Chapters 5-6. Dynamics: Forces and Newton s Laws of Motion. Applications

Transcription

1 Chapters 5-6 Dynamics: orces and Newton s Laws of Motion. Applications That is, describing why objects move orces Newton s 1 st Law Newton s 2 nd Law Newton s 3 rd Law Examples of orces: Weight, Normal, Tension, riction 1D Dynamics of Single Bodies Systems of interacting objects The free body diagram technique Examples: Masses interacting via Normal forces Masses interacting via Tensions in ropes. Ideal pulleys 2D Dynamics: Circular motion

2 orces A force is the manifestation of an interaction: it is a push or a pull onto an object by another object an agent direct interactions contact forces interactions via a certain medium long-range forces (force fields) forces are vectors, with magnitude measured in newtons (N). Ex: order of magnitude: SI newton (N) 1 kg ~ 10 N. 150 lb ~ 666 N If several forces act simultaneously on the same object, the result is a net force: net i i

3 y net _ y 1y 1 Net orces Components net 1 2 Adding force vector arrows is done using the components: 1. x-component of the resultant: 2. y-component of the resultant: net _ x 1x 2x net _ y 1y 2 y 2 y 1x 2 2x 3. magnitude of the resultant: 4. direction of the resultant: 1 net _ y tan 2 2 net net _ x net _ y x net _ x net _ x If there are n forces acting on the particle, n components are to be added in each direction: net i net xi, yi i i i

4 Problem 1. Calculating net force: Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is 60. If dog A exerts a force of 270 N and dog B exerts a force of 300 N, find the magnitude of the resultant force and the angle it makes with respect with dog A s rope.

5 Newton s 1 st Law of Motion If the net force acting on an object is zero, then the object will tend to keep its state of motion with constant or zero velocity. If all the forces are in the same plane (2D case), we can express this as: x 0 0 v const. (or zero) y 0 This is one of the notations we ll use for a net force Comments: Zero net force does not mean that the forces in the system must be zero: it just means that the forces cancel each other out When the net force is zero the object is called in translational equilibrium The value of the constant velocity (including zero) depends on the system of coordinates associated with any inertial frame of reference

6 What are Inertial rames? An inertial frame is a frame of reference where Newton s 1 st Law is valid (and, as we shall see, 2 nd Law as well.) Ex: The Earth s surface can be considered as a good approximation for an inertial frame for low scale events. (Astronomically, it is not.) A frame moving with constant speed relative to an inertial frame is also inertial. Therefore, any system of reference moving with constant velocity with respect to the ground (Earth) is an inertial frame Quiz: In the example above I said that the Earth is only an approximated inertial frame. Why? Once we chose an inertial frame (at least approximately) a frame which accelerates (including rotation) in the inertial frame is non-inertial. In a non-inertial frame objects can accelerate with a zero net force or move with constant velocity (or be at rest) under a finite net force. In this class, we are working only with inertial frames.

7 Newton s 2 nd Law of Motion Newton s second law describes what happens if a finite net force acts on an object: If a net force acts on an object of mass m, then the object will accelerate with an acceleration proportional to the net force. The proportionality constant is the mass m of the body, such that ma x y ma ma x y (2D motion) cause effect m Comments about the mass: m kg newton kg SI SI s 2 Mass in the expression above is the measure of inertia of an object, called inertial mass, as opposed to the gravitational mass that determines gravitational attraction: the masses within the two concepts are equivalent Mass and weight are different physical quantities! Mass is a scalar property of the quantity of matter in an object Weight is a vector: the force of gravitational attraction exerted by Earth Ex: On the moon the gravitational acceleration is about g/6, so you will weight six times less. Your mass, however, will stay the same as on Earth.

8 Ex 1: Consider the linear motion of a puck on a stretch of ice: Ex 2: Consider a freely falling object. Its acceleration is g determined by its weight which is given by Newton s 2 nd Law as W mg Ex 3: If a particle performs circular motion, the centripetal acceleration is due to a centripetal net force v a) If no net force acts on the puck: 0 a 0 v const. b) If a force acts in the direction of motion the puck speeds up: v speed up c) If a force acts against the direction of motion, the puck decelerates: Descending freely: W v slowdown g mg Ascending freely: v W 2 v a a m r R v mg g

9 Problem 2. Calculating a uniform acceleration from force: A hockey puck with mass m = 0.50 kg is at rest at the origin (x 0 = 0) on the horizontal, frictionless surface of the rink. At time t 0 = 0 a player applies a time dependent force = At to the puck (where A = 4.0 m/s is a constant and t is the time), parallel to the x-axis, and he continues to apply this force until time t 1 = 2.00 s when the force is removed. a) What is the time dependent acceleration of the puck? b) Calculate the position and velocity of the puck at t 1 = 2.00 s? c) Calculate the position and velocity of the puck at t 2 = 3.00 s?

