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Chapter 5
More Applications of
Newton’s Laws
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5.1 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
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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
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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
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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
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Active Figure5.1
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Some Coefficients of Friction
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Fig 5.2
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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
Fig 5.3(a)
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Fig 5.3(b)&(c)
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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
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Fig 5.4
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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
Fig 5.5
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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
Fig 5.6
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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
Fig 5.6
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Fig 5.7
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5.2 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
Fig 5.8
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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
Fig 5.9
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Active Figure5.9
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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
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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
Fig 5.11
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Banked Curve These are designed
with friction equaling zero
There is a component of the normal force that supplies the centripetal force
Fig 5.11
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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
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Fig 5.13
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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
Fig 5.14
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Loop-the-Loop, Part 2
At the top of the circle (c), the force exerted on the object is less than its weight
Fig 5.14
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Non-Uniform Circular Motion The acceleration and
force have tangential components
produces the centripetal acceleration
produces the tangential acceleration
Fig 5.15
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Active Figure5.15
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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
Fig 5.17
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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
Fig 5.17
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5.4 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
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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
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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
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R Proportional To v, Example
Analyzing the motion results in
Fig 5.18(a)
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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
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Terminal Speed To find the terminal speed,
let a = 0
Solving the differential equation gives
is the time constant and = m/b
Fig 5.18(b)
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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
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R Proportional To v2, example Analysis of an object
falling through air accounting for air resistance
Fig 5.19
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R Proportional To v2, Terminal Speed The terminal speed
will occur when the acceleration goes to zero
Solving the equation gives
Fig 5.19
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Some Terminal Speeds
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5.5 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
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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
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Fig 5.21
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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
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Fig 5.22
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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
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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
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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
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5.6 Drag Coefficients of Automobiles
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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
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Fig 5.23