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

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 F 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 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

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

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

is the time constant and = m/b

For objects moving at high speeds through air, the resistive force is approximately proportional to the square of the speed

R = 1/2 DAv2

D is a dimensionless empirical quantity that is called the drag coefficient

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

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

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