Ch2. Force and Motion 2.1 Newtonian Mechanics 2.2 Newton’s First Law of Motion 2.3 Newton’s Second Law of Motion 2.4 Newton’s Third Law of Motion 2.5 Some Particular Forces 2.6 Applications of Newton’s Laws

2.1 Newtonian Mechanics:

2.1 Newtonian Mechanics Three laws of motion (empirical) – basis for classical mechanics Describe the relation between a force and the motion it causes Can be applied to the motion of objects ranging in size from the very small to astronomical. Isaac Newton (1642 – 1727)

2.2 Newton’s First Law of Motion:

2.2 Newton’s First Law of Motion If no force acts on a body, the body’s velocity cannot change; that is, the body cannot acceleration. (HRW) Translation of Newton’s original statement: “Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.”

2.2 Newton’s First Law of Motion:

2.2 Newton’s First Law of Motion Newton's first law of motion : all objects resist change in their state of motion. All objects have this tendency, i.e. they have inertia . The mass of an object is a measure of its inertia. Larger mass more inertia larger tendency to resist changes in its state of motion The mass of an object is usually given in the unit of g (gram) or kg (kilogram) .

2.2 Newton’s First Law of Motion:

2.2 Newton’s First Law of Motion Newton's first law of motion is not true for all reference frames. But we can always find reference frames in which it (as well as the rest of Newtonian mechanics) is true inertial reference frame , or simply inertial frames . Examples of non-inertial frame : a linearly accelerating frame, a rotating frame. In short, all inertial frames are non-accelerating .

2.3 Newton’s Second Law of Motion:

2.3 Newton’s Second Law of Motion The net force on a body is equal to the product of the body’s mass and its acceleration. (HRW) Translation of Newton’s original statement: “The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed.”

2.3 Newton’s Second Law of Motion:

2.3 Newton’s Second Law of Motion Definition of Force According to the modern version of Newton’s Second Law of Motion, one may express it in equation form: For deeper understanding of the above equation, we rewrite the R.H.S. as:

2.3 Newton’s Second Law of Motion:

2.3 Newton’s Second Law of Motion Definition of Force The last equality is true only when the mass ( m ) of the object is a constant. The more general form of the Newton’s Second Law of Motion is given by: where is the linear momentum (see Section 5.1) of the object and the equation may be regarded as the definition of Force .

2.3 Newton’s Second Law of Motion:

2.3 Newton’s Second Law of Motion Unit of Force The SI unit of force is the newton ( N ). It is defined directly from equation: in terms of the units for mass and acceleration: For example, we can exert a 2 N force on a 1 kg object by pulling it so that its measured acceleration of 2 m/s 2 , and so on.

2.3 Newton’s Second Law of Motion:

2.3 Newton’s Second Law of Motion Net Force 10 N 10 N 10 N 20 N These forces balance . The net force is zero and the block does not move. These forces do not balance. The block moves as if it were being pulled by a net force of 10 N. 5 N 5 N Net force = 10 N 6 N 6 N

2.4 Newton’s Third Law of Motion:

2.4 Newton’s Third Law of Motion When two bodies interact, the forces on the bodies from each other are always equal in magnitude and opposite in direction. (HRW) Translation of Newton’s original statement: “To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.”

2.4 Newton’s Third Law of Motion:

downward press from the cow upward support from the ground 2.4 Newton’s Third Law of Motion In the above system, the upward support on the cow from the floor and the downward press on the floor from the cow form a third-law force pair (action and reaction pair) Action and Reaction

2.5 Some Particular Forces:

2.5 Some Particular Forces In the following, we focus on the physics of three common types of force: Frictional Force, Drag Force, and Centripetal Force An engineer preparing a racing motor car must consider all three types.

