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Chapter 5 Outline
Applying Newton’s Laws
• Statics
• Dynamics
• Friction
• Kinetic friction
• Static friction
• Fluid resistance
• Circular Motion
• Fundamental forces
Statics
• When a body is not accelerating, we say that it is in
equilibrium.
• Statics is the study of bodies in equilibrium.
• Since the acceleration is zero, we use Newton’s first law to solve
these problems.
𝑭=0
𝐹𝑥 = 0;
𝐹𝑦 = 0;
𝐹𝑧 = 0
•
Carefully draw a free body diagram that shows all forces
acting on the body.
•
Choose an appropriate coordinate system.
Normal Force
•
When an object is resting on a surface, the surface is
exerting a force on the object, called the normal force.
•
The normal force is always perpendicular to the surface.
•
The textbook uses the letter 𝑛 for the normal force. It is also
common to use 𝐹𝑁 .
Statics Example
Dynamics
• Now we consider a body that is accelerating.
• Dynamics is the study of bodies not in equilibrium.
• Since the acceleration is not zero, we use Newton’s second law to
solve these problems.
𝑭 = 𝑚𝒂
𝐹𝑥 = 𝑚𝑎𝑥 ;
𝐹𝑦 = 𝑚𝑎𝑦 ;
𝐹𝑧 = 𝑚𝑎𝑧
•
Carefully draw a free body diagram that shows all forces
acting on the body.
•
Choose an appropriate coordinate system.
Apparent Weight
•
Consider the case of a person standing on a scale in an
elevator.
•
If the elevator is still or moving at a constant velocity, the scale
will read the person’s actual weight, 𝑤 = 𝑚𝑔.
•
If the elevator is acceleration, the scale will read the apparent
weight.
𝑛 = 𝑚(𝑔 + 𝑎𝑦 )
• If the body is accelerating downward at the acceleration
due to gravity, that is 𝑎𝑦 = −𝑔, it experiences apparent
weightlessness.
• This is the case during free fall, or while in orbit.
Dynamics Example
Frictional Forces
•
So far, we have ignored one of the most important types
of force: the force of friction.
•
Contact force
•
Parallel to surface
Kinetic Friction
•
When the surfaces are moving relative to each other, we
have kinetic friction.
𝑓k = 𝜇k 𝑛
•
The force of kinetic friction is proportional to the normal force,
𝑛, and the coefficient of kinetic friction, 𝜇k .
•
𝜇k depends on both surfaces.
•
Does not depend on contact area!
Static Friction
•
When the surfaces are still with respect to each other, we
have static friction.
𝑓s ≤ 𝜇s 𝑛
•
The force of static friction is proportional to the normal force, 𝑛,
and the coefficient of static friction, 𝜇s .
•
𝜇s depends on both surfaces.
•
Generally, 𝜇s > 𝜇k for a given pair of surfaces.
Static and Kinetic Friction
Coefficient of Rolling Friction
•
In practice, we often use wheels to reduce the frictional
forces involved in moving objects.
•
•
Still, we have to consider the frictional force involved.
The coefficient of rolling friction, 𝜇r :
𝑓r = 𝜇r 𝑛
•
The frictional force takes into account the deformation of the
wheels.
•
For steel on steel (trains), values of 𝜇r are typically 0.002 to
0.003.
Friction Example
Fluid Resistance
•
In the case of the contact between solid surfaces, we find
that the frictional force is roughly independent of the
relative speed or contact area.
•
For motion through fluids, the resistive force is quite
sensitive to the speed.
•
For small objects moving at very low speeds, the fluid resistance
force, 𝑓, is approximately proportional to the speed.
𝑓 = 𝑘𝑣
•
At higher speeds, the fluid resistance force, 𝑓, is approximately
proportional to the square of the speed.
𝑓 = 𝐷𝑣 2
Terminal Speed
•
Since the fluid resistance force is
speed dependent, there will be
some speed at which the weight
of a falling body is balanced by
the drag force.
•
•
At this point, the acceleration is
zero, and the body has reached its
terminal speed.
From 𝑓 = 𝐷𝑣 2 , the terminal
speed is:
𝑚𝑔
𝑣𝑡 =
𝐷
Fluid Resistance Example
Circular Motion
• Recall from Chapter 3 that circular motion is produced by a
centripetal acceleration towards the center of curvature.
𝑣2
𝑎rad =
𝑅
• The period, 𝑇, is the time for one full circle.
2𝜋𝑅
𝑇=
𝑣
• The acceleration towards the center must be provided by
some force.
• Planet in orbit: Gravity
• Ball on a string: Tension
• Car going around turn?
Circular Motion Example
Consider a racecar going around a turn with a radius of
curvature of 250 m. The static and kinetic coefficients of
friction between the track and the tires are 𝜇𝑠 = 0.96 and
𝜇𝑘 = 0.68, respectively.
• What is the maximum speed at which the car can take the
corner without skidding if the track is (a) flat or (b) banked
at 18°?
Fundamental Forces
• We have talked about a lot of different kinds of forces.
• Gravitational, friction, normal, fluid resistance, tension…
• Are these all actually different in nature?
• Four fundamental forces (we think):
• Gravitational
• Electromagnetic
• Strong
• Weak
• All interactions arise through one of these fundamental
forces.
• All of the interactions we have dealt with so far were
electromagnetic or gravitational.
Chapter 5 Outline
Applying Newton’s Laws
• Statics:
𝑭=0
• Dynamics:
𝑭 = 𝑚𝒂
• Friction
• Kinetic friction: 𝑓k = 𝜇k 𝑛
• Static friction: 𝑓s ≤ 𝜇s 𝑛
• Rolling resistance: 𝑓r = 𝜇r 𝑛
• Fluid resistance: 𝑓 = 𝑘𝑣 or 𝑓 = 𝐷𝑣 2
• Terminal velocity: 𝑣𝑡 =
• Circular Motion
• Fundamental forces
𝑚𝑔
𝐷