10 A Catalog of orces Contact forces: Tension orce T Long-range forces: Gravitational orce or Weight g Normal orce n Electromagnetic orce (in PHYS 155) riction orce f (superficial and drag) Elastic (or Spring) orce e Comments: The contact forces listed above are actually macroscopic manifestations of longrange forces: we say that they are not fundamental However, differentiating them depending on the specific mechanical manifestation is useful when analyzing phenomena at macroscopic scales Ex: rictions between surfaces are electric, the reaction of a wall you push against is also electric, the force that keeps you on a erris wheel is ultimately gravitational and electric, etc.

11 Weight, g The weight of an object is the force of gravitational attraction exerted onto the object by Earth Close to the surface of the Earth, where the gravitational acceleration is nearly constant, the weight is: g mg 2 Magnitude: mg m9.8 m/s m Direction: vertically down mg Ex: If a person steps on a scale which shows 130 lb, her mass m given by g 130 lb 580 N 2 mg m g 580 N 9.8 m/s 60 kg g g

12 Normal orce, n According to Newton s laws, an object at rest must have no net force acting on it or instance, if a book is sitting on a table, the force of gravity is still there, so there must be another force to cancel it: the force must be perpendicular (or normal ) on the table to balance the weight The forces that appear at the surface of contact between objects pushing against each other due to molecular interactions are called normal forces So, surfaces can intermediate the interaction between objects in direct contact via normal forces Magnitude: depending on the effect of the other forces in the system Direction: perpendicular on the surface of contact Ex: A box at rest on a surface will be kept in equilibrium by a normal force equal in magnitude with the weight n+mg ma 0 n mg 0 n mg mg m n

13 Tension in ropes, T If an object is pulled by a rope, the action of the rope can be associated with a force that is transmitted along its length We say that the rope is under tension manifested as a tension force. So, cords can intermediate the interaction between objects via tension forces. Magnitude: depending on the effect of the other forces along the cord Direction: along the cord, away from the object Ex: An object suspended by a wire will be kept in equilibrium by a tension force equal in magnitude with the weight T +mg ma 0 T mg 0 T mg T m mg

14 Newton s 3 rd Law of Motion Any time a force is exerted on an object, that force is caused by another object. So the forces always act in pairs The interaction between two bodies always involves two forces: if a body A acts on a body B with a force (action), body B will act onto body A with a force (reaction) equal in magnitude and opposite in direction A on B B on A So, be cautious, the action and reaction always act on different objects Make sure you don t use them as if they were acting on the same object and don t identify two forces as action-reaction pair if they act onto the same object. Ex 1: consider a box at rest on the ground. There are two pairs actionreaction in this situation both describing the interaction between the box and planet Earth (Normal) E on B Box B Earth E B on E E on B (Weight) B on E Ex 2: consider a box at rest suspended by a cord. The interaction between the box and ceiling is described by two actionreaction tensions (besides the box-earth interaction) Ceiling C (Tension) B on C C on B Box B

15 Problems: Pair action-reaction elevator 3. Normal force: A person of mass m stands on a bathroom scale in an elevator accelerating uniformly. The scale reads an apparent weight of only 0.75 of the person s regular weight. What is the acceleration of the elevator? n scale on person scale n person on scale person n scale Exercise 1: Tension A cord is used to pull vertically upward a body with mass m = 10 kg, such that the upward acceleration is a = 7.0 m/s 2. T g mg a) Is the tension in the rope still equal to the weight? b) If not, is the tension larger or smaller than the weight? If the rope breaks, what will be the acceleration of the object? m a mg

