class assignmentIf 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)Newtons Law Universal of GravitationnotesReview Tomorrow Free Response Test on ThursdayNewtons Universal Law of GravitationGravitational force is the mutual force of attraction between particles of matterThere is a gravitational force between any 2 objectsThe larger the objectthe more pull it hasFor example, there is a gravitational force between 2 pencilsSince they have the same mass, they have the same gravitational pullIf the objects are larger, they will have more gravitational forceOur gravitational force is extremely small (due to our small mass) relative to the earths gravitational force (which has a much larger mass) As a result, our gravitational force is cancelled out due to earths 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 forceIt depends on the distance between two objects andThe magnitude of the masses

Increase the distance-decrease the gravityGravitational 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 gravityInverse square of distance complete the chartExample problem #1Satellites 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 Satellitewrite 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 doesnt matter.

This assumes circular orbits.33Finding the speed of an Earth Satellite (Continued)Solving for v:

This analysis assumes a circular orbit, but it is generalizable to any orbit. 34Newtons CannonNewton 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, http://en.wikipedia.org/wiki/File:Newton_Cannon.svg35Gravitational 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

Satellites in Circular OrbitsExample #2: Orbital Speed of the Hubble Space Telescope

Determine the speed of the Hubble Space Telescope orbitingat a height of 598 km above the earths surface.

Satellites in Circular Orbits

Keplers contributions to astronomyUsed Tychos 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 planets period is proportional to the cube of its distance from the sun. T2 ~ r3

40Point out that Kepler and Galileo corresponded and were contemporaries.Keplers First and Second LawHere is an animated visualization of Keplers first two laws.

http://www.surendranath.org/Applets/Dynamics/Kepler/Kepler1Applet.html