Motion Motion is all around us. How something moves is probably the first thing we notice about some process. Quantifying motion is the were we learn how objects fall and thus gravity. Even our understanding of the atom was based on studying the motion of particles, starting with Rutherford s experiment with α particles and very thin gold foils. It is said he remarked It is like shooting 14 inch shells at a piece of tissue paper and having one bounce back at you. Speed Speed is just how fast something is moving. It is measured in how far something moves in some time. The units are always distance divided by time: Speed = distance time We might measure speed in miles/hour, meters/second etc. We said speed was how far something moves in some time. Obviously this depends on the time period we use. Things rarely move at constant speed. We measure the speed of something over a very short time, we can talk about the instantaneous speed. Instantaneous speed is like taking a snap shot of the speed. The average speed is what we have above for some time period that may not be very small. By the way, figuring out what the instantaneous speed (or other quantities) is the purpose of differential calculus. Notice we can re-write this in other useful forms you probably use every time you take a long car trip: distance = speed time time = distance speed Ifitis60milestoRichmondandyouaretravelingat30miles/hour, itwilltake2hourstogetthere. Tobecorrect,speedisrelative. Ifyou
gently toss a ball to another person standing in the park, they have no difficulty catching it. If you are moving in a car at 50 miles/hour and toss thenthesameballtheywillprobablynotbeyourfriendfor long! We will talk about relativity later and see how all this comes into play at normal speeds and very large speeds. Velocity If you tell someone to travel at 50 miles/hour for 1 hour to get to Norfolk, you would be correct but that person might end up in near Richmond. When we specify a speed and a direction, we are giving a velocity. Velocity is a vector. Like all vectors it has magnitude (how much) and direction (which way). Another example is moving in a circle on a merry-go-round. Your speed is constant but your velocity is not since the direction constantly changes as you go around the circle. Acceleration You know speed can change. If you are driving down I-64 and someone stops in front of you, you need to change your velocity. Just as the position changing gives us velocity, when velocity changes we have an acceleration. Acceleration is: Acceleration = change in velocity time interval Acceleration has units of distance divided by time 2. For example, if a car goes from 0 to 60 mi/hr in 10 seconds the acceleration is: Acceleration = 60mi/hr 10s = 6 mi/hr/s but those are ugly units. We can re-write the acceleration definition a way that is sometime useful: velocity acquired = Acceleration x time
Note this is velocity acquired. It adds to whatever velocity I already have when the acceleration starts. If my acceleration is 1 m/s 2 and I am traveling at 10 m/s, after 2 second of this acceleration, I will have a velocity of 12 m/s. Of course, if you start from rest (velocity equal to zero) then the velocity acquired is all of the velocity. Accelerationisreallyavectorsoitdoesnothavetobeinthesame direction as the velocity. When you come to a stop in your car, your acceleration is opposite to your velocity and so you stop. Remember the merry-go-round example of constant speed but changing velocity? There is an acceleration on the merry-go-round. The acceleration is pointed inward towards center of the circle so the velocity gets constantly pushes to point tangent to the path of the rider. We will talk more about motion in a circle later. Newton s Law of Motion Galileo define a property of all matter called inertia. Newton redefined this into what we now call Newton s 1 st Law of Motion. Inertia is the property of matter to stay in motion if it is moving and tostayatrestifatrest. Saidanotherway,thingsdon tjustjumpinto motion without something happening or without friction stop if they are moving. Newton s 1 st Law of Motion is stated as: Every material object continues in its state of rest or uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it. Mass is a measure of inertia. More mass means more inertia. Don t confuse weight and mass. Mass is a measure of an object s inertia i.e. its sluggishness to start moving or stop if it is in motion.
Weight is the gravitational force exerted on an object by the earth (the planet most of us live on). This can be confusing in the United States using the English units because we normally speak of pounds when we talk about weight and mass. A pound is a unit of force. Of course when we say a pound of mass, we mean an amount of inertia equal to an amount of material having a gravitational force of one pound on the surface of the earth. Believe it or not, the unit of mass in the English unit system is the slug! In the metric system, the unit of mass is the kilogram (kg). It is approximately the mass of a liter of water (1000 cm 3 ). The unit of force is the newton (N). On the surface of the earth, a 1 kg mass has a weight of 9.8 N. On the moon, the same object still has a mass of 1 kg but its weight is about 1 of 9.8 N or 1.6 N. 6 Newton s 2 nd Law of Motion First we should define force. Force is simply a push or a pull. It can be from gravity i.e. gravity pulls on a mass, a magnetic pull between a north and south magnetic pole or simply you pushing a book across a table. Newton s 2 nd law relates mass, acceleration and force. In words: The acceleration of an object is directly proportional to the net force acting on the object, is in the direction of the net force and is inversely proportional to the mass of the object. We can can write this as: Force acceleration = mass net force mass acceleration
SometimeswerearrangetheformulaasF=ma. Thereareseveral things we should notice about this statement. Notice the acceleration isinthesamedirectionasthenetforce. Thistellsusforceisavector like velocity or acceleration. When you push something, you push in a direction e.g, north or down, as well as some amount e.g. 5 pounds or 10 N. We can use Newton s 2 nd law to figure out what a newton (N) is in terms of mass, time and distance. Since force is mass times acceleration, the units of a newton can be broken down to kg m/s 2. Finally, notice the statement of Newton s 2 nd law says net force. This means all the forces added together acting on an object. Since they are vectors, we have to add all the forces vectors. Newton s 3 rd Law of Motion So far we have discussed what happens when one or more forces are applied to an object. You naturally tend to think of a force being exerted, say by a hammer, on a receiving object like a nail. Newton s 3 rd law says: When one object exerts a force on a second object, the second object exerts an equal and opposite force on the first object. Note we are talking about forces on different objects. You push on a cart, the cart pushes back with an equal and opposite force. A common mistake is to take the forces on the same object. Since they are equal and opposite, the cancel out (with this reasoning) and nothing would ever move. The flaw in this reasoning is that Newton s 3 rd law tells us the forces are on two different objects: F 12 = -F 21. Considerabulletbeingfiredfromagun. Theexplosionoftheshell pushes the bullet forward. and pushes the(gun + person) backwards (recoil). Of course, the bullet s mass is very small compared to the
