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Transcript
Chapter 7 Rotational Motion and The Law of Gravity The radian • The radian is a unit of angular measure • The angle in radians can be defined as the ratio of the arc length s along a circle divided by the radius r s r 360 1 rad 57.3 2 [rad] [deg rees] 180 Rotation of a rigid body • We consider rotational motion of a rigid body about a fixed axis • Rigid body rotates with all its parts locked together and without any change in its shape • Fixed axis: it does not move during the rotation • This axis is called axis of rotation • Reference line is introduced Angular position • Reference line is fixed in the body, is perpendicular to the rotation axis, intersects the rotation axis, and rotates with the body • Angular position – the angle (in radians or degrees) of the reference line relative to a fixed direction (zero angular position) Angular displacement • Angular displacement – the change in angular position. • Angular displacement is considered positive in the CCW direction and holds for the rigid body as a whole and every part within that body f i Angular velocity • Average angular velocity avg f i t f ti t • Instantaneous angular velocity – the rate of change in angular position lim t 0 t Angular acceleration • Average angular acceleration avg f i t f ti t • Instantaneous angular acceleration – the rate of change in angular velocity lim t 0 t Uniform circular motion • A special case of 2D motion • An object moves around a circle at a constant speed • Period – time to make one full revolution 2r T v • An object traveling in a circle, even though it moves with a constant speed, will have an acceleration Centripetal acceleration • Centripetal acceleration is due to the change in the direction of the velocity v a lim t 0 t • Centripetal acceleration is directed toward the center of the circle of motion Centripetal acceleration • The magnitude of the centripetal acceleration is given by 2 v ac r Centripetal acceleration During a uniform circular motion: • the speed is constant • the velocity is changing due to centripetal (“center seeking”) acceleration • centripetal acceleration is constant in magnitude (v2/r), is normal to the velocity vector, and points radially inward Rotation with constant angular acceleration • Similarly to the case of 1D motion with a constant acceleration we can derive a set of formulas: Chapter 7 Problem 5 A dentist’s drill starts from rest. After 3.20 s of constant angular acceleration, it turns at a rate of 2.51 × 104 rev/min. (a) Find the drill’s angular acceleration. (b) Determine the angle (in radians) through which the drill rotates during this period. Relating the linear and angular variables: position • For a point on a reference line at a distance r from the rotation axis: s r • θ is measured in radians Relating the linear and angular variables: speed s (r ) v lim lim r lim r t 0 t t 0 t t 0 t • ω is measured in rad/s • Period 2r 2 T v Chapter 7 Problem 2 A wheel has a radius of 4.1 m. How far (path length) does a point on the circumference travel if the wheel is rotated through angles of 30°, 30 rad, and 30 rev, respectively? Relating the linear and angular variables: acceleration v (r ) at lim lim r lim r t 0 t t 0 t 0 t t • α is measured in rad/s2 • Centripetal acceleration v (r ) 2 r ac r r 2 2 Total acceleration • Tangential acceleration is due to changing speed • Centripetal acceleration is due to changing direction • Total acceleration: a at ac 2 2 Centripetal force • For an object in a uniform circular motion, the centripetal acceleration is 2 v ac R • According to the Newton’s Second Law, a force must cause this acceleration – centripetal force mv Fc mac R 2 • A centripetal force accelerates a body by changing the direction of the body’s velocity without changing the speed Centripetal force • Centripetal forces may have different origins • Gravitation can be a centripetal force • Tension can be a centripetal force • Etc. Centripetal force • Centripetal forces may have different origins • Gravitation can be a centripetal force • Tension can be a centripetal force • Etc. Free-body diagram Chapter 7 Problem 28 A roller-coaster vehicle