The Laws of Motion
The Concept of Force Newton s First Law and Inertial Frames Mass Newton s Second Law The Gravitational Force and Weight Newton s Third Law Analysis Models using Newton s Second Law Forces of Friction
Forces in everyday experience Push on an object to move it Throw or kick a ball May push on an object and not be able to move it Forces are what cause any change in the velocity of an object. Newton s definition A force is that which causes an acceleration Section 5.1
Contact forces involve physical contact between two objects Examples a, b, c Field forces act through empty space No physical contact is required Examples d, e, f Section 5.1
Gravitational force Between objects Electromagnetic forces Between electric charges Nuclear force Between subatomic particles Weak forces Arise in certain radioactive decay processes Note: These are all field forces. Section 5.1
A spring can be used to calibrate the magnitude of a force. Doubling the force causes double the reading on the spring. When both forces are applied, the reading is three times the initial reading. Section 5.1
The forces are applied perpendicularly to each other. The resultant (or net) force is the hypotenuse. Forces are vectors, so you must use the rules for vector addition to find the net force acting on an object. Section 5.1
5.2) Newton s First Law and Inertial Frames Newton s First law of motion = In the absence of external forces, an object at rest remains at rest and an object in motion continues in motion with a constant velocity (that is, with a constant speed in a straight line). When no force acts on an object, the acceleration of the object is zero. If nothing acts to change the object s motion, then its velocity does not change. Inertia of the object = The tendency of an object to resist any attempt to change its velocity. Inertial Frames A moving object can be observed from any number of reference frames. An inertial frame of reference is one that is not accelerating. Any reference frame that moves with constant velocity reletive to an inertial frame is itself an inertial frame.
Newton s First Law Reviewed Newton s First Law: An object at rest or an object in motion at constant speed will remain at rest or at constant speed in the absence of a resultant force. A glass is placed on a board and the board is jerked quickly to the right. The glass tends to remain at rest while the board is removed. Assume glass and board move together at constant speed. If the board stops suddenly, the glass tends to maintain its constant speed.
The tendency of an object to resist any attempt to change its velocity is called inertia. Mass is that property of an object that specifies how much resistance an object exhibits to changes in its velocity. Masses can be defined in terms of the accelerations produced by a given force acting on them: m a 1 2 m a 2 1 The magnitude of the acceleration acting on an object is inversely proportional to its mass. Section 5.3
Mass is an inherent property of an object. Mass is independent of the object s surroundings. Mass is independent of the method used to measure it. Mass is a scalar quantity. Obeys the rules of ordinary arithmetic The SI unit of mass is kg. Section 5.3
Mass and weight are two different quantities. Weight is equal to the magnitude of the gravitational force exerted on the object. Weight will vary with location. Example: w earth = 180 lb; w moon ~ 30 lb m earth = 2 kg; m moon = 2 kg Section 5.3
5.4) Newton s Second Law Newton s 2nd law what happens to an object that has a nonzero resultant force acting on it. The acceleration of an object is directly proportional to the resultant force acting on it (a F). The magnitude of the acceleration of an object is inversely proportional to its mass (a 1/m) Newton s 2nd law = The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (a F/m) Relate mass and force through the mathematical statement of Newton s 2nd law: F m a (5.2) This equation is a vector expression and hence is equivalent to three component equations : F ma (5.3) Fx ma x y y Fz ma z Unit of force = the newton, which is defined as the force that, when acting on a 1-kg mass, produces an acceleration of 1 m/s 2. 1 N 1 kg m/s 2 (5.4)
Second Law: Whenever a resultant force acts on an object, it produces an acceleration: an acceleration that is directly proportional to the force and inversely proportional to the mass. a F m
Pushing the cart with twice the force produces twice the acceleration. Three times the force triples the acceleration.
F F a a/2 Pushing two carts with same force F produces one-half the acceleration. The acceleration varies inversely with the amount of material (the mass).
Example (5.1) : An Accelerating Hockey Puck A hockey puck having a mass of 0.30 kg slides on the horizontal, frictionless surface of an ice rink. Two forces act on the puck, as shown in Figure (5.5). The force F 1 has a magnitude of 5.0 N, and the force F 2 has a magnitude of 8.0 N. Determine both the magnitude and the direction of the puck s acceleration.
5.5) Gravitational Force and Weight All objects are attracted to the Earth. The attractive force exerted by the Earth on an object = the force of gravity F g. This force is directed toward the center of the Earth, and its magnitude = the weight of the object. A freely falling object experiences an acceleration g acting toward the center of the Earth. Applying Newton s 2nd law F = ma to a freely falling object of mass m, with a = g and F = F g, we obtain : F g = mg (5.6) The weight of an object = the magnitude of F g = mg. Because it depends on g, weight varies with geographic location. g decreases with increasing distance from the center of the Earth bodies weigh less at higher altitudes than at sea level.
In Newton s Laws, the mass is the inertial mass and measures the resistance to a change in the object s motion. In the gravitational force, the mass is determining the gravitational attraction between the object and the Earth. Experiments show that gravitational mass and inertial mass have the same value. Section 5.5
If two objects interact, the force exerted by object 1 on object 2 (action force)is equal in magnitude and opposite in direction F to the 21 force exerted by object 2 on object 1(reaction force). F12 F21 Note on notation: B. F 12 is the force exerted by A on The action force is equal in magnitude to the reaction force and opposite in direction. One of the forces is the action force, the other is the reaction force F AB Section 5.6
Third Law: For every action force, there must be an equal and opposite reaction force. Forces occur in pairs. Action Reaction Reaction Action
The force F 12 exerted by object 1 on object 2 is equal in magnitude and opposite in F 21 direction to exerted by object 2 on object 1. F F 12 21 Section 5.6
The normal force (table on monitor) is the reaction of the force the monitor exerts on the table. Normal means perpendicular, in this case The action (Earth on monitor) force is equal in magnitude and opposite in direction to the reaction force, the force the monitor exerts on the Earth. Section 5.6
The particle model is used by representing the object as a dot in the free body diagram. The forces that act on the object are shown as being applied to the dot. The free body helps isolate only those forces acting on the object and eliminate the other forces from the analysis. Section 5.6
Assumptions Objects can be modeled as particles. Interested only in the external forces acting on the object Can neglect reaction forces Initially dealing with frictionless surfaces Masses of strings or ropes are negligible. The force the rope exerts is away from the object and parallel to the rope. When a rope attached to an object is pulling it, the magnitude of that force is the tension in the rope. Tension When a rope attached to an object is pulling on the object the rope exerts a force T on the object, and the magnitude of that force is called the tension in the rope. Tension is the magnitude of a vector quantity, so tension is a scalar quantity. Section 5.7
If the acceleration of an object that can be modeled as a particle is zero, the object is said to be in equilibrium. The model is the particle in equilibrium model. Mathematically, the net force acting on the object is zero. F 0 F x 0 and F 0 y Section 5.7
A lamp is suspended from a chain of negligible mass. The forces acting on the lamp are: the downward force of gravity the upward tension in the chain Applying equilibrium gives Fy 0T Fg 0T Fg Section 5.7
If an object that can be modeled as a particle experiences an acceleration, there must be a nonzero net force acting on it. Model is particle under a net force model. Draw a free-body diagram. Apply Newton s Second Law in component form. Section 5.7
Forces acting on the crate: A tension, acting through the rope, is the magnitude of force T The gravitational force, F g The normal force, n, exerted by the floor Section 5.7
n rope y T crate (a) Figure (5.8) x (b) F g Apply Newton s second law in component form to the crate. Force in x direction Force in the x direction is only T. Applying F x ma x to the horizontal motion gives : F x T ma x or a x T m Force in y direction No acceleration occures in the y direction. Applying F y ma y with a y =0 : n ( Fg ) 0 or n F g
If T is a constant force, then the acceleration a x = T/m also is constant. Apply the constant-acceleration equations of kinematics from Chapter 2 to obtain the crate s displacement x and velocity v x as functions of time. Because a x = T/m is constant, Equation (2.8) and (2.11) can be written as : v xf v xi T m t x In the situation just described, the magnitude of the normal force n is equal to the magnitude of F g, but this is not always the case. v xi t 1 2 T m t 2
Apply Newton s Second Law in component form: F T ma x F n F 0 n F y g g x Solve for the unknown(s) If the tension is constant, then a is constant and the kinematic equations can be used to more fully describe the motion of the crate.
The normal force is not always equal to the gravitational force of the object. For example, in this case F n F F 0 y and may also be less than g n n mg F F g Section 5.7
The case where the magnitude of n the magnitude of F g. 2) A lamp suspended from a light chain fastened to the ceiling (Figure (5.10a). A free-body diagram for the lamp (Figure (5.10b) shows that the forces acting on the lamp are the downward force of gravity F g and the upward force T exerted by the chain. T T = T Figure (5.10) (a) T (b) F g (c) Apply the second law to the lamp, a = 0 (no forces in the x direction), F x 0. The condition Fy ma y 0 gives : Fy T Fg 0 or T F g T and T (Figure (5.10c)) - not include in the problems.
Conceptualize Draw a diagram Choose a convenient coordinate system for each object Categorize Is the model a particle in equilibrium? If so, SF = 0 Is the model a particle under a net force? If so, SF = m a Section 5.7
Analyze Draw free-body diagrams for each object Include only forces acting on the object Find components along the coordinate axes Be sure units are consistent Apply the appropriate equation(s) in component form Solve for the unknown(s) Finalize Check your results for consistency with your freebody diagram Check extreme values Section 5.7
Example (5.4) : A Traffic Light at Rest A traffic light weighing 125 N hangs from a cable tied to two other cables fastened to a support. The upper cables makes angles of 37.0 o and 53.0 o with the horizontal. Find the tension in the three cables. y T 3 T 2 37.0 o T 1 53.0 o T 3 T 1 T 2 37.0 o 53.0 o x (a) (b) F g T 3 (c) Figure (5.11)
A car of mass m is on an icy driveway inclined at an angle θ as in figure below. a) Find the acceleration of the car, assume the driveway is frictionless b) Suppose the car is released from rest at the top of the incline and the distance from the car s front bumper to the bottom of the incline is d. How long does it take the front bumper to reach the bottom of the hill. What is the car s speed as it arrives there. Section 5.7
Example (5.7) : One Block Pushes Another Two blocks of masses m 1 and m 2 are placed in contact with each other on a frictionless horizontal surface. A constant horizontal force F is applied to the block of mass m 1. (a)determine the magnitude of the acceleration of the two-block system. (b) Determine the amgnitude of the contact force between the two blocks.
Example (5.8) : Weighing a Fish in an Elevator A person weighs a fish o mass m on a spring scale attached to the ceiling of an elevator, as illustrated in Figure (5.13). Show that if the elevator accelerates either upward or downward, the spring scale gives a reading that is different from the weight of the fish.
When two or more objects are connected or in contact, Newton s laws may be applied to the system as a whole and/or to each individual object. Whichever you use to solve the problem, the other approach can be used as a check. Section 5.7
Example (5.9) : Atwood s Machine When two objects of unequal mass are hung vertically over a frictionless pulley of negligible mass, as shown in Figure (5.14), the arrangement is called an Atwood machine. The device is sometimes used in the laboratory to measure the free-fall acceleration. Determine the magnitude of the acceleration of the two objects and the tension in the lightweight cord.
Vary the masses and observe the values of the tension and acceleration. Note the acceleration is the same for both objects The tension is the same on both sides of the pulley as long as you assume a massless, frictionless pulley. What if? The mass of both objects is the same? One of the masses is much larger than the other? Section 5.7
Example (5.10) : Acceleration of Two Objects Cnnected by a Cord A ball of mass m 1 and a block of mass m 2 are attached by a lightweight cord that passes over a frictionless pulley of negligible mass, as shown in Figure (5.16a). The block lies on a frictionless incline of angle. Find the magnitude of the acceleration of the two objects and the tension in the cord.
When an object is in motion on a surface or through a viscous medium, there will be a resistance to the motion. This is due to the interactions between the object and its environment. This resistance is called the force of friction.
Friction is proportional to the normal force. ƒ s µ s n and ƒ k = µ k n μ is the coefficient of friction These equations relate the magnitudes of the forces; they are not vector equations. For static friction, the equals sign is valid only at impeding motion, the surfaces are on the verge of slipping. Use the inequality for static friction if the surfaces are not on the verge of slipping. Section 5.8
The coefficient of friction depends on the surfaces in contact. The force of static friction is generally greater than the force of kinetic friction. The direction of the frictional force is opposite the direction of motion and parallel to the surfaces in contact. The coefficients of friction are nearly independent of the area of contact.
Static friction acts to keep the object from moving. As long as the object is not moving, ƒ s = F If F F increases, so does If decreases, so does ƒ s ƒ s ƒ s µ s n Remember, the equality holds when the surfaces are on the verge of slipping. Section 5.8
The force of kinetic friction acts when the object is in motion. Although µ k can vary with speed, we shall neglect any such variations. ƒ k = µ k n Section 5.8
Vary the applied force Note the value of the frictional force Compare the values Note what happens when the can starts to move Section 5.8
Section 5.8
Friction is a force, so it simply is included in the F in Newton s Laws. The rules of friction allow you to determine the direction and magnitude of the force of friction. Section 5.8
Example (5.11) : Experimental Determination of s and k The following is a simple method of measuring coefficients of friction : Suppose a block is placed on a rough surface inclined relative to the horizontal, as shown in Figure (5.19). The incline angle is increased until the block starts to move. Let us show that by measuring the critical angle c at which this slipping just occurs, we can obtain s.
Example (5.12) : The Sliding Hockey Puck A hockey puck on a frozen pond is given an initial speed of 20.0 m/s. If the puck always remains on the ice and slides 115 m before coming to rest, determine the coefficient of kinetic friction between the puck and ice.
Example (5.13) : Acceleration of Two Connected Objects When Friction is Present A block of mass m 1 on a rough, horizontal surface is connected to a ball of mass m 2 by a lightweight cord over a lightweight, frictionless pulley, as shown in Figure (5.21a). A force of magnitude F at an angle with the horizontal is applied to the block as shown. The coefficient of kinetic friction between the block and surface is k. Determine the magnitude of the acceleration of the two objects.