Transcript Chapter 6 - SFSU Physics & Astronomy

Applications of Newton’s
Laws
Strings and Springs
When you pull on a string or rope, it becomes
taut. We say that there is tension in the string.
Strings and Springs
The tension in a real rope will vary along its
length, due to the weight of the rope.
Here, we will assume that
all ropes, strings, wires,
etc. are massless unless
otherwise stated.
Strings and Springs
An ideal pulley is one that simply changes the
direction of the tension:
Strings and Springs
Hooke’s law for springs states that the
force increases with the amount the
spring is stretched or compressed:
The constant k is called the spring
constant.
Translational Equilibrium
When an object is in translational equilibrium,
the net force on it is zero:
(6-5)
This allows the calculation of unknown forces.
Translational Equilibrium
Connected Objects
When forces are exerted on connected objects,
their accelerations are the same.
If there are two objects connected by a string,
and we know the force and the masses, we can
find the acceleration and the tension:
Connected Objects
We treat each box as a separate system:
Connected Objects
If there is a pulley, it is easiest to have the
coordinate system follow the string:
Frictional Forces
Friction has its basis in surfaces that are not
completely smooth:
Frictional Forces
The kinetic frictional force is also independent
of the relative speed of the surfaces, and of their
area of contact.
Frictional Forces
Kinetic friction: the friction experienced by
surfaces sliding against one another
The kinetic frictional force depends on the
normal force:
(6-1)
The constant
kinetic friction.
is called the coefficient of
Frictional Forces
The static frictional force keeps an object from
starting to move when a force is applied. The
static frictional force has a maximum value, but
may take on any value from zero to the maximum,
depending on what
is needed to keep
the sum of forces
zero.
Frictional Forces
(6-2)
where
(6-3)
The static frictional force is also independent
of the area of contact and the relative speed of
the surfaces.
Frictional Forces
Newton’s Law of Universal Gravitation
Newton’s insight:
The force accelerating an apple downward is
the same force that keeps the Moon in its orbit.
Hence, Universal Gravitation.
Newton’s Law of Universal Gravitation
The gravitational force is always attractive, and
points along the line connecting the two
masses:
The two forces shown are an action-reaction
pair.
Newton’s Law of Universal Gravitation
G is a very small number; this means that the
force of gravity is negligible unless there is a
very large mass involved (such as the Earth).
If an object is being acted upon by several
different gravitational forces, the net force on it
is the vector sum of the individual forces.
This is called the principle of superposition.
Gravitational Attraction of Spherical Bodies
Gravitational force between a point mass and a
sphere: the force is the same as if all the mass
of the sphere were concentrated at its center.
Gravitational Attraction of Spherical Bodies
What about the gravitational force on objects at
the surface of the Earth? The center of the Earth is
one Earth radius away, so this is the distance we
use:
Therefore,
Gravitational Attraction of Spherical Bodies
The acceleration of gravity decreases slowly with
altitude:
Gravitational Attraction of Spherical Bodies
Once the altitude becomes comparable to the
radius of the Earth, the decrease in the
acceleration of gravity is much larger:
Gravitational Attraction of Spherical Bodies
The Cavendish experiment allows us to measure
the universal gravitation constant:
Gravitational Attraction of Spherical Bodies
Even though the gravitational force is very small,
the mirror allows measurement of tiny deflections.
Measuring G also allowed the mass of the Earth to
be calculated, as the local acceleration of gravity
and the radius of the Earth were known.
Circular Motion
An object moving in a circle must have a force
acting on it; otherwise it would move in a straight
line.
The direction of the
force is towards the
center of the circle.
Circular Motion
Some algebra gives us the magnitude of the
acceleration, and therefore the force, required
to keep an object of mass m moving in a circle
of radius r.
The magnitude of the force is given by:
(6-15)
Circular Motion
This force may be provided by the tension in a
string, the normal force, or friction, among
others.
Circular Motion
Circular Motion
An object may be changing its speed as it
moves in a circle; in that case, there is a
tangential acceleration as well: