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Circular motion

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Have you ever been to an amusement park?

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Fig. 05.19

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Why somebody needs to know about circular motion Otherwise we wouldnt go on those rides? Space station, satellites, live from overseas Planets, Comets and Moons

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Uniform Circular Motion Velocity is a vector, it has both direction and magnitude Simplest case: direction constant, magnitude changes. Linear Motion We will now study another simple case: magnitude is constant and direction continually changes. Uniform Circular Motion

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7 Circular Motion X θ = 0 y i f How do we locate something on a circle? Give its angular position θ What is the location of 90 0 or π/ 2

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8 Circular Motion is the angular position. Angular displacement : Note : angles measured Clockwise (CW) are negative and angles measured (CCW) are positive. is measured in radians. 2 radians = 360 = 1 revolution x y i f

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What is a radian? C =2πr = (total angle of one rotation) x r Radian measure relates displacement along circumference to angular displacement

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10 Circular Motion X θ = 0 y i f How do we locate something on a circle? Give its angular position θ What is the location of 90 0 or π/ 2 From θ =0 to θ = 90 0 we have gone ¼ of way around ¼ of 2π =π/2

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11 x y i f r arclength = s = r is a ratio of two lengths; it is a dimensionless ratio! This is a radian measure of angle If we go all the way round s =2πr and Δθ =2 π

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Question A child runs around the circular ledge of a fountain which has a diameter of 6.5 m. After the child has run 9.8 m, her angular displacement is A)0.48 rad. B)6.0 rad. C)1.5 rad. D)3.0 rad.

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13 If we move around a circle we change θ. How fast does θ change? The average and instantaneous angular velocities ω are: is measured in rads/sec.

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v and ω For circular motionHow far do you travel during Δt if you are spinning at an angular velocity ω? How far? s = Δθ r = ω Δt r How fast? (speed) v = s/Δt =ωr v =ωr

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Question A child runs around the circular ledge of a fountain which has a diameter of 6.5 m. If it takes her 72 seconds to run all the way around, her angular velocity is A)0.087 rad/s B)0.57 m/s C)0.17 rad/s D)0.044 rad/s

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Question A CD makes one complete revolution every tenth of a second. The angular velocity of point 4 is: A)the same as for pt 1. B)twice that of pt 2. C)half that of pt 2. D)1/4 that of pt 1. E)four times that of pt 1.

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Question A CD makes one complete revolution every tenth of a second. Which has the largest linear (tangential) velocity? A)point 1 B)point 2 C)point 3 D)point 4

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18 The time it takes to go one time around a closed path is called the period (T). Comparing to v = r : ω = v/r f is called the frequency, the number of revolutions (or cycles) per second. f = 1/T

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19 Centripetal Acceleration Here, v 0. The direction of v is changing. If v 0, then a 0. The net force cannot be zero. Consider an object moving in a circular path of radius r at constant speed. x y v v v v

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20 Conclusion: to move in a circular path, an object must have a nonzero net force acting on it. It is always true that F = ma. What acceleration do we use?

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21 The velocity of a particle is tangent to its path. For an object moving in uniform circular motion, the acceleration is radially inward.

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22 All three angles =Δθ = Δr/r =vΔt/r Δv=vΔθ = v 2 Δt/r

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23 The magnitude of the radial acceleration is: Δv=vΔθ = v 2 Δt/r a r = Δv/Δt = v 2 Δt/rΔt

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What makes it go round? A particle in circular motion is accelerating so there must be a force providing this acceleration F =ma = mv 2 /r

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Question A spider is sitting on a turntable that is rotating at 33rpm. The acceleration a of the spider is A) greater the closer the spider is to the central axis. B) greater the farther the spider is from the central axis. C) nonzero and independent of his location. D) zero.

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27 Example (text problem 5.12): The rotor is an amusement park ride where people stand against the inside of a cylinder. Once the cylinder is spinning fast enough, the floor drops out. ( a) What force keeps the people from falling out the bottom of the cylinder? Draw an FBD for a person with their back to the wall: x y w N fsfs It is the force of static friction.

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28 Example (text problem 5.12): The rotor is an amusement park ride where people stand against the inside of a cylinder. Once the cylinder is spinning fast enough, the floor drops out. (a) What force keeps the people from falling out the bottom of the cylinder? x y w N fsfs (b) If s = 0.40 and the cylinder has r = 2.5 m, what is the minimum angular speed of the cylinder so that the people dont fall out?

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29 Example (text problem 5.79): A coin is placed on a record that is rotating at 33.3 rpm. If s = 0.1, how far from the center of the record can the coin be placed without having it slip off? Apply Newtons 2 nd Law : x y w N fsfs Draw an FBD for the coin:

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Centrifuge Salad spinners, spin drying, Uranium centrifuges Need force to keep spinning. If object overcomes that force it moves out of circle.(spinners, drying) More massive particles are harder to rotate F r = Mv 2 /r (uranium) If there is no normal force or friction (which depend on M) more massive particles will be thrown out of circle.

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Question An object is in uniform circular motion. Which of the following statements is true? A)Velocity = constant B)Speed = constant C)Acceleration = constant D) Both B and C are true

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Fig. 05.12

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Fig. 05.13

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35 Unbanked and Banked Curves Example (text problem 5.20): A highway curve has a radius of 825 m. At what angle should the road be banked so that a car traveling at 26.8 m/s has no tendency to skid sideways on the road? (Hint: No tendency to skid means the frictional force is zero.) Take the cars motion to be into the page.

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36 Nonuniform Circular Motion Here, the speed is not constant. There is now an acceleration tangent to the path of the particle. The net acceleration of the body is atat v arar a Recall v is tangential =ωr

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37 atat arar a a t changes the magnitude of v. a r changes the direction of v. Can write:

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Fig. 05.19

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39 Example: What is the minimum speed for the car so that it maintains contact with the loop when it is in the pictured position? FBD for the car at the top of the loop: Nw y x Apply Newtons 2 nd Law: r

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40 The apparent weight at the top of loop is: N = 0 when This is the minimum speed needed to make it around the loop. Example continued:

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A boy swings on a tire swing. When is the tension in the rope the greatest? A)At the highest point of the swing. B)Midway between his highest and lowest points. C)At the lowest point in the swing. D)It is constant.

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42 Tangential and Angular Acceleration The average and instantaneous angular acceleration are: is measured in rads/sec 2.

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43 Recall that the tangential velocity is v t = r. This means the tangential acceleration is

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44 The kinematic equations : Linear Angular With

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r=67.5m T=30 min t to cruise =20s α = ? Δθ =?

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Gravity and Circular Motion What keeps the Moon going around the Earth? Why do planets go around the Sun? What determines the orbits of satellites? How can we weigh the Earth? the Sun? What is the Universe made of?

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Circular Orbits Consider an object of mass m in a circular orbit about the Earth. Earth r The only force on the satellite is the force of gravity: The speed of the satellite :

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How much does the Earth weigh? Solve for M e We measure v of satellite, we measure how high it is (r), we can measure G in the lab, hence we calculate M e. We have weighed the Earth!

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Weigh the Sun What about the Earth going around the sun? Gravity keeps the Earth moving around Sun. Same equations as for the satellite. Gravity is universal r R (distance from Sun to Earth) m s M s

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Weighing the Sun How fast are we traveling around the Sun? (v) v=2πR/T T = 1 year From astronomy we can measure R We have weighed the Sun

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Gravitational Orbits We can do this for any objects in gravitational orbit around each other. Independent of the mass of the orbiting object. Depends only on the mass of the central object. Planets around Sun, Moons around Planets ….. General formula

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What is the Universe made of ?

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We see mostly galaxies. We can measure the velocity of stars in a galaxy Most of the bright stuff (luminous matter, stars) is in the center. We are made of the same stuff as stars. We expect

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56 Example: How high above the surface of the Earth does a satellite need to be so that it has an orbit period of 24 hours? From previous problem : Also need, Combine and solve for r:

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57 Example: How high above the surface of the Earth does a satellite need to be so that it has an orbit period of 24 hours? From previous slide: Also need, Combine these expressions and solve for r:

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