Download Holt Physics Chapter 7: Rotational Motion and the Law of Gravity

Holt Physics Chapter 7: Rotational Motion and the Law of Gravity
I.
Section 7-1: Measuring Rotational Motion
A. When something spins it undergoes “rotational motion”. When
something spins around a single point it is called “circular motion”.
B. We measure how fast something spins not in m/s (different points
on the object are spinning at different velocities) but by
measuring the angle described in a given time period.
C. Angles can be measured in radians (rad)
1. The radian is the ratio of the arc length (s) to the radius
(r) of a circle
(insert fig. 7-1 here)
2. The radian is a “pure number” with no units (the
abbreviation “rad” is always used)
3. Conversions:
360o = 2π rad
360o = 6.28 rad
Θ(rad) = π/180o Θ(deg)
Θ(rad) = .0174533 Θ(deg)
(insert fig. 7-3)
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D. Angular displacement describes how much an object has rotated
relative to a reference line
(insert fig 7-4)
Angular Displacement
ΔΘ = Δs/r
angular displacement = change in arc length/radius
E. Watch your sign! Θ is considered positive when rotating
COUNTERclockwise (when viewed from above). Therefore an
angle of ½π rad = -1½π rad
F. Angular Speed (ω = “omega”) describes the rate of rotation.
Average angular speed is measured in radians per second.
Angular Speed
ωavg = ΔΘ/Δt
average angular speed = angular displacement/time
G. Angular Acceleration (α = “alpha”) occurs when angular speed
changes. Remember acceleration? a = velocity/time ??
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Angular Acceleration
αavg = ω2 – ω1/t2 – t1 = Δω/Δt
average angular acceleration = change in speed/time
II.
H. “All points on a rotating rigid object have the same angular
acceleration and angular speed.” P.250
Section 7-2: Tangential and Centripetal Acceleration
A. Relationships between angular and linear quantities
1. Objects in circular motion have a “tangential speed”
2. This is not the “tangent” from trigonometry but the
“tangent” that intersects a circle at exactly one point
tangent
circle
3. “tangential speed” is an object’s speed along a line tangent
to its circular path
4. Objects further from the center of the circle have a
higher tangential speed (these quantities are directly
related, i.e. as one gets larger so does the other)
5. Think of horses flying off a carousel on tangential paths
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Tangential Speed
vt = rω
tangential speed = radius x angular speed
6. Tangential Acceleration is tangent to the circular path
Tangential Acceleration
at = rα
tangential acceleration = radius x angular acceleration
B. Centripetal Acceleration
1. A change in an object’s direction must be described as an
acceleration
a. since a = v/t and velocity is a vector with magnitude and
direction…
b. by changing an object’s direction we change its velocity
and, hence its acceleration!
2. An object moving at a constant speed around a circular path
has an acceleration due to its constantly changing direction.
An acceleration of this type is called “centripetal
acceleration”
Centripetal Acceleration
ac = vt2/r
centripetal acceleration = (tangential speed)2/ radius
ac = rω2
centripetal acceleration = radius x (angular speed)2
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3. Tangential and centripetal accelerations are perpendicular
because centripetal acceleration always points towards the
center of the circle
4. This fact enables us to “Find the total acceleration using
the Pythagorean theorem” (p.259)
Insert fig. 7-9
III. Section 7-3: Causes of Circular Motion
A. “Force That Maintains Circular Motion”
1. Whatever force is preventing an object from traveling a
straight line (gravity or a tether perhaps) can be measured
in Newtons and is called the “force that maintains circular
motion”
Force That Maintains Circular Motion
Fc = mvt2/r
Force = mass x (tangential speed2/radius)
Fc = mrω2
Force = mass x radius x angular speed2
B. Newton’s Law of Universal Gravitation states that gravitational
force depends on the distance between two masses
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Newton’s Law of Universal Gravitation
Fg = (G)(m1m2/r2)
gravitational force = gravitational constant x (product of masses/radius2)
C. The gravitational constant (G) is defined as 6.673 x 10-11
Newton meters squared per kilogram squared and has been proven
experimentally
D. Newton’s Law of Universal Gravitation is often referred to as the
“inverse square law” as force decreases when distance increases
(we say these two are inversely related)
E. Gravitational force is localized at the center of a spherical mass
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