Download (the terminal velocity is smaller for larger cross

Physics 218: Mechanics
Instructor: Dr. Tatiana Erukhimova
Lectures 10,11
Falling with air resistance
dv
2
a=
= g - kv
dt
Terminal Velocity with Coffee Filters
mg - Fr = ma
where Fr is the resistance force.
Fr
a=gm
1. A penny and a quarter dropped from a ladder land at the
same time (air resistance is negligible).
2. A coin dropped in a coffee filter from a ladder lands later
than a coin without coffee filter (the terminal velocity is
smaller for larger cross-section area).
3. A quarter dropped in a coffee filter will land faster than a
penny in a coffee filter (the terminal velocity is larger for
larger mass)
4. Two identical coins dropped in coffee filters of different
diameters land at different times (the terminal velocity is
smaller for larger cross-section area).
Resistance force: Fr = gAv
2
A – area of the projectile
For a spherical projectile in air at g = 0.25 N ´ s /m
STP:
2
4
Terminal velocity:
Fr
a=g=0
m
Fr = mg
gAv = mg
mg
vT =
gA
2
A 70-kg man with a parachute: vT ~ 5 m/s
A 70-kg man without a parachute: vT ~ 70
m/s
Dynamics
Connection between force and motion
The concept of force gives us a quantitative
description of the interaction between two
bodies or between a body and its environment
Newton’s Laws
1st Law: A body acted on by no net force moves with
constant velocity (which may be zero) and zero acceleration
2nd Law: The acceleration of an object is directly
proportional to the net force acting on it and is inversely
proportional to its mass. The direction of the acceleration
is in the direction of the net force acting on the object.
3rd Law: For every action there is an equal, but
opposite reaction
Newton’s law of gravitation
Newton’s
st
1
Law
A body acted on by no net force moves
with constant velocity (which may be zero)
and zero acceleration
Aristotle: a natural state of
an object is at rest; a force
is necessary to keep an
object in motion. It follows
from common sense.
384-322 B.C.
Galileo: was able to
identify a hidden force of
friction behind commonsense experiments
1564-1642
Galileo: If no force is applied to
a moving object, it will continue
to move with constant speed in
a straight line
Inertial reference frames
Galilean principle of relativity: Laws of
physics (and everything in the Universe)
look the same for all observers who move
with a constant velocity with respect to
each other.
2nd Law
From experiments we know:
1.Force is a vector
2.The direction of acceleration vector is the
same as the direction of the force vector
3.The magnitude of the force and
acceleration are related by a constant
which depends on number of blocks
involved.
Newton’s second law


F  ma
The vector acceleration of an object is in the same
direction as the vector force applied to the object
and the magnitudes are related by a constant called
the mass of the object.


F  mg
Gravitational force
Normal force
Force exerted by a spring:
Hooke’s law: If spring is stretched or compressed
by some small amount it exerted a force which is
linearly proportional to the amount of stretching
or compressing. The constant of proportionality is
called the spring constant

Fspring  k x ,
x -is
deviation from the natural
length
The force resisting the pull of the spring –
friction
There is some maximum value the friction force can
achieve, and once we apply a force greater than this
maximum there is a net force on the object, so it
accelerates.
The maximum of the force of friction varied linearly
with the amount that the block pushes on the table.


Ffriction   N

 - coefficient of friction, N is the vertical force exerted by
the block on the table
The friction force only exists when there is another
force trying to move an object
Kinetic Friction
• For kinetic friction, it turns out that the
larger the Normal Force the larger the
friction. We can write
FFriction = KineticN
Here  is a constant
• Warning:
– THIS IS NOT A VECTOR EQUATION!
Static Friction
• This is more complicated
• For static friction, the friction force can vary
FFriction  StaticN
Example of the refrigerator:
– If I don’t push, what is the static friction
force?
– What if I push a little?
A Recipe for Solving Problems
1. Sketch
Isolate the body (only external forces but not forces
that one part of the object exert on another part)
2. Write 
down 2nd Newton’s law

F  ma
Choose a coordinate system
Write 2nd Newton’s law in component form:





F  Fx i  Fy j  max i  ma y j
Fx  max , Fy  ma y
3. Solve for acceleration
Pulling Against Friction
A box of mass m is on a surface with coefficient
of kinetic friction . You pull with constant
force FP at angle Q. The box does not leave
the surface and moves to the right.
1. What is the magnitude of the acceleration?
2. What angle maximizes the acceleration?
Q
Is it better to push or pull a
sled?
You can pull or push a sled with the same force
magnitude, FP, and angle Q, as shown in the figures.
Assuming the sled doesn’t leave the ground and has a
constant coefficient of friction, , which is better?
FP
FP
Coefficient of friction: 
H
θ
1) Find the force of friction if the block is at
rest.
2) The block slides down the incline. What
is the velocity of the block when it reaches
the bottom?
Quiz
a) A crate of mass m is on the flat bed of a pick up
truck. The coefficient of friction between the crate
and the truck is . The truck is traveling at the
constant velocity of magnitude V1. Draw the free
body diagram for the crate.
b) The truck starts to accelerate with an
acceleration ac. Draw the free body diagram for
the crate, if the crate does not slip.
Have a great day!
Reading: Chapter 5,6
Hw: Chapter 6 problems and
exercises