Chapter 7 Rotational Motion

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Information about Chapter 7 Rotational Motion
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Published on July 28, 2008

Author: mwarner1968

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Announcements : Announcements Homework for tomorrow… (Ch. 7, Probs 5, 19, & 30) Office hours… MTWRF 11-noon Chapter 7 : Chapter 7 Rotational Motion and the Law of Gravity Section 7.1: Angular Speed & Angular Acceleration : Section 7.1: Angular Speed & Angular Acceleration Kinematics: Translational & Rotational Translating Body: Displacement: Velocity: Acceleration: Rotating Body: Angular Displacement: Angular Velocity: Angular Acceleration: Angular Displacement : Angular Displacement  (theta) is the angle subtended w.r.t the x-axis 1 rev = 360° = 2 rad v r  x y Angular Displacement : Angular Displacement How much distance, s , is traversed? s is the arc length Notice:   must be in radians Check: For C = 2r r  x y s Angular Displacement : Angular Displacement Angular Displacement,  , is the difference in the object’s final and initial angles CCW is +, CW is - What are the SI units? i x y f Definition of the Radian : Definition of the Radian Units? The radian is dimensionless! Angular Velocity : Angular Velocity Average angular speed, av, is the ratio of the angular displacement to the time interval for that angular displacement What are the SI units? Angular Acceleration : Angular Acceleration Average angular acceleration is the ratio of the angular velocity to the time interval for that angular velocity What are the SI units? Direction of Angular Acceleration : Direction of Angular Acceleration When  is increasing… … is parallel to  When  is decreasing… … is anti-parallel to  Quiz Question 1 : Quiz Question 1 How does the angular velocity of the green tape, g, compare to the angular velocity of the blue tape, b? They’re the same g > b g < b Quiz Question 2 : Quiz Question 2 How does the angular acceleration of the green tape, g, compare to the angular acceleration of the blue tape, b? They’re the same g > b g < b Demo Questions: : Demo Questions: Estimate  and .. What is the direction of ? What is the direction of ? Section 7.2: Rotational Motion under Constant Angular Acceleration : Section 7.2: Rotational Motion under Constant Angular Acceleration Kinematic equations with constant angular acceleration: Look familiar? Compare to 1D rectilinear motion… : Compare to 1D rectilinear motion… Kinematic equations (1D) with constant acceleration: Analogy between Translational and Rotational Kinematics : Analogy between Translational and Rotational Kinematics Problem 1 : Problem 1 A wheel initially has an angular velocity of 18 rad/s but is slowing down at a rate of 2.0 rad/s2. By the time it stops it will have turned through: 13 rev 26 rev 39 rev 52 rev 65 rev Section 7.3: Relations between Angular and Linear Quantities : Section 7.3: Relations between Angular and Linear Quantities Can we relate angular & linear velocity? Yes! Arc length is Dividing through by t The tangential velocity, vt, of a point on a rotating object equals the distance of that point from the axis of rotation, r, multiplied by the angular speed,  r  s vt vt Relationship between Translational and Rotational Acceleration : Relationship between Translational and Rotational Acceleration Likewise… The tangential acceleration, at, of a point on a rotating object equals the distance of that point from the axis of rotation, r, multiplied by the angular acceleration,  Relations between translational and rotational quantities : Relations between translational and rotational quantities Summarizing… Problem 2 : Problem 2 A child riding on a playground merry-go-round rotates at a rate of 1.0 rad/s in a circle of radius 3.0 m. The (linear) speed of the child is: 3.0 m/s 0.33 m/s 19 m/s 0.48 m/s 9.4 m/s Quiz Question 3 : Quiz Question 3 A ladybug sits at the outer edge of a record rotating at a constant 45 r.p.m. A gentleman bug sits halfway between her and the axis of rotation. Which of the following is/are true? The ladybug’s linear speed is the same as the gentleman bug’s The ladybug and the gentleman bug will have the same angular displacement after 1.0 s. The ladybug and the gentleman bug have the same tangential acceleration. I only I & II II &III I, II, & III None of these Demo Question: : Demo Question: I twirl a ball attached to the end of a rope around in a circle with a constant speed. Does the ball have an acceleration? YES! If the speed is constant, what is changing to cause the acceleration? v v Section 7.4:Centripetal Acceleration : Section 7.4:Centripetal Acceleration The rotating body undergoes a centripetal acceleration Direction: towards the center of the circle (in the direction of ) Magnitude: r Centripetal Acceleration : Centripetal Acceleration For constant vt… Q: Is positive or negative? Tangential & Centripetal Acceleration : Tangential & Centripetal Acceleration For an increasing vt… What is the magnitude of the total acceleration? Quiz Question 4 : Quiz Question 4 An object moves in a circular path with constant speed v. Which of the following statements is true concerning the object? Its velocity is constant, but its acceleration is changing. Its acceleration is constant, but its velocity is changing. Both its velocity and acceleration are changing. Its velocity and acceleration remain constant. Centripetal Force : Centripetal Force - Any force that causes circular motion Direction: towards the center of the circle Magnitude: - Newton’s 2nd Law Concept Questions : Concept Questions What supplies the centripetal force for… a ball on a string whirled around in a circle? a satellite in orbit? a car racing on a circular track? Example 1 : Example 1 A bored astronaut in outer space… twirls a ball on a string whirled around at constant speed in a circle. The length of the string is 30 cm, the mass of the ball is 50 g, and the ball’s angular speed is 100 rev/min. What is the tension in the string? Example 2 : Example 2 A bored astronaut on Earth… twirls a ball on a string whirled around at constant speed in a circle. The length of the string is 30 cm, the mass of the ball is 50 g, and the ball’s angular speed is 100 rev/min. What is the tension in the string when the ball is at the bottom of the path? What about the top of the path? “Jackass” : “Jackass” What is the minimum velocity such that the bike will stay on the track? v R “Weightlessness” implies what? : “Weightlessness” implies what? n mg Artificial Gravity : Artificial Gravity One way… Another way… “2001: A Space Odyssey” a = 9.8 m/s2 r v Section 7.5:Newtonian Gravitation : Section 7.5:Newtonian Gravitation The force of gravity between any two bodies is proportional to the product of their masses and inversely proportional to the square of their separation Where Direction: Attractive only Law of Gravitation - Restrictions : Law of Gravitation - Restrictions Masses must be small compared to their separation (“point-like”) or Masses must be uniform spheres OK OK Not OK From the MCAT… : From the MCAT… Which of the following expressions gives the acceleration due to gravity at the surface of the Earth? (Note: M is the mass of Earth, m is the mass of an observer at the surface of the Earth, and r is the radius of the Earth.) GM/r GM/r2 GmM/r GmM/r2 From the MCAT… : From the MCAT… If the gravitational force between Earth and the space station could suddenly be eliminated, which of the following diagrams would best describe the path taken by the station? Example 1 : Example 1 What is the orbital period of the International Space Station? Kepler’s 3rd Law!! Principle of Superposition : Principle of Superposition What is ? m1 m2 m3 m4 Principle of Superposition : Principle of Superposition This is the Principle of Superposition. m1 m2 m3 m4 Quiz Question 5 : Quiz Question 5 Four point masses, each with mass m, are arranged symmetrically about the origin on the x-axis. A fifth point mass, with mass M, is n the y-axis. The direction of the gravitational force on M is: Up Down Up and to the left Down and to the right Depends whether m>M or m<M m M m m m y x Gravitational Potential Energy, revisited… : Gravitational Potential Energy, revisited… For small displacements near the Earth’s surface… In general… The gravitational potential energy associated with an object of mass m at a distance r from the center of Earth is Quiz Question 6 : Quiz Question 6 An object is dropped from an altitude of one Earth radius above the Earth’s surface. If M is the mass of the Earth and R is its radius, then the speed of the object just before it hits Earth is (ignoring air resistance): Question : Question What speed is needed for an object of mass m to soar off into space and never return? This speed is commonly known as an object’s Escape Velocity Problem 1 : Problem 1 What is the minimum initial height needed to “Loop the Loop”? h r Problem 2 : Problem 2 A small object of mass m, on the end of a light cord is held horizontally at a distance r from a fixed support as shown. The object is then released. What is the tension in the cord when the object is at the lowest point of its swing? r

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