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Physics: Chapter 7--Rotational Motion and the Law of Gravity
I.
Measuring Rotational Motion
A. Rotational Quantities
1. rotational motion is defined as the motion of a body that spins about an axis; the
axis of rotation is the line about which the rotation occurs
2. a point on an object that rotates about a single axis undergoes circular motion
3. describing rotational or circular motion is done by measuring angle in radians,
rather than degrees
a. A radian is an arc length (or length on the circle) equal to the radius of the
circle
b. Any angle  measured in radians is equal to s (arc length) divided by r
(radius);  = s/r
c. Length divided by length gives no dimension, so the dimension radians or
rads is used
d. Radian/revolution/degrees relationship: 2 radians = 1 revolution or 360°
4. angular displacement is the angle through which a point, line, or body is rotated in
a specified direction and about a specified axis
a. in other words, angular displacement is the change in arc length/distance from
the axis (the radius)
b. ∆ = ∆s/r
c. angular displacement is measured in radians
d. In our text and lots of others, counterclockwise rotation is positive, clockwise
rotation is negative
5. angular velocity (or speed) is the rate at which a body rotates about an axis,
usually expressed in radians per second
a. equationally with words: average angular velocity = angular
displacement/time interval
b. equationally with symbols: avg = ∆ /∆t
6. angular acceleration is the time rate of change of angular velocity, expressed in
radians per second per second
a. equationally with words: average angular acceleration = change in angular
velocity/time interval
b. equationally with symbols: avg = ∆ /∆t
7. all points on a rotating rigid object have the same angular acceleration and
angular speed
B. Comparing Angular and Linear Quantities
Angular substitutes for linear quantities
Linear
Angular
x
v
a



Physics: Chapter 7--Rotational Motion and the Law of Gravity
Rotational and linear kinematic equations
Rotational motion with constant angular
Linear motion with constant
acceleration
acceleration
1
( i   f )t
2
 f   i   (t )
 
1
   i (t )   (t ) 2
2
2
2
 f   i  2
II.
III.
1
(vi  v f ) t
2
 vi  a ( t )
x 
vf
1
a (t)2
2
 2ax
 x  vi ( t ) 
vf
2
 vi2
Tangential and Centripetal Acceleration
A. Tangential speed
1. the tangential speed of an object traveling in a circle is measured by a tangent line
to its circular path.
2. example: The horses on a carousel all have the same angular speed, but the
horses on the outside have a greater tangential speed than the horses on the inside
because they have a greater distance to travel in making one rotation
3. equationally with words: tangential speed = distance from axis (the radius) 
angular speed
4. equationally with symbols: vt = r • 
B. Tangential acceleration
1. tangential acceleration is the instantaneous linear acceleration of an object
directed along the tangent to the object's circular path
2. tangential acceleration is also measured by a tangent line to the circle
3. equationally with words: tangential acceleration = distance from the axis (the
radius) × angular acceleration
4. equationally with symbols: at = r • 
C. Centripetal Acceleration
1. centripetal acceleration (ac) is acceleration directed toward the center of a circular
path (centripetal means "center-seeking")
2. centripetal acceleration can be found using these equations: ac = vt2/r and ac = r2
3. objects have continuously changing centripetal acceleration even if angular
velocity remains constant; they are technically continuously changing direction,
and acceleration is a vector quantity
D. Tangential and Centripetal Acceleration
1. tangential and centripetal acceleration are not the same
2. the tangential component of acceleration is due to changing speed; the centripetal
component of acceleration is due to changing direction
3. tangential acceleration is perpendicular to centripetal acceleration; therefore, total
acceleration can be found using the Pythagorean theorem (they are two legs of a
right triangle and are related to each other by the tangent function)
Causes of Circular Motion
A. Force That Maintains Circular Motion
Physics: Chapter 7--Rotational Motion and the Law of Gravity
1. the force of circular motion is the net force directed towards the center of the
circle that keeps an object moving in a circular path (often referred to as
centripetal force)
2. force that maintains circular motion can be found with these two equations:
Fc=mvt2/r or Fc = mr2 where m is the mass of the object
3. A force directed toward the center is necessary for circular motion; if this force is
suddenly gone, the object leaves the circular motion on a line-tangent-to-thecircle path
B. Centrifugal force doesn't exist; what's noticed as centrifugal force is inertia (Newton's
second law
C. Newton's Universal Law of Gravitation
1. There exists a gravitational force between all objects regardless of their mass and
because of their mass
2. This gravitational force between masses is directly proportional to the mass of the
objects and inversely (and squared) proportional to the distance separating the
masses
m m
3. Newton's Universal Law of Gravitation: Fg  G 1 2 2 where G is the constant
r
of universal gravitation and is equal to 6.673 x 10-11 N•m2/kg2
4. When using distances between large spherical masses in this equation, you have
to consider the gravitational force coming from the center of the mass (distance
between objects would include the radii of the objects)