Chapter 5 & 6 Force and Motion-I & II

Study of relation between force and acceleration of a body: 5.2 Newtonian Mechanics Study of relation between force and acceleration of a body: Newtonian Mechanics. Newtonian Mechanics does not hold good for all situations. Examples: Relativistic or near-relativistic motion Motion of atomic-scale particles

If no force acts on a body, the body’s velocity cannot change; 5.3 Newton’s First Law Newton’s First Law: If no force acts on a body, the body’s velocity cannot change; that is, the body cannot accelerate. If the body is at rest, it stays at rest. If it is moving, it continues to move with the same velocity (same magnitude and same direction).

5.6 Newton’s Second Law The net force on a body is equal to the product of the body’s mass and its acceleration. In component form, The acceleration component along a given axis is caused only by the sum of the force components along that same axis, and not by force components along any other axis.

5.8 Newton’s Third Law When two bodies interact, the forces on the bodies from each other are always equal in magnitude and opposite in direction. The minus sign means that these two forces are in opposite directions The forces between two interacting bodies are called a third-law force pair.

Inertial Reference Frames An inertial reference frame is one in which Newton’s laws hold. (a) The path of a puck sliding from the north pole as seen from a stationary point in space. Earth rotates to the east. (b) The path of the puck as seen from the ground. If a puck is sent sliding along a short strip of frictionless ice—the puck’s motion obeys Newton’s laws as observed from the Earth’s surface. If the puck is sent sliding along a long ice strip extending from the north pole, and if it is viewed from a point on the Earth’s surface, the puck’s path is not a simple straight line. The apparent deflection is not caused by a force, but by the fact that we see the puck from a rotating frame. In this situation, the ground is a noninertial frame.

Motion of an object relative to some frame of reference( Inertial Reference Frame ) P.3

Special case : S S' u t x' x P.3

Special case : S S' u t x' x Galilean Transformation P.3

Accelerated Noninertial Frame Galilean Transformation  Ax u u(t)

Accelerated Noninertial Frame Suppose that no horizontal forces acting on mass m m Fx= 0 A A

Sample Problem, Part a 5.9 Applying Newton’s Laws The reading is equal to the magnitude of the normal force on the passenger from the scale. We can use Newton’s Second Law only in an inertial frame. If the cab accelerates, then it is not an inertial frame. So we choose the ground to be our inertial frame and make any measure of the passenger’s acceleration relative to it. Sample Problem, Part a

The physiological effects of acceleration on the human body. Body Orientation Effect 2g Upright parallel to a Walking becomes strenuous 3g Walking is impossible 4g-6g Progressive dimming of vision due to decrease of blood to retina, ultimate blackout 9g-12g Reclining perpendicular to a Chest pain, fatigue, some loss of peripheral vision, but one is still conscious and can move hands and fingers. 14

Atwood's machine is a device where two blocks with masses m1 and m2 are connected by a cord ( of negligible mass) passing over a frictionless pulley ( also of negligible mass). Assume that m2 > m1 . Atwood's machine is attached to the ceiling of an elevator, as shown in Figure. When the elevator accelerates downword with an accerelation “ a ” ( relative to an inertial frame ). Find the magnitude of blocks’ acceleration relative to the pulley. Find the tension in the cord.

Coriolis Force(科里奧利) (fictitious force) Inertial Frame S

Coriolis Force(科里奧利) Noninertial Frame ( for a person P in a rotating frame )

The Centrifugal Force Inertial Frame Noninertial Frame

The Centrifugal Force

Motion in a Noninertial Reference Frame P.3

A block of mass m is placed on a wedge of mass M that is on a horizontal table . All surface are frictionless. Find the acceleration of the wedge. y m x M θ Hints:

Chapter 6 Force and Motion-II

Frictional forces are very common in our everyday lives. Examples: If you send a book sliding down a horizontal surface, the book will finally slow down and stop. If you push a heavy crate and the crate does not move, then the applied force must be counteracted by frictional forces. Friction

Friction If we either slide or attempt to slide a body over a surface, the motion is resisted by a bonding between the body and the surface. The resistance is considered to be single force called the frictional force, f. This force is directed along the surface, opposite the direction of the intended motion.

fs is the static frictional force 6.2 Frictional Force: motion of a crate with applied forces There is no attempt at sliding. Thus, no friction and no motion. NO FRICTION Finally, the applied force has overwhelmed the static frictional force. Block slides and accelerates. WEAK KINETIC FRICTION Force F attempts sliding but is balanced by the frictional force. No motion. STATIC FRICTION To maintain the speed, weaken force F to match the weak frictional force. SAME WEAK KINETIC FRICTION Force F is now stronger but is still balanced by the frictional force. No motion. LARGER STATIC FRICTION Static frictional force can only match growing applied force. Force F is now even stronger but is still balanced by the frictional force. No motion. EVEN LARGER STATIC FRICTION Kinetic frictional force has only one value (no matching). fs is the static frictional force fk is the kinetic frictional force

