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## Chapter 7 Rotational Motion and The Law of Gravity

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**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**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**Angular velocity**• Average angular velocity • Instantaneous angular velocity – the rate of change in angular position**Angular acceleration**• Average angular acceleration • Instantaneous angular acceleration – the rate of change in angular velocity**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 • 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 • Centripetal acceleration is directed toward the center of the circle of motion**Centripetal acceleration**• The magnitude of the centripetal acceleration is given by**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: • θis measured in radians**Relating the linear and angular variables: speed**• ωis measured in rad/s • Period**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**• αis measured in rad/s2 • Centripetal acceleration**Total acceleration**• Tangential acceleration is due to changing speed • Centripetal acceleration is due to changing direction • Total acceleration:**Centripetal force**• For an object in a uniform circular motion, the centripetal acceleration is • According to the Newton’s Second Law, a force must cause this acceleration – centripetal force • 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.**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) • Gravitational constantG= 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**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 • 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 gis 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:**Escape speed**• Accounting for the shape of Earth, projectile motion (Ch. 3) has to be modified:**Escape speed**• Escape speed: speed required for a particle to escape from the planet into infinity (and stop there)**Escape speed**• If for some astronomical object • Nothing (even light) can escape from the surface of this object – a black hole**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.**Johannes Kepler**(1571-1630) Tycho Brahe/ Tyge Ottesen Brahe de Knudstrup (1546-1601) • Kepler’s laws • 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 • From the definition of a period**Satellites**• For a circular orbit and the Newton’s Second law • Kinetic energy of a satellite • Total mechanical energy of a satellite**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.**Answers to the even-numbered problems**Chapter 7 Problem 10 50 rev**Answers to the even-numbered problems**• Chapter 7 • Problem 30 • 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 • 5.59 × 103 m/s • (b) 3.98 h • (c) 1.47 × 103 N