Download Circular Motion

Circular Motion
Any object that revolves about a single axis undergoes
circular motion. The line about which the rotation
occurs is called the axis of rotation. Ex. Spinning a
ferris wheel
Tangential Speed, vt
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Tangential speed can be used to describe the
speed of an object in circular motion. When the
tangential speed is constant, the motion is called
uniform circular motion.
The tangential speed depends on the distance
from the object to the center of the circular path.
E.g. pair of horses side by side on a carousel.
The outside horse has a greater tangential
speed.
Centripetal acceleration, ac
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Centripetal acceleration is due to a change
in direction.
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The acceleration of a ferris wheel car moving in
a circular path and at constant speed is due to a
change in direction, and is termed centripetal
acceleration.
Centripetal acceleration-the acceleration of an
object in uniform circular motion.
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ac=vt2/r
Sample Problem A:
A test car moves at a constant speed around
a circular track. If the car is 48.2m from
the track’s center and has an ac of
8.05m/s2, what is the car’s tangential
speed?
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ac
Given:
 r=48.2m
ac=8.05m/s2
 vt-=?
 8.05m/s2=vt2/48.2m=19.7m/s
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Tangential Acceleration
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Tangential acceleration is acceleration due to a
change in speed.
A car moving in a circle has ac. If the speed of
the car changes, it also has tangential
acceleration.
Fc, centripetal force
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Centripetal force is the net force directed toward
the center of an object’s circular path. Newton’s
Second Law applies.
Fc=mvt2/r
Sample Problem B:
A pilot is flying a small plane at 56.6m/s in a
circular path with a radius of 188.5m. The
centripetal force needed to maintain the plane’s
circular motion is 1.89x104N. What is the
plane’s mass?
Solution:
Given: r=188.5m
Fc=1.89x104N
 Vt=56.5m/s
m=?
 1.89x104N=m(56.6m/s)/188.5m=1110kg
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More on Fc
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It acts at right angles to an object’s circular
motion, so the force changes the direction
of the object’s velocity. Without centripetal
force, the object stops moving in a circular
path and leads to a straight path that is
tangent to the circle.
Newton’s Law of Universal
Gravitation
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Gravitational force is the mutual force of
attraction between particles of matter.
Orbiting objects are in free fall- Newton
observed that if an object were projected at just
the right speed, the object would fall down
toward Earth in just the same way that Earth
curved out from under it. So, it would orbit the
Earth. A gravitational attraction between Earth
and our sun keeps Earth in its orbit around the
sun.
What does Fgrav depend on?
It depends on the masses and the
distance.
 Newton’s Law of Universal Gravitation:
Fg=G m1m2/r2; G=constant of universal
gravitation
 What is the value of G?
6.673x10-11Nxm2/kg2

Universal Gravitation con’t…
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Newton demonstrated
that the gravitational force
that a spherical mass
exerts on a particle
outside the sphere would
be the same if the entire
mass of the sphere were
concentrated at the
sphere’s center.
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Gravitational force acts
between all masses. It
always attracts objects to
one another. E.g. the
force that the moon
exerts on the Earth is
equal and opposite to the
force that Earth exerts on
the moon. (Example of
Newton’s Third Law).
Universal Gravitation…
 Earth’s acceleration is so small that it cannot be
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detected for its mass is so large and
acceleration is inversely proportional to mass,
the Earth’s acceleration is negligible.
Sample Problem C: Find the distance between
a 0.300-kg billiard ball and a 0.400-kg billiard
ball if the magnitude of the gravitational force
between them is 8.92x10-11N.
Sample Problem C con’t…
Given: m1=0.300-kg
m2=0.400-kg
Fg=8.92x10-11N
G=6.673x10-11Nxm2/kg2
8.92x10-11N=6.673x10-11Nxm2/kg2 (0.300kg)(0.400-kg)/r2
r=0.30m
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Newton’s Law of Universal Gravitation
accounts for ocean tides.
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High and low tides are partly due to the
gravitational force exerted on Earth by its
moon.
Henry Cavendish
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In 1798, determined the value
of G via experimentation. He
took two small spheres that
were fixed to the ends of a
suspended light rod, and
attracted to two large spheres
by gravitational force. Once
you have the value of G, it can
be used to determine Earth’s
mass.
Gravitational Field
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A gravitational field is an interaction between a mass and
the gravitational field created by other masses. Earth’s
gravitational field can be explained by gravitational field
strength, g. The value of g is equal to the magnitude of
the gravitational force exerted on a unit mass at that
point, or g=Fg/m.
Gravitational filed strength is equal to free fall
acceleration, however, they are not the same thing. E.g.
object hanging from a spring scale.
Weight changes with location.
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Here, weight is mass times gravitational field
strength.
Fg=GmmE/r2
g=Fg/m=GmE/r2
What does gravitational field strength depend
on? Mass and distance
On the surface of any planet, the value of g will
depend on the planet’s m and r, and so will your
weight.
Motion in Space
Claudius Ptolemy’s view on motionPlanets travel in small circles called
epicycles while simultaneously traveling in
larger circular orbits.
 Nicolaus Copernicus published a book and
proposed that Earth and other planets
orbit the sun in perfect circles.
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Kepler and planetary motion
Tycho Brahe-an astronomer who made
precise observations about the planets
and stars. Some of his data did not have
face validity with the model of Copernicus.
 Johannes Kepler (astronomer): did work to
reconcile Copernican theory with the data
of Brahe. He developed three laws of
planetary motion:
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Kepler’s Laws
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First Law: Each planet travels in an elliptical orbit
around the sun, and the sun is at one of the
focal points.
Second Law: An imaginary line drawn from the
sun to any planet sweeps out equal areas in
equal time intervals.
Third Law: The square of a planet’s orbital
period (T2) is proportional to the cube of the
average distance between the planet and the
sun or T2 is roughly r3.
What did Kepler find?
He concluded the 1st law while examining
Mars and found that the planet’s orbits are
ellipses and not circles.
 The 3rd law relates to the orbital period
and mean distance for two orbiting planets
as follows:
 T=Period, T12/T22=r13/r23
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Kepler continued…
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Kepler’s Third law applies to satellites orbiting
Earth, including our moon. (r is the distance
between the orbiting satellite and the Earth in
this situation).
Newton utilized Kepler’s laws to support and
validate his law of gravitation. Newton proved
that if force is inversely proportional to the
distance squared, the resulting orbit must be an
ellipse or circle. He also portrayed that his law
could be utilized to validate Kepler’s third law.
Rotational Motion and Torque

The motion of a rotating rigid object (e.g.
a football spinning as it flies through the
air). If the only force acting on the football
is gravity, the football spins around a point
called its center of mass. As it moves
through the air, its center of mass follows a
parabolic path.
Torque is the quantity that measures the ability of a
force to rotate an object around some axis.
Example: a cat flap door
 The perpendicular distance from the axis
of rotation to a line drawn along the
direction of force is called the lever arm.
The lever arm depends on the angle.
 A force applied to an extended object can
produce torque. This torque, in turn,
causes the object to rotate.
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Torque calculation:
Torque=Fd sin theta
 Net torque = the sum of all torques
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Machines and Efficiency
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Machine-a device used to multiply forces or simply
change the direction of forces. The law of conservation
of energy underlies every machine.
Lever-a simple machine.
Fulcrum-pivot point of a lever.
Mechanical advantage-the ratio of output force to input
force for a machine.
Pulley-a kind of lever that can be used to change the
direction of a force.
No machine can put out more energy than is put into it.