今日課程內容 CH11 滾動 力矩 角動量 陀螺儀 CH13 重力 牛頓萬有引力 重力位能 行星運動與衛星軌道運動

11.6: Torque( 力矩 ) Revisited Fig (a) A force F, lying in an x-y plane, acts on a particle at point A. (b) This force produces a torque  = r x F on the particle with respect to the origin O. By the right-hand rule for vector (cross) products, the torque vector points in the positive direction of z. Its magnitude is given by in (b) and by in (c).

11.7 Angular Momentum( 角動量 )

Sample problem: Angular Momentum

11.8: Newton’s 2 nd Law in Angular Form The (vector) sum of all the torques acting on a particle is equal to the time rate of change of the angular momentum of that particle.

Sample problem: Torque, Penguin Fall

11.9: The Angular Momentum of a System of Particles The total angular momentum L of the system is the (vector) sum of the angular momenta l of the individual particles (here with label i): With time, the angular momenta of individual particles may change because of interactions between the particles or with the outside. Therefore, the net external torque acting on a system of particles is equal to the time rate of change of the system’s total angular momentum L.

11.10: Angular Momentum of a Rigid Body Rotating About a Fixed Axis

11.10: More Corresponding Variables and Relations for Translational and Rotational Motion a

11.11: Conservation of Angular Momentum If the net external torque acting on a system is zero, the angular momentum L of the system remains constant, no matter what changes take place within the system.

11.11: Conservation of Angular Momentum( 角動量守恆 ) If the component of the net external torque on a system along a certain axis is zero, then the component of the angular momentum of the system along that axis cannot change, no matter what changes take place within the system. (a)The student has a relatively large rotational inertia about the rotation axis and a relatively small angular speed. (b)By decreasing his rotational inertia, the student automatically increases his angular speed. The angular momentum of the rotating system remains unchanged.

Sample problem Figure a shows a student, sitting on a stool that can rotate freely about a vertical axis. The student, initially at rest, is holding a bicycle wheel whose rim is loaded with lead and whose rotational inertia I wh about its central axis is 1.2 kg m 2. (The rim contains lead in order to make the value of I wh substantial.) The wheel is rotating at an angular speed  wh of 3.9 rev/s; as seen from overhead, the rotation is counterclockwise. The axis of the wheel is vertical, and the angular momentum L wh of the wheel points vertically upward. The student now inverts the wheel (Fig b) so L wh that, as seen from overhead, it is rotating clockwise. Its angular momentum is now - L wh.The inversion results in the student, the stool, and the wheel’s center rotating together as a composite rigid body about the stool’s rotation axis, with rotational inertia I b = 6.8 kg m 2. With what angular speed  b and in what direction does the composite body rotate after the inversion of the wheel?

Sample problem

11.12: Precession of a Gyroscope (b) A rapidly spinning gyroscope, with angular momentum, L, precesses around the z axis. Its precessional motion is in the xy plane. (c) The change in angular momentum, dL/dt, leads to a rotation of L about O. (a) A non-spinning gyroscope falls by rotating in an xz plane because of torque .

習題 Ch11: 6, 11, 24, 26, 35, 40, 54, 61, 68

Chapter 13 Gravitation ( 重力 )

13.2 Newton’s Law of Gravitation Here m 1 and m 2 are the masses of the particles, r is the distance between them, and G is the gravitational constant. G =6.67 x10 11 Nm 2 /kg 2 =6.67 x10 11 m 3 /kg s 2. Fig (a) The gravitational force on particle 1 due to particle 2 is an attractive force because particle 1 is attracted to particle 2. (b) Force is directed along a radial coordinate axis r extending from particle 1 through particle 2. (c) is in the direction of a unit vector along the r axis.

13.2 Newton’s Law of Gravitation A uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell’s mass were concentrated at its center.

12.3 Gravitation and the Principle of Superposition For n interacting particles, we can write the principle of superposition for the gravitational forces on particle 1 as Here F 1,net is the net force on particle 1 due to the other particles and, for example, F 13 is the force on particle 1 from particle 3, etc. Therefore, The gravitational force on a particle from a real (extended) object can be expressed as: Here the integral is taken over the entire extended object.

Example, Net Gravitational Force: Figure 13-4a shows an arrangement of three particles, particle 1 of mass m 1 = 6.0 kg and particles 2 and 3 of mass m 2 =m 3 =4.0 kg, and distance a =2.0 cm. What is the net gravitational force 1,net on particle 1 due to the other particles?

13.4: Gravitation Near Earth’s Surface( 地表附近的重力 ) If the particle is released, it will fall toward the center of Earth, as a result of the gravitational force, with an acceleration we shall call the gravitational acceleration a g. Newton’s second law tells us that magnitudes F and a g are related by If the Earth is a uniform sphere of mass M, the magnitude of the gravitational force from Earth on a particle of mass m, located outside Earth a distance r from Earth’s center, is Therefore,

13.4: Gravitation Inside Earth( 地球內部的重力 ) A uniform shell of matter exerts no net gravitational force on a particle located inside it. Calculations: The force magnitude depends linearly on the capsule’s distance r from Earth’s center. Thus, as r decreases, F also decreases, until it is zero at Earth’s center.

13.6: Gravitational Potential Energy( 重力位能 ) The gravitational potential energy of the two-particle system is: U(r) approaches zero as r approaches infinity and that for any finite value of r, the value of U(r) is negative. If the system contains more than two particles, consider each pair of particles in turn, calculate the gravitational potential energy of that pair with the above relation, as if the other particles were not there, and then algebraically sum the results. That is,

13.6: Gravitational Potential Energy

13.6: Gravitational Potential Energy: Potential Energy and Force The minus sign indicates that the force on mass m points radially inward, toward mass M.

13.6: Gravitational Potential Energy: Escape Speed( 脫離速度 ) If you fire a projectile upward, there is a certain minimum initial speed that will cause it to move upward forever, theoretically coming to rest only at infinity. This minimum initial speed is called the (Earth) escape speed. Consider a projectile of mass m, leaving the surface of a planet (mass M, radius R) with escape speed v. The projectile has a kinetic energy K given by ½ mv 2, and a potential energy U given by: When the projectile reaches infinity, it stops and thus has no kinetic energy. It also has no potential energy because an infinite separation between two bodies is our zero-potential-energy configuration. Its total energy at infinity is therefore zero. From the principle of conservation of energy, its total energy at the planet’s surface must also have been zero, and so This gives the escape speed

13.6: Gravitational Potential Energy: Escape Speed

Example:

13.7: Planets and Satellites: Kepler’s 1 st Law( 克普勒行星第一運動定律 ) 1. THE LAW OF ORBITS: All planets move in elliptical orbits, with the Sun at one focus.

13.7: Planets and Satellites: Kepler’s 2 nd Law 2. THE LAW OF AREAS: A line that connects a planet to the Sun sweeps out equal areas in the plane of the planet’s orbit in equal time intervals; that is, the rate dA/dt at which it sweeps out area A is constant. Angular momentum, L: 角動量守恆

13.7: Planets and Satellites: Kepler’s 3 rd Law 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. Consider a circular orbit with radius r (the radius of a circle is equivalent to the semimajor axis of an ellipse). Applying Newton’s second law to the orbiting planet yields Using the relation of the angular velocity, , and the period, T, one gets: 對橢圓而言， r = a( 半長軸 )

Example, Halley’s Comet

13.8: Satellites( 衛星 ): Orbits( 軌道 ) and Energy As a satellite orbits Earth in an elliptical path, the mechanical energy E of the satellite remains constant. Assume that the satellite’s mass is so much smaller than Earth’s mass. The potential energy of the system is given by For a satellite in a circular orbit, Thus, one gets: For an elliptical orbit (semimajor axis a),

Example, Mechanical Energy of a Bowling Ball

13.9: Einstein and Gravitation The fundamental postulate of Einstein’s general theory of relativity about gravitation (the gravitating of objects toward each other) is called the principle of equivalence, which says that gravitation and acceleration are equivalent.

13.9: Einstein and Gravitation: Curvature of Space

習題 Ch13: 10, 13, 20, 22, 28, 35, 40, 53, 65