2.5 Some Particular Forces:

2.5 Some Particular Forces 14 Frictional Force A car will eventually stop if the engine is turned off: the friction slows down the cars’ motion. Friction arises whenever an object slides or tends to slide over another object and always acts against the motion, actual or intended. PUSH To Move FRICTIONAL FORCE to Resist Motion

2.5 Some Particular Forces:

2.5 Some Particular Forces 15 Frictional Force Friction is due to the irregularities on a surface. Even a smooth surface has these “peaks and valleys”. If two surfaces rub over each other, the irregularities on the two surfaces would crush together, resulting in an opposing force. PUSH To Move FRICTIONAL FORCE to Resist Motion

2.5 Some Particular Forces:

2.5 Some Particular Forces 16 Frictional Force What does friction depend on? Friction does not depend on the area of contact, but depends only on the nature of the contact surfaces. Friction is much reduced if the contact surfaces are lubricated with water or oil, and if rollers and bearings are placed between the contact surfaces. The greater the weight of the object, the greater is the friction.

2.5 Some Particular Forces:

Properties of Frictional Force Experiments showed that when a dry and unlubricated body presses against a surface of the same condition, and a force attempts to slide the body along the surface, the resulting frictional force has three properties. 2.5 Some Particular Forces 17

2.5 Some Particular Forces:

2.5 Some Particular Forces 18 Property 1: If the body does not move, then the static frictional force and the component of that is parallel to the surface balance each other. They are equal in magnitude, and is directed opposite to that component of .

2.5 Some Particular Forces:

2.5 Some Particular Forces 19 Property 2: The magnitude of has a maximum value f s, max that is given by where μ s is the coefficient of static friction and F N is the magnitude of the normal force on the body from the surface. If the magnitude of the component of that is parallel to the surface exceeds f s, max then the body begins to slide along the surface.

2.5 Some Particular Forces:

2.5 Some Particular Forces 20 Property 3: If the body begins to slide along the surface, the magnitude of the frictional force rapidly decreases to a value f k given by where μ k is the coefficient of kinetic friction . Thereafter, during the sliding, a kinetic frictional force with magnitude f k given by the above equation opposes the motion.

2.5 Some Particular Forces:

2.5 Some Particular Forces Friction and Moving Car According to Newton’s Third Law of motion, the car is driven by the forward frictional push of the road on the tyres . wheel turning tyre pushes backwards against the road forward frictional force pushes the car forwards

2.5 Some Particular Forces:

2.5 Some Particular Forces Friction and Moving Car If we apply the brakes (very hard), the wheels are locked and thus prevent from rotating. As a result, the car skids and decelerates. The decelerating force is actually the kinetic frictional force – a backward push of the ground on tyres . wheels locked backward frictional force decelerates the car

2.5 Some Particular Forces:

2.5 Some Particular Forces Example A car skids with all four wheels locked for 38.5m and then runs into a tree. The impact speed of the car is estimated from the damage to be 40kmh -1 . The coefficient of friction (kinetic) for the tyre /road contact is found to be 0.76. Estimate the speed of the car just prior to skidding.

2.5 Some Particular Forces :

2.5 Some Particular Forces Example The car decelerates on skidding. The decelerating force F is given by If the average deceleration over a skidding distance is a , then (Newton ’ s Second Law)

2.5 Some Particular Forces :

2.5 Some Particular Forces Example Applying the equation of uniform acceleration,

2.5 Some Particular Forces :

2.5 Some Particular Forces Drag Force When there is a relative velocity between a fluid ( either a gas or a liquid ) and a body, the body experiences a drag force that opposes the relative motion and points in the direction in which the fluid flows relative to the body.

2.5 Some Particular Forces :

2.5 Some Particular Forces Drag Force In particular, if air is the fluid, the body is blunt (like a ball), and the relative motion is fast enough so that the air becomes turbulent (breaks up into swirls) behind the body, the magnitude of the drag force is related to the relative speed v by an experimentally determined drag coefficient C according to

2.5 Some Particular Forces :

2.5 Some Particular Forces Drag Force where ρ 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 . The drag coefficient C (typical values range from 0.4 to 1.0) is not truly a constant for a given body because if v varies significantly, the value of C can vary as well. In this course, we ignore such complications.