16 Kinetic friction rictional forces, Appears at the surface of contact between two rough surfaces moving relative to each other Direction: in the surface of contact (perpendicular on the normal force), against the relative motion Magnitude: proportional to the magnitude of the normal. The coefficient of proportionality µ k is called coefficient of kinetic friction and is a property characterizing the surfaces in contact: f k n k It is a good approximation to assume that the kinetic friction does not depend on the relative velocity of the surfaces in contact, or on the area of contact Ex: The kinetic friction experienced by a box dragged on a certain surface does not depend on the face in contact with the surface f k f v f k n m f k mg v v

17 Static friction Appears at the surface of contact between two surfaces at rest relative to each other when there is a force trying to move them Direction: in the surface of contact (perpendicular on the normal force), against the force trying to move the object. Magnitude: equal to the force trying to move the object (such that the object is at rest). It increases with the increasing applied force up to a maximum value proportional to the normal. The coefficient of proportionality µ s is called coefficient of static friction: n f s n s f s Comment: The maximum static friction is larger than the kinetic friction Therefore, if an object is acted by a force, the static friction increases as the applied force increases, until it reaches its maximum, the object starts to move, and the kinetic friction takes over. Then, if the applied force remains the same, the object accelerates. riction, f f s f k f s no motion mg fk sliding Applied force,

18 Rolling friction Appears at the flattening surface of contact between a wheel and a surface The direction and magnitude are similar to the kinetic friction f r n However, the coefficient of rolling friction µ r is smaller than in the kinetic one for the same pair of materials in contact, explaining why it is easier to roll an object rather than dragging it: Drag r s k r Is the resistance opposed by a fluid against the motion of an object submerged in it luid dynamics is a rather complex field of physics, so here we ll model the drag only very approximately Direction: against the direction of object s velocity Magnitude: depends on the speed of the object. Then, using a simple model, f n cv vˆ f r where c is a constant

19 more about drag When small objects move with low speed, we are dealing with Stokes Drag: or objects moving with high speed, we are dealing with Rayleigh Drag: f f kvvˆ Dv v 2 ˆ f f kv Dv The coefficients k and D are constants that depends on the geometry of the object Exercise 2: Terminal velocity. A large ball is released from rest and accelerates vertically downward. The ball experiences a drag due to the air resistance which increases with velocity until it becomes equal to the weight and the object moves with a terminal velocity. a) Using the notation on the diagram, what is the relationship between the weight and drag before the terminal velocity? b) ind the terminal velocity v t both in the Stokes and Rayleigh drag cases: f g v 2 mg

20 Problem: 5. 1D Dynamics Acceleration of one body on a flat rough surface: A box of mass m is pulled by a force making an angle θ with the horizontal so that it slides across the floor with kinetic friction coefficient µ k. The box starts from rest. a) Draw the force vector diagram including all the forces acting on the box. b) Write Newton s 2 nd law along horizontal and vertical axes c) What is the acceleration of the box in terms of given quantities? d) Calculate the velocity of the box after a time t. e) Say that the velocity of the box is v 0 when the force is removed. What is the ulterior acceleration of the box? f) Calculate the stopping distance of the box after the force is removed. n f k m θ mg

21 Problem: 6. 1D Dynamics Acceleration of an object on a rough incline: A box of mass m slide down an incline of angle θ, with the coefficient of kinetic friction is μ k. The object was released from the top of the incline from rest and reaches the bottom of the incline in a time t. a) Draw the force vector diagram including all the forces acting on the box. b) Write Newton s 2 nd law along axes parallel and perpendicular on the incline c) ind the acceleration of the object down the incline in terms of g, θ and μ k d) ind the length of the incline e) ind the velocity of the box at the bottom of the incline f k m n mg θ

22 Systems of Interacting Objects The ree Body Diagram Technique In general objects are not isolated: they move as part of systems of objects In the previous lecture set we ve already introduced two contact forces that can intermediate the interaction between the various parts of a system: normal forces and tensions in wires When exerted between parts of a system, these forces are called internal forces: they cannot modify the motion of a system as a whole, but they can redistribute the motion between the different parts The ree Body Diagram technique offers a strategy to analyze systematically the motion of such a system. In general we ll follow its steps: 1. Impart the system into free bodies 2. Build the force diagram for each free body 3. Select a system of coordinate for each free body and write out Newton s 2 nd Law for each one of them 4. Solve the resulting simultaneous equations for the unknowns of the problem: in 2D, a complex of n bodies will result in 2n independent equations: n along x and n along y. These can be completed with specific definitions. The equations will contain forces, masses, accelerations and constant parameters In our analysis we ll make as usually simplifying assumptions: for instance, we ll assume that the free bodies and ropes cannot be deformed, and we ll neglect the mass of ropes and pulleys called ideal pulleys

