More Applications of Newton’s Laws Chapter 5 More Applications of Newton’s Laws

Forces of Friction 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

Forces of Friction, cont. The force of static friction, ƒs, is generally greater than the force of kinetic friction, ƒk The coefficient of friction (µ) depends on the surfaces in contact Friction is proportional to the normal force ƒs £ µs n and ƒk= µk n These equations relate the magnitudes of the forces, they are not vector equations

Forces of Friction, final 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 Static friction acts to keep the object from moving If increases, so does If decreases, so does ƒs  µs n where the equality holds when the surfaces are on the verge of slipping Called impending motion

Active Figure AF_0501 static and kinetic frictional forces.swf

Kinetic Friction The force of kinetic friction acts when the object is in motion Although µk can vary with speed, we shall neglect any such variations ƒk = µk n

Some Coefficients of Friction

Friction in Newton’s Laws Problems Friction is a force, so it simply is included in the SF in Newton’s Laws The rules of friction allow you to determine the direction and magnitude of the force of friction

Friction Example, 1 The block is sliding down the plane, so friction acts up the plane This setup can be used to experimentally determine the coefficient of friction µ = tan q For µs, use the angle where the block just slips For µk, use the angle where the block slides down at a constant speed

Friction Example 2 Image the ball moving downward and the cube sliding to the right Both are accelerating from rest There is a friction force between the cube and the surface

Friction Example 2, cont Two objects, so two free body diagrams are needed Apply Newton’s Laws to both objects The tension is the same for both objects

Uniform Circular Motion A force, , is directed toward the center of the circle This force is associated with an acceleration, ac Applying Newton’s Second Law along the radial direction gives

Uniform Circular Motion, cont A force causing a centripetal acceleration acts toward the center of the circle It causes a change in the direction of the velocity vector If the force vanishes, the object would move in a straight-line path tangent to the circle

Active Figure AF_0509 tangential velocity.swf

Centripetal Force The force causing the centripetal acceleration is sometimes called the centripetal force This is not a new force, it is a new role for a force It is a force acting in the role of a force that causes a circular motion

Conical Pendulum The object is in equilibrium in the vertical direction and undergoes uniform circular motion in the horizontal direction v is independent of m

Horizontal (Flat) Curve The force of static friction supplies the centripetal force The maximum speed at which the car can negotiate the curve is Note, this does not depend on the mass of the car

Banked Curve These are designed with friction equaling zero There is a component of the normal force that supplies the centripetal force

Loop-the-Loop This is an example of a vertical circle At the bottom of the loop (b), the upward force experienced by the object is greater than its weight

Loop-the-Loop, Part 2 At the top of the circle (c), the force exerted on the object is less than its weight

Non-Uniform Circular Motion The acceleration and force have tangential components produces the centripetal acceleration produces the tangential acceleration

Vertical Circle with Non-Uniform Speed The gravitational force exerts a tangential force on the object Look at the components of Fg The tension at any point can be found

Top and Bottom of Circle The tension at the bottom is a maximum The tension at the top is a minimum If Ttop = 0, then

Active Figure AF_0515 tangential and radial forces.swf

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

Motion with Resistive Forces, cont The magnitude of can depend on the speed in complex ways We will discuss only two is proportional to v Good approximation for slow motions or small objects is proportional to v2 Good approximation for large objects

R Proportional To v The resistive force can be expressed as b depends on the property of the medium, and on the shape and dimensions of the object The negative sign indicates is in the opposite direction to

R Proportional To v, Example Analyzing the motion results in

R Proportional To v, Example, cont Initially, v = 0 and dv/dt = g As t increases, R increases and a decreases The acceleration approaches 0 when R ® mg At this point, v approaches the terminal speed of the object

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

Active Figure AF_0518 terminal speed.swf

R Proportional To v2 For objects moving at high speeds through air, the resistive force is approximately proportional to the square of the speed R = 1/2 DrAv2 D is a dimensionless empirical quantity that is called the drag coefficient r is the density of air A is the cross-sectional area of the object v is the speed of the object

R Proportional To v2, example Analysis of an object falling through air accounting for air resistance

R Proportional To v2, Terminal Speed The terminal speed will occur when the acceleration goes to zero Solving the equation gives

Some Terminal Speeds

Fundamental Forces Gravitational force Electromagnetic forces Between two objects Electromagnetic forces Between two charges Nuclear force Between subatomic particles Weak forces Arise in certain radioactive decay processes

Gravitational Force Mutual force of attraction between any two objects in the Universe Inherently the weakest of the fundamental forces Described by Newton’s Law of Universal Gravitation

Electromagnetic Force Binds atoms and electrons in ordinary matter Most of the forces we have discussed are ultimately electromagnetic in nature Magnitude is given by Coulomb’s Law

Nuclear Force The force that binds the nucleons to form the nucleus of an atom Attractive force Extremely short range force Negligible for r > ~10-14 m For a typical nuclear separation, the nuclear force is about two orders of magnitude stronger than the electrostatic force

Weak Force Tends to produce instability in certain nuclei Short-range force About 1034 times stronger than gravitational force About 103 times stronger than the electromagnetic force

Unifying the Fundamental Forces Physicists have been searching for a simplification scheme that reduces the number of forces 1987 – Electromagnetic and weak forces were shown to be manifestations of one force, the electroweak force The nuclear force is now interpreted as a secondary effect of the strong force acting between quarks

Drag Coefficients of Automobiles

Reducing Drag of Automobiles Small frontal area Smooth curves from the front The streamline shape contributes to a low drag coefficient Minimize as many irregularities in the surfaces as possible Including the undercarriage