Fapplied Stick-slip Smooth sliding Static region Kinetic region

6.3 Properties of Friction Property 1. If the body does not move, then the static frictional force and the component of F that is parallel to the surface balance each other. They are equal in magnitude, and is fs directed opposite that component of F. Property 2. The magnitude of has a maximum value fs,max that is given by where ms is the coefficient of static friction and FN is the magnitude of the normal force on the body from the surface. If the magnitude of the component of F that is parallel to the surface exceeds fs,max, then the body begins to slide along the surface. Property 3. If the body begins to slide along the surface, the magnitude of the frictional force rapidly decreases to a value fk given by where mk is the coefficient of kinetic friction. Thereafter, during the sliding, a kinetic frictional force fk opposes the motion. 32

CENTRIPETAL (center-seeking) ACCELERATION Uniform Circular Motion As the direction of the velocity of the particle changes, there is an acceleration!!! CENTRIPETAL (center-seeking) ACCELERATION Here v is the speed of the particle and r is the radius of the circle.

Uniform circular motion θ

Uniform circular motion θ θ θ

Uniform circular motion: Examples: When a car moves in the circular arc, it has an acceleration that is directed toward the center of the circle. The frictional force on the tires from the road provide the centripetal force responsible for that. In a space shuttle around the earth, both the rider and the shuttle are in uniform circular motion and have accelerations directed toward the center of the circle. Centripetal forces, causing these accelerations, are gravitational pulls exerted by Earth and directed radially inward, toward the center of Earth. Uniform circular motion: A body moving with speed v in uniform circular motion feels a centripetal acceleration directed towards the center of the circle of radius R.

Car in flat circular turn

Car in banked circular turn (1) Frictionless road

Car in banked circular turn (2) With Friction ( s ) R

Homework Chapter 5 ( page 110 ) 33 , 43, 51, 55, 58, 59, 61, 65, 90 12, 29, 30, 33, 34, 51, 54, 56, 58, 59, 63, 68 Due date 2012/10/05

6.4 The Drag Force and Terminal Speed When there is a relative velocity between a fluid and a body (either because the body moves through the fluid or because the fluid moves past the body), the body experiences a drag force, D, that opposes the relative motion and points in the direction in which the fluid flows relative to the body.

Motion with Resistive Forces Motion can be through a medium Either a liquid or a gas The medium exerts a resistive force, , on an object moving through the medium The magnitude of depends on the medium The direction of is opposite the direction of motion of the object relative to the medium nearly always increases with increasing speed Slide 42

Motion with Resistive Forces The magnitude of can depend on the speed in complex ways We will discuss only two cases is proportional to v Good approximation for slow motions or small objects is proportional to v2 Good approximation for large objects ; b is a constant C is the drag coefficient ; ρ is the density of medium (air) A is the cross-sectional area of the object v is the speed of the object Slide 43

D Proportional to v, Example Analyzing the motion results in Slide 44

Terminal Speed To find the terminal speed, let a = 0 Solving the differential equation gives τ is the time constant and τ = m/b Slide 45

D Proportional to v2, Example Analysis of an object falling through air accounting for air resistance C is the drag coefficient ; ρ is the density of medium (air) A is the cross-sectional area of the object v is the speed of the object Slide 46

求 v(t) = ? 當t >> b/2g 時， v(t) = ? ( Terminal speed) 當 t 趨近於 0 時， v 對 t 的變化為何?

6.4 The Drag Force and Terminal Speed When a blunt body falls from rest through air, the drag force is directed upward; its magnitude gradually increases from zero as the speed of the body increases. From Newton’s second law along y axis where m is the mass of the body. Eventually, a = 0, and the body then falls at a constant speed, called the terminal speed vt . For cases in which air is the fluid, and the body is blunt (like a baseball) rather than slender (like a javelin), and the relative motion is fast enough so that the air becomes turbulent (breaks up into swirls) behind the body, where r is the air density (mass per volume), A is the effective cross-sectional area of the body (the area of a cross section taken perpendicular to the velocity), and C is the drag coefficient . 48

Homework Chapter 5 ( page 110 ) 33 , 43, 51, 55, 58, 59, 61, 65, 90 12, 29, 30, 33, 34, 51, 54, 56, 58, 59, 63, 68 Due date 2012/10/05

靠強力的氣體推進..利用對空氣的反作用力.. 還有風阻角度的調整... 上到天空之後 靠氣流就可以維持不往下墜了

Friction If we either slide or attempt to slide a body over a surface, the motion is resisted by a bonding between the body and the surface. The resistance is considered to be single force called the frictional force, f. This force is directed along the surface, opposite the direction of the intended motion.

6.2 Frictional Force: motion of a crate with applied forces There is no attempt at sliding. Thus, no friction and no motion. NO FRICTION Finally, the applied force has overwhelmed the static frictional force. Block slides and accelerates. WEAK KINETIC FRICTION Force F attempts sliding but is balanced by the frictional force. No motion. STATIC FRICTION To maintain the speed, weaken force F to match the weak frictional force. SAME WEAK KINETIC FRICTION Force F is now stronger but is still balanced by the frictional force. No motion. LARGER STATIC FRICTION Static frictional force can only match growing applied force. Force F is now even stronger but is still balanced by the frictional force. No motion. EVEN LARGER STATIC FRICTION Kinetic frictional force has only one value (no matching). fs is the static frictional force fk is the kinetic frictional force