23 Problem 7. Multiple bodies interacting by surface contact: Three crates with masses m 1, m 2 and m 3 sit on a rough surface with coefficient of kinetic friction µ. The crates are pushed by a force horizontally from left. The system accelerates horizontally. a) Sketch complete free body diagrams for each crate. b) Based on these diagrams, write Newton s 2nd Law symbolically for each crate, along some xy-directions. c) Calculate the horizontal acceleration of the system in terms of given quantities, using the equations written in part b). Analyze the final equation you obtained for, and suggest an argument that could have given you the respective relationship without the intermediary step of considering the free body diagrams. m 1 m 2 m 3

24 Problem: 8. Multiple bodies interacting by ropes passing around pulleys: Two boxes with masses m 1 and m 2 are connected by a massless rope passing around three ideal pulleys, as in the figure. When the system is released from rest (configuration A), the mass m 1 moves upward and m 2 downward. a) Sketch complete free body diagrams for each mass, and the tensions on the pulleys b) Based on these diagrams, write Newton s 2nd Law symbolically for each mass, along some xy-directions. c) Calculate the acceleration of the two masses m 1 m 2

25 Dynamics of Circular Motion Centripetal orces The centripetal acceleration is caused by a net centripetal force. By Newton s 2 nd Law v 2 r mar m r path r a r v The centripetal force is the resultant of the radial components of all the forces acting on the object r Therefore, to calculate it, add the radial components of all forces acting on the particle moving in a circle, taking the inward components to be positive and outward components negative Then, the centripetal acceleration is given by Newton s 2 nd Law applied along the radius as above Ex: 3 forces acting on a mass m moving in a circle v 2 r 2r 1 r m r 3 2r Note that, if the force is perpendicular on the radius, it doesn t contribute to the centripetal force 1r 1 2

26 Exercise 3: How Angelina Died by Bad Physics In the movie Wanted, bullets are curved by skilled tattooed assassins. or instance, Angelina kills herself by firing a bullet in a circle passing through the skulls of some bald dudes before hitting her. Say that the bullet has a mass m = 4.2 g and a muzzle speed of 600 m/s. Also, say that the dudes arranged themselves conveniently in a circle with radius r = 5.0 m. a) How large should be the centripetal force keeping the bullet on the circular trajectory? Meditate about the possible origin of such a force and how realistic is such a scenario b) How fast should Angelina toss the gun to the sweaty guy in the middle for the scene to make sense?

27 Dynamics of Circular Motion Uniform and Nonuniform motion If the tangent components of the forces acting on a particle moving in a circle cancel each other out, only the centripetal force is non-zero, the motion is uniform circular (that is, the speed v = constant) Otherwise, if besides the centripetal force there is a net tangent force component, the motion is nonuniform circular (that is, the speed v is not constant) In a non-uniform circular motion, by Newton s 2 nd Law, the net tangent for is responsible for the tangent acceleration Hence, the tangent force is given by dv t mat m dt r r a r t a t v Variable speed Ex: Going back to the example on the previous slide, we see that the mass m performs nonuniform circular motion, since besides centripetal force it is acted by a net tangent force responsible for a change in speed dv t 1 t 2 t 3 m dt 1 3 1t 2t 2

28 Exercise 4: A ball of mass m is connected by a string of length r and moved into a circular trajectory with constant speed v. Let s estimate the force a person must exert on the string to make the ball revolve in a horizontal circle. a) What kind of centripetal force is exerted on the ball? r v b) Based on Newton s 2 nd Law, what is this force in terms of given quantities? Problem 9. Tension as a centripetal force: Now let s assume that the ball from the exercise above is swung in a vertical circle, still with a constant speed. a) Determine the tension in an arbitrary point of the circle where the radius makes an angle θ with respect to the horizontal b) Use the result from part (a) to determine the tension in the string when the string is horizontal, on top of the circle, and at the bottom of the circle.

29 Problem: 10. Conical pendulum: A bob of mass m is suspended from a fixed point with a massless string of length L (i.e., it is a pendulum). What tangential speed v must the bob have, so that it moves in a horizontal circle with the string always making an angle θ from the vertical?