has a mass of 500 kg when fully loaded with passengers. (a) If the vehicle has a speed of 20.0 m/s at point A, what is the force exerted by the track on the car at this point? (b) What is the in maximum speed the vehicle can have at point B and still remain on the track? Newton’s law of gravitation • Any two (or more) massive bodies attract each other • Gravitational force (Newton's law of gravitation) m1m2 F G 2 rˆ r • Gravitational constant G = 6.67*10 –11 N*m2/kg2 = 6.67*10 –11 m3/(kg*s2) – universal constant Gravitation and the superposition principle • For a group of interacting particles, the net gravitational force on one of the particles is n F1,net F1i i2 Chapter 7 Problem 33 Objects with masses of 200 kg and 500 kg are separated by 0.400 m. (a) Find the net gravitational force exerted by these objects on a 50.0-kg object placed midway between them. (b) At what position (other than infinitely remote ones) can the 50.0-kg object be placed so as to experience a net force of zero? Gravity force near the surface of Earth • Earth can be though of as a nest of shells, one within another and each attracting a particle outside the Earth’s surface • Thus Earth behaves like a particle located at the center of Earth with a mass equal to that of Earth mEarthm1 ˆ GmEarth ˆ F1, Earth G 2 j 2 m1 j g m1 ˆj REarth REarth g = 9.8 m/s2 • This formula is derived for stationary Earth of ideal spherical shape and uniform density Gravity force near the surface of Earth In reality g is not a constant because: Earth is rotating, Earth is approximately an ellipsoid with a non-uniform density Gravitational potential energy • Gravitation is a conservative force (work done by it is path-independent) • For conservative forces potential energy can be introduced • Gravitational potential energy: Gm1m2 U (r ) r Gravitational potential energy Gm1m2 U (r ) r Escape speed • Accounting for the shape of Earth, projectile motion (Ch. 3) has to be modified: 2 v ac g v gR R Escape speed • Escape speed: speed required for a particle to escape from the planet into infinity (and stop there) Ki U i K f U f 2 m1v Gm1m planet 00 2 R planet vescape 2Gm planet R planet Escape speed • If for some astronomical object vescape 2Gmobject Robject 3 10 m / s c 8 • Nothing (even light) can escape from the surface of this object – a black hole Escape speed Chapter 7 Problem 56 Show that the escape speed from the surface of a planet of uniform density is directly proportional to the radius of the planet. Kepler’s laws Tycho Brahe/ Tyge Ottesen Brahe de Knudstrup (1546-1601) Johannes Kepler (1571-1630) Three Kepler’s laws • 1. The law of orbits: All planets move in elliptical orbits, with the Sun at one focus • 2. The law of areas: A line that connects the planet to the Sun sweeps out equal areas in the plane of the planet’s orbit in equal time intervals • 3. The law of periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit First Kepler’s law • All planets move in elliptical orbits with the Sun at one focus, whereas the second focus is empty • Any object bound to another by an inverse square law will move in an elliptical path Second Kepler’s law • A line drawn from the Sun to any planet will sweep out equal areas in equal times • Area from A to B and C to D are the same Third Kepler’s law • For a circular orbit and the Newton’s Second law GMm GM 2 2 (m)( r ) 3 2 r r F ma • From the definition of a period T 2 T 2 4 2 2 4 3 T r GM 2 2 Satellites • For a circular orbit and the Newton’s Second law 2 GMm v (m) F ma 2 r r • Kinetic energy of a satellite 2 GMm U mv K 2r 2 2 • Total mechanical energy of a satellite GMm GMm GMm E K U K 2r r 2r Chapter 7 Problem 45 The Solar Maximum Mission Satellite was placed in a circular orbit about 150 mi above Earth. Determine (a) the orbital speed of the satellite and (b) the time required for one complete revolution. Questions? Answers to the even-numbered problems Chapter 7 Problem 10 50 rev Answers to the even-numbered problems Chapter 7 Problem 30 (a) 4.39 × 1020 N toward the Sun (b) 1.99 × 1020 N toward the Sun (c) 3.55 × 1020 N toward the Sun Answers to the even-numbered problems Chapter 7 Problem 36 (a) 5.59 × 103 m/s (b) 3.98 h (c) 1.47 × 103 N