2.5 Some Particular Forces :

2.5 Some Particular Forces 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. This upward force opposes the downward gravitational force on the body. We can relate these forces to the body’s acceleration by using Newton’s second law applied along the vertical y axis ( ) as where m is the mass of the body. Unneeded subscript y is dropped

2.5 Some Particular Forces :

2.5 Some Particular Forces Terminal Speed If the body falls for a sufficiently long time, D eventually equals F g and the acceleration a becomes zero – the body’s speed no longer increases. The body then falls at a constant speed, called the terminal speed v t . To determine v t , we set a = 0 :

2.5 Some Particular Forces :

2.5 Some Particular Forces Terminal Speed [left] the pumpkin when it has just begun to fall through air and, [middle] the free-body diagram a little later, after a drag force has developed. [right] The drag force has increased until it balances the gravitational force on the body. The pumpkin now falls at its constant terminal speed . v small, a large v increased, a reduced v = v t , a =0

2.5 Some Particular Forces :

2.5 Some Particular Forces Terminal speed of falling raindrops How long does it take to reach the terminal speed? Solve If there were no drag, the speed would be: at ground level. A rain drop with radius R =1.5 mm falls from a cloud that is at height h =120 m above the ground. Assume that the drop is spherical throughout its fall. The density of water r w is 1000kg/m3, and the density of air r a is 1.2. What is its terminal speed, if the drag coefficient is 0.6?

2.5 Some Particular Forces :

2.5 Some Particular Forces Centripetal Force An object is in uniform circular motion, if it travels around a circle or a circular arc at constant (uniform) speed. Although the speed does not vary, the object is accelerating because the velocity changes direction. The acceleration is always directed radially inward and hence called a centripetal acceleration . By Newton’s second law, a force must cause the centripetal acceleration and must also be directed toward the center of the circle – Centripetal Force.

2.5 Some Particular Forces :

2.5 Some Particular Forces Centripetal Force R String An overhead view of a ball moving with constant speed v in a circular path of radius R on a horizontal frictionless surface. The centripetal force on the ball is , the pull from the string, directed toward the center of the circle .

2.5 Some Particular Forces :

2.5 Some Particular Forces Centripetal Force r p x p y p x y θ θ The magnitude of the centripetal acceleration a is given by and can be proven as follows:

2.5 Some Particular Forces :

2.5 Some Particular Forces Centripetal Force

2.5 Some Particular Forces :

2.5 Some Particular Forces Centripetal Force and the magnitude of the centripetal force F is hence given by where m is the mass of the object.

2.5 Some Particular Forces :

38 2.5 Some Particular Forces Car Travelling Round a Bend Centripetal force Centre of circular path r v A car travelling round a bend on a level road can be viewed as moving along a circular path The centripetal force is pointing towards the centre of the circular path

2.5 Some Particular Forces :

Car Travelling Round a Bend The centripetal force is provided by the sideway friction between the tyres and the road surface: Equating the two forces, we get v is called the critical curve speed for the bend 2.5 Some Particular Forces ??? static or kinetic coeff . ???

2.5 Some Particular Forces :

2.5 Some Particular Forces Car Travelling Round a Bend When the speed of the car is smaller than the critical curve speed of a bend, the car has no difficulty in negotiating the bend . When the car is just at the critical curve speed , it is travelling at the limit of adhesion to the road. It cannot brake or steer onto a tighter course to avoid an unexpected hazard without risking side-slipping. When the speed of the car is greater than the critical curve speed, the frictional force is not large enough to provided the necessary centripetal force. As a result, the car side-slips .

2.5 Some Particular Forces :

2.5 Some Particular Forces Car Travelling Round a Bend If the bend is banked, the car may negotiate a bend at a higher speed because a component of the normal reaction contributes to the centripetal force as shown: Why? Which way to bank?

2.5 Some Particular Forces :

N mg F 2.5 Some Particular Forces Car Travelling Round a Bend When a bicycle (or motorcycle) travel round a bend, it is necessary to lean it into the turn. Otherwise the centrifugal forces will throw the bicycle (or motorcycle) over on its side.

2.5 Some Particular Forces :

2.5 Some Particular Forces Analysis of a car travelling round a bend (neglect friction) If the car mass moves at a constant speed v of 20 m/s around a banked circular track of radius R = 190 m. The angle required to prevent skidding regardless of m is at least

2.5 Some Particular Forces :

2.5 Some Particular Forces Analysis of a car travelling round a bend (include) friction f ) The car just begin to skid At the same degree of banking and radius, friction allows higher speed. v becomes 24.4 m/s, when . For safety, it is better not to depend on friction.

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