(gun + person) mass so the same force applied to the bullet accelerates the bullet to a high speed while the force on the (gun + person) produces a much smaller acceleration on the (gun + person) which has the much larger mass. Free Fall One of the most obvious examples of acceleration is free fall. When you drop something, it s velocity constantly increases (neglecting air resistance). The velocity of a free falling object, starting from rest, is just: v = g t where g is acceleration of gravity (9.8 m/s 2 ). Time of Fall (s) Instantaneous Speed (m/s) 0 0 1 9.8 2 19.6 3 29.4 4 39.2 Thatisprettyeasybuthowfardoessomethingfall. Sincethespeedis constantly increasing, the distance traveled each second is increasing also. Still assuming we start from rest (v o =0), the distance traveled is given by: distance traveled = 1 2 g t2 Let s redo our table using this formula to add the distance traveled:
Time of Fall (s) Instantaneous Speed (m/s) distance traveled (m) 0 0 0 1 9.8 4.9 2 19.6 19.6 3 29.4 44.1 4 39.4 78.4 Notice the distance between each second gets bigger and bigger. A common observation is that some objects don t fall with the same acceleration. A leaf does not fall like a marble. Air resistance changes how the leaf falls and to some very much smaller extent the marble. If we could turn off air resistance e.g. put both in a vacuum, they would fall in the same manner. The basic two equations to figure out things with constant acceleration are: v = v o + a t x = x o + v o t + 1 2 a t2 v o is the initial velocity and x o is the initial position. If we are discussing an object falling, then a = g = 9.8 m/s 2. A couple of points to note about free falling objects. When we throw something vertically up (Note we are not starting fromrest), theobject goesupforawhile, momentarilystopsat some maximum altitude and then comes back down. If you work it out, you will find it comes back to your hand with the same speed into your hand. Of course the velocity is different since the velocity is now directed down and not up. Also note that the time to reach the maximum height is the same to fall back down to the original position (1/2 of the total time).
Equilibrium - Adding up Forces Consider the common situation when there is more than one force acting on an object. When the net force on an object is zero, there is no acceleration. We say the object is in equilibrium. If the object is at rest, then it stays at rest. You are currently in equilibrium setting in a chair. Gravity is pulling you down with a force equal to your weight. The chair is pushingbackupwithaforceequaltoyourweight(newton s3 rd law). If the chair pushes too hard, you will accelerate up into the air. It it does not push hard enough, you would accelerate down to the ground. That is one clever chair! Let s think about the example of an ramp (inclined plane). There are several forces acting when we push (apply a force) a crate up a ramp. Theforcesarethepush, theforceofgravity(mg)downonthe crate and the reaction force of the ramp on the crate (Newton s 3 rd law). For now we assume the ramp is frictionless. The reaction force of the ramp is called the normal force because it is perpendicular to the ramp. The weight of the crate can be resolved into two components which are perpendicular and parallel to the ramp. (To do this we would use trig. Note that these two components add up the the weight.). The perpendicular component is just balanced by the normal force. If the crate is motionless, the parallel component of theweightisjustbalancedbyourpush. Ifwepushjustalittleharder than the parallel component of the weight, the crate accelerates up the ramp. Push mg N
Energy - A First Look Probably the single most important concept in physics is that of energy. The concept of work did not exist in Newton s time. The nature of energy was still being debated in the 1850 s. To introduce the idea of energy, let s look at a related idea of work. When we multiply (Force x time) we get change in momentum. When we multiply (Force x distance) we get Work (change in energy). The definition of work is: Work = force x distance where the force is the component of the force in the direction of the displacement (distance). In the case of the ramp, this is just the push force times the sine of the angle the ramp makes with the horizontal direction. If you carry two suitcases up one flight of stairs you do the same amount of work as moving one suitcase up two floors. The force involved is the gravitational force on the suitcase. Note that you have to move some distance against a force. If you push really hard on a wall, you might think you are doing a lot of work but, in fact, you have done no work since the wall does not move. The unit of work is a (newton x meter) which is called a joule (J). Work is not a vector but a direction less number or scalar. Forms of Mechanical Energy Energy can take many forms. The first we should consider is gravitational potential energy. This is energy which depends on position and is stored energy. If you stretch a rubber band, you store energy in the rubber. Chemical energy is another form of potential energy. Gasoline is a store of energy because of the chemical
bonds in the hydrocarbon. The most obvious example of potential energy is gravitational potential energy. When we lift something up, like our suitcase to the second floor, we are storing energy in the object. The gravitational potential energy is given by: gravitational potential energy = weight x height GPE = mgh Note that the units are joules since it is a force (weight) times a distance. Also note it depends on the height you raise the object. As an example, consider our suitcase. Let s say its mass is 10 kg. We know the force of gravity on it is (mass x g) = 10 kg x 9.8 m/s 2 = 98 N. Let s say we move it upstairs which is 3 m above the ground floor. The gravitational potential energy the suitcase has after we move it is (98 N x 3 m) or 294 J. For the case of the ramp, the change in the gravitational potential energy is still mgh but the h is not the distance we move along the ramp but the difference in height of the mass from its initial position and its final position (h 2 - h 1 ). Push